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+{"text":"\\section{Introduction}\\label{section_intro}\n\nLet $T$ be a complete discrete valuation ring with fraction field $K$\nand residue field $k$. Let $X$ be a smooth, projective, geometrically \nintegral curve over $K$. Let $F=K(X)$ be the function field of $X$ and let $t$ be an uniformizing parameter of $T$.\n Under a mild assumption on the characteristic $p$\n of the residue field of $T$, we prove \n a conjecture by Colliot-Th\\'el\\`ene, Suresh and the second\n author \\cite[conjecture 1]{CTPS}.\n \n \n \\begin{stheorem} \\label{thm_patching0}\n (see Th. \\ref{thm_patching}) Let $G$ be a reductive \n $F$--group and assume that $p$ does not divide the order of the automorphism group of the absolute root system of \n $G_{ad}$.\n Let $Z$ be a twisted flag $F$--variety of $G$.\n Then $Z(F) \\not = \\emptyset$ if and only if $Z(F_v) \\not = \\emptyset$\n for all discrete valuations of $F$ arising from models of $X$.\n \\end{stheorem}\n\nThe conjecture was known for the cases of smooth projective\nquadrics and Severi-Brauer varieties \\cite{CTPS}.\nIf we use all rank one valuations, the\nresult holds unconditionnally (Cor. \\ref{cor_main_hhk3}).\nOn the other hand, if $G$ extends to a reductive group \nscheme over a smooth model $\\goth X$ of  $X$, \nthe result is unconditional (Cor. \\ref{cor_main_hhk2}).\n \n \\smallskip\n \nLet us review the contents.\nGiven a henselian couple $(A,I)$ and \na reductive $A$-group scheme $H$,\nsection 2 deals with generation of groups $H(A)$ by \nunipotent elements and also with the quotient \nof $H(A)\/RH(A)$ by the (normal) subgroup of $R$-trivial elements\ndefined in \\cite{GS}. Section 3 is devoted to mild improvements\nof patching techniques of Harbater-Hartmann-Krashen \ninvolving more analytical functions and leads to various\ngroup decompositions (prop. \\ref{prop_R_eq});\nthis permits to obtain a patching theorem\nfor twisted flag varieties (th. \\ref{thm_key}).\nSection 4  deals with the setting\nof the beginning of the introduction and \nprovides a weaker version of  the main result.\nWhat remains to do is mostly a local study  for\na two dimensional complete regular local ring $R$ with parameters $t,s$\nand torsors over its localization $R[t^{-1},s^{-1}]$.\nThis is achieved in  section 5 by making use of the \nloop torsors over $R[t^{-1},s^{-1}]$ and of the results of \n\\cite{Gi2}.   \n\n\n \n\\medskip\n\n\n\\noindent{\\bf Acknowledgements.}  We thank Gabriel  Dospinescu\nfor telling us about Schneider's book and \nDavid Harbater for comments on a preliminary version of the paper.  \n \n \\bigskip  \n \n \\smallskip\n\n \\noindent{\\bf Conventions and notation.}\n$(a)$ A variety $V$ over a field $F$ means\na separated $F$--scheme of finite type which is \n integral \\cite[Tag 020D]{St}.\n \n \\smallskip\n \n\\noindent $(b)$ Let $G$ be a reductive $F$--algebraic group and let $F_s$ be a separable closure of $F$.\nLet $(B,T)$ be a Killing couple of $G_{k_s}$ and denote\n$\\Delta(G_{F_s})= \\Delta(G_{F_s},B,T)$  the associated Dynkin diagram\n(there is a canonical bijection $\\Delta(G_{F_s},B,T) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_s},B',T')$\nfor another choice of Killing couple).\nThis diagram \n$\\Delta(G_{F_s})$  is equipped \n with the star action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ \\cite[\\S 6.4]{BoT65}.\n We recall that there is a increasing bijection $I \\to P_I$ between\n the subsets of $\\Delta(G_{F_s})$ and the $F_s$--parabolic subgroups of \n $G_{F_s}$ containing $B$ and the $G(F_s)$--conjugacy classes of parabolic subgroups of $G_{F_s}$.\n Since minimal $F$--parabolic subgroups are $G(F)$-conjugated \n we denote by  $\\Delta_0(G) \\subset \\Delta(G_{F_s})$ be conjugacy class of \n $P_0 \\times_F F_s$ where $P_0$ is a minimal $F$--parabolic subgroup of $G$ (it is stable under the star action).\n The triple $(\\Delta(G_{F_s}), \\hbox{star action}, \\Delta_0(G))$ is called \n the Tits index of $G$. By abuse of notation, $\\Delta_0(G)$ alone is called the Tits index. \n By a twisted $F$--flag variety of $G$, we mean \n a $G$-variety $X$ such becomes isomorphic over $F_s$\n to some $G_{F_s}\/P_I$ as $G_{F_s}$-variety \n (such an $I$ is unique).\nAccording to \\cite[prop. 1.3]{MPW},  \n$I$ is invariant under the star action and \n$X$ is $G$-isomorphic to the variety of $F$--parabolic\nsubgroups of type $I$ which is denoted in this paper\nby $\\text{\\rm{Par}}_I(G)$. \n \n Those  varieties are called sometimes projective \n homogeneous $G$--varieties (as in \\cite{CTPS})\n but we warn the reader of the danger of this terminology\n since there exist in positive characteristic\n $F$--subgroups $Q$ which are not smooth such that \n $G\/Q$ is a projective $F$--variety \\cite{W}. \n In the appendix \\ref{app_parabolic}, we show that \n a homogeneous $G$--variety $X$ is a variety of parabolic subgroups\n if and only if the action is transitive in the sense\n that $G(E)$ acts transitively on $X(E)$ for each $F$ field $E$ (Prop. \\ref{prop_homog}).\n \n \n \n\n \\smallskip\n \n\\noindent $(c)$ \nWe use mainly  the terminology and notation of Grothendieck-Dieudonn\\'e \\cite[\\S 9.4  and 9.6]{EGA1}\n which agrees with that  of Demazure-Grothendieck used in \\cite[Exp. I.4]{SGA3}.Let $S$ be a scheme and let $\\mathcal E$ be a quasi-coherent sheaf over $S$.\n For each morphism  $f:T \\to S$, \nwe denote by $\\mathcal E_{T}=f^*(\\mathcal E)$ the inverse image of $\\mathcal E$ \nby the morphism $f$.\n We denote by $\\mathbf{V}(\\mathcal E)$ the affine $S$--scheme defined by \n$\\mathbf{V}(\\mathcal E)=\\mathop{\\rm Spec}\\nolimits\\bigl( \\mathrm{Sym}^\\bullet(\\mathcal E)\\bigr)$;\nit is affine  over $S$ and \nrepresents the $S$--functor $Y \\mapsto \\mathop{\\rm Hom}\\nolimits_{\\mathcal{O}_Y}(\\mathcal E_{Y}, \\mathcal{O}_Y)$  \\cite[9.4.9]{EGA1}. \n  We assume now that $\\mathcal E$ is locally free of finite rank and denote by $\\mathcal E^\\vee$ its dual.\nIn this case the affine $S$--scheme $\\mathbf{V}(\\mathcal E)$ is  of finite presentation\n(ibid, 9.4.11); also\nthe $S$--functor $Y \\mapsto H^0(Y, \\mathcal E_{Y})= \n\\mathop{\\rm Hom}\\nolimits_{\\mathcal{O}_Y}(\\mathcal{O}_Y, \\mathcal E_{Y} )$ \nis representable by the  affine $S$--scheme $\\mathbf{V}(\\mathcal E^\\vee)$\nwhich is also denoted by  $\\mathbf{W}(\\mathcal E)$  \\cite[I.4.6]{SGA3}.\n \n \n \n \\section{Around Kneser-Tits' problem} \n Let $A$ be a semilocal ring, $I$ its radical. \n \n \\subsection{Subgroups of elementary elements}\\label{subsec_PS}\n Let  $H$ be a reductive $A$-group scheme which   is strictly $A$--isotropic, that is, \n all factors of $H_{ad}$ are isotropic.\nLet  $H$ be a strictly proper parabolic $A$--subgroup of $H$ and let $P^{-}$ be \na $A$--parabolic subgroup opposite to $P$. \nThe subgroup  of $H(A)$ generated by $R_u(P)(A)$ and $R_u(P^-)(A)$ does not depend \nof the choice of $P$ and $P^{-}$, this is a normal subgroup of $H(A)$ \nwhich is denoted by $H(A)^+$ \\cite[Thm. 1  and comments]{PS}.\n\n\n\\begin{slemma}\\label{lem_GS0}\nWe have  $H(A)=H(A)^+ \\,  P(A)$. \n\\end{slemma}\n\n\\begin{proof}\nWe put $U=R_u(P)$,  $U^{-}=R_u(P^{-})$.\n According to \\cite[XXVI.5.2]{SGA3}, we have a decomposition \n $H(A)  =U(A)  \\, U^{-}(A) P(A)$ whence $H(A)  = H(A)^+ \\, P(A)$. \n\\end{proof}\n\n\n\nA special  case of \\cite[prop. 7.7]{GS} is the following.\n\n\n\\begin{slemma}\\label{lem_GS}\nAssume that $(A,I)$ is an henselian couple and that  $H$ is  semisimple simply connected.\nThen the map  $H(A)\/H(A)^+ \\to H(A\/I)\/ H(A\/I)^+$ is an isomorphism. \n\\end{slemma}\n\n\\subsection{R-equivalence}\n\nWe assume now that  $A$ is a nonzero finite $k$--algebra for  a field $k$.\nThen $A$ is an Artinian $k$--algebra. We denote by $I$ its Jacobson radical\nand remind  the reader that $(A,I)$ is an henselian couple.\n\n\nLet $H$ be an $A$--reductive group scheme and \n consider the Weil restriction \\break $G=R_{A\/k}(H)$, this is a smooth  affine connected algebraic \n $k$--group \\cite[A.5.9]{CGP}. We have $G(k)=H(A)$. \n We use $R$-equivalence for group schemes over rings as defined in \n \\cite{GS}.\n \n \n \\begin{slemma}\\label{lem_R_eq}\n We decompose $A=A_1 \\times \\dots \\times A_d$ where each $A_i$ is a local artinian \n $k$--algebra of residue field $k_i$.\n  \n  \\smallskip\n  \n  \\noindent  (1) The map $H(A) \\to \\prod_i H(k_i)$ induces  isomorphisms\n $$\n  G(k)\/RG(k) \\buildrel\\sim\\over\\lgr   H(A)\/RH(A) \\buildrel\\sim\\over\\lgr  \\prod_i H(k_i)\/RH(k_i) .\n $$\n \n \n  \\smallskip\n  \n  \\noindent  (2) If $H$ is furthermore simply connected and strictly isotropic, \n  we have a commutative diagram  of isomorphisms\n \n\\[\\xymatrix{\n  H(A)\/H(A)^+  \\ar[r]^\\sim  \\ar[d]^\\wr &    H(A)\/RH(A) \\ar[d]^\\wr  & \\ar[l]^\\sim    G(k)\/R \\\\ \n \\prod_i H(k_i)\/H(k_i)^+  \\ar[r]^\\sim & \\prod_i H(k_i)\/RH(k_i)  . \n}\\]\n \n \n\n \\end{slemma}\n\n \\begin{proof} \n (1) The first isomorphism $G(k)\/RG(k) \\buildrel\\sim\\over\\lgr H(A)\/RH(A)$  is a formal fact \\cite[lemma 2.4]{GS}.\n Let $n$ be the smallest positive integer such that $I^n=0$.\n We put $A_j=A\/I^{j+1}$ for $j=0, \\dots, n-1$, it comes \n with the ideal $J_j= I^{j}\/I^{j+1}=  I \\, A_j $ which satisfies $J_j^2=0$. We have \n $A_j\/J_j= A_{j-1}$ for $j=1, \\dots, n-1$.\n \n We put $G_j= R_{A_j\/k}(H_{A_j})$ for $j=0,\\dots , n-1$; \n we have a sequence of homomorphisms \n $G=G_{n} \\to G_{n-1} \\to \\dots \\to G_1 \\to G_0 = \\prod_i R_{k_i\/k}(H_{k_i})$.\n It induces homomorphisms\n $$\n G(k)\/R = G_{n}(k)\/R\\to G_{n-1}(k)\/R \\to \\dots \\to G_1(k)\/R \\to G_0(k)\/R = \\prod_i H(k_i)\/R\n $$\n and we will show by a d\\'evissage argument that all of them are  isomorphisms.\n \n \n Let $j$ be a an integer satisfying $1 \\leq j \\leq n-1$.\n According to a variant of \\cite[Lemma 8.3]{GPS}, we have an exact sequence of fppf (resp.\\ \\'etale, Zariski)\nsheaves on $\\mathop{\\rm Spec}\\nolimits(k)$\n$$\n0 \\to \\mathbf{W}\\Bigl( (t_j)_*\\bigl( \\mathop{\\rm Lie}\\nolimits(H)(A_{j-1}) \\otimes_{A_{j-1}} J_j \\bigr) \\Bigr)\n\\to R_{A_j\/k}(H_{A_j}) ) \\to R_{A_i\/k}(H_{A_{j-1}} ) \\to 1  .\n$$\n where $t_j: \\mathop{\\rm Spec}\\nolimits(A_j) \\to  \\mathop{\\rm Spec}\\nolimits(k)$ is the structural morphism.\n We have  then a sequence of $k$--algebraic groups\n $$\n 0 \\to (\\mathbb{G}_a)^{m_j} \\to G_j \\to G_{j-1} \\to 1.\n $$\n The map $G_j \\to G_{j-1}$ is a $(\\mathbb{G}_a)^{m_j}$-torsor\n which is trivial since $G_{j-1}$ is affine. We have then \n a decomposition of $k$-schemes $ G_j \\buildrel\\sim\\over\\lgr G_{j-1}  \\times_k  (\\mathbb{G}_a)^{m_j}$ \n which induces an isomorphism \n $G_j(k)\/R \\to G_{j-1}(k)\/R$. Thus the map $G(k)\/R \\to  G_0(k)\/R = \\prod_i H(k_i)\/R$\n is an isomorphism as desired.\n \n\n\\smallskip\n\n\\noindent (2) We have $A\/I=k_1 \\times \\dots \\times k_d$.\nLemma \\ref{lem_GS} provides an isomorphism $H(A)\/H(A)^+ \\buildrel\\sim\\over\\lgr H(A\/I)\/H(A\/I)^+\n=\\prod_i H(k_i)\/H(k_i)^+$ and we have an isomorphism $H(k_i)\/H(k_i)^+ \\buildrel\\sim\\over\\lgr H(k_i)\/RH(k_i)$\nfor each $i$ \\cite[thm. 7.2]{Gi1}.\nThis completes the proof by chasing diagram.\n\\end{proof}\n\n \n \\section{Patching, R-equivalence and twisted flag schemes}\n \n \\subsection{Using the implicit function theorem}\n We consider  a variation of the framework of \\cite[2.4]{HHK}.\n Let $T$ be a complete DVR of fraction field $K$ and $t$ be an unformizing parameter. Let $\\widehat R_0$ be ring\n containing $T$ and  which is also a complete discrete valuation ring having uniformizer $t$. We denote by $F_0$ the fraction field of $\\widehat R_0$.\n \nLet $\\alpha>1$ be a real number and we define the absolute value on $F_0$ by \n$\\mid\\! t^n u \\!\\mid= \\alpha^{-n}$ for $n \\in \\mathbb{Z}$ and $u \\in \n(\\widehat R_0)^\\times$.\n \n Let $F_1$, $F_2$ be subfields of $F_0$ containing $T$.\n We further assume that we are given \n  $t$-adically complete $T$-submodules $V \\subset F_1 \\cap  \\widehat R_0$\n  and $W \\subset  F_2 \\cap  \\widehat R_0$ \nsatisfying the following conditions:\n\n\n\\begin{equation}\\label{cond_I}\n V + W = \\widehat R_0;\n\\end{equation}\n\n\\vskip-4mm\n\n\n\\begin{equation}\\label{cond_II} \nV \\cap t\\widehat R_0= t V \\quad \\hbox{ and} \\quad\nW \\cap t\\widehat R_0= t W.\n\\end{equation}\n\n\n \\noindent Note that Condition \\eqref{cond_II}  is equivalent to \n\n\n\\begin{equation}\\label{cond_IIbis} \n V \\cap t^n\\widehat R_0= t^n V \\quad \\hbox{and} \\quad \nW \\cap t^n\\widehat R_0= t^n W \\quad \\hbox{ for \\,  each } \\quad  n \\geq 1.\n\\end{equation}\n\n\n\n\n\\begin{sremark} {\\rm Condition \\eqref{cond_II} above is added there compared with  \\cite[2.4]{HHK}\nbut we do not require at this stage that $F_1$ is dense in $F_0$.\n }\n\\end{sremark}\n\n\nWe equip the submodules $V[\\frac{1}{t}]$ of $F_0$ \nof the induced metric (and similarly for $W[\\frac{1}{t}]$).\n\n\n\\begin{slemma}\\label{lem_banach} (1) $V$ is closed in $\\widehat R_0$.\n\n\\smallskip\n\n\\noindent (2) For $v \\in V[\\frac{1}{t}] \\setminus \\{0\\}$, we have  $$\n\\mid v\\mid= \\mathrm{Inf}\\{ \\alpha^{n} \\mid t^n v \\in V \\} .\n$$\n\n\n\\smallskip\n\n\\noindent (3) We have $V= \\mathrm{Inf}\\{ v \\in V[\\frac{1}{t}]  \\mid \n\\enskip \\mid v \\mid  \\geq 0 \\} $ and $V$ is a clopen submodule of\n$V[\\frac{1}{t}]$.\n\n\n\\smallskip\n\n\\noindent (4) $V[\\frac{1}{t}]$ is closed in $F_0$ and is a Banach $K$-space. \n \n\\end{slemma}\n\n\n\n\\begin{proof}\n (1) Our assumption is that the map $V \\to \\limproj V\/t^{m+1} V$ is an isomorphism. Let $(x_n)$ be a sequence of $V$ \n which converges in  $\\widehat R_0$. For each $m \\geq 0$, condition \\eqref{cond_IIbis} shows that \n the map $V\/t^{m+1} V \\to \\widehat R_0\/t^{m+1} \\widehat R_0$ \n is injective so that the sequence $(x_n)$ modulo $t^{m+1} V$ is stationary to some $v_m  \\in V\/t^{m+1} V$.\n The $v_m$'s define a point $v$ of $V$ and the sequence $(x_n)$ converges to $v$.\n \n \\smallskip\n \n \n \\noindent (2) We are given $v = t^m v'\\in  V[\\frac{1}{t}]$  with $v' \\in V \\setminus t V$\n and we have \\break \n  $\\mid\\! v \\! \\mid= \\mathrm{Inf}\\{  \\alpha^{n} \\mid t^n v \\in \\widehat R_0 \\}\n  = \\alpha^{m} \\mathrm{Inf}\\{  \\alpha^{n} \\mid t^n v' \\in \\widehat R_0 \\}$.\n  Condition (II) implies that  $v' \\in    \\widehat R_0 \\setminus t  \\widehat R_0$ so that \n  $\\mid\\! v \\! \\mid=0$. We conclude that  $\\mid\\! v \\! \\mid= \\mathrm{Inf}\\{ \\alpha^{n} \\mid t^n v \\in V \\}$.\n  \n  \n \n \\smallskip\n \n \\noindent (3) It readily follows from the assertion (2).\n \n \n \n \\smallskip\n \n \\noindent (4) Let $(x_n)$ be a sequence of $V[\\frac{1}{t}]$ converging to some $x \\in F_0$.\n We want to show that $x$ belongs to $V[\\frac{1}{t}]$ so that we can assume that $x\\not = 0$\n and that $\\mid\\! x_n \\! \\mid = \\mid\\! x \\! \\mid = \\alpha^m$ for all $n \\geq 0$.\n Assertion (2) shows that $t^m x_n$ is a sequence of $V$ and (1) shows that its limit\n $t^m x$ belongs to $V$. Thus $x \\in V[\\frac{1}{t}]$. We have shown that $V[\\frac{1}{t}]$ is closed in $F_0$.\n \n Finally since $F_0$ is a Banach $K$--space so is $V[\\frac{1}{t}]$.\n\\end{proof}\n\n\n\nThe following statement extends partially \\cite[th. 2.5]{HHK} and \\cite[prop. 4.1]{HHK2}.\n\n\n\\begin{sproposition}\\label{prop_analytic}\nLet $a,b,c$ be  positive integers.\nLet $\\Omega \\subset (F_0)^a \\times (F_0)^b$ be an open\nneighborhood of $(0,0)$ and let \n$f: \\Omega \\to (F_0)^c$ be an analytic map.\nWe denote by $f^a: \\Omega \\cap (F_0)^a \\to (F_0)^c$\nand $f^b: \\Omega \\cap (F_0)^b \\to (F_0)^c$. We assume that \n\n\n\\smallskip\n\n(i) $f(0,0)=0$;\n\n\\smallskip\n\n(ii) the differentials $Df^a_{0}: (F_0)^{a} \\to (F_0)^c$ and   $Df^b_{0}: (F_0)^{a} \\to (F_0)^c$\nsatisfy $$\nDf^a_{0}\\Bigl( V[\\frac{1}{t}]^a \\Bigr) + Df^b_{0}\\Bigl( V[\\frac{1}{t}]^b\\Bigr)=(F_0)^c.\n$$\n\n\n\\smallskip\n\n\\noindent Then there is a real number\n$\\epsilon  > 0$ such that for all $y \\in (F_0)^c$ with $\\mid \\! y \\! \\mid \\, \n \\leq \\epsilon$, there exist\n$v \\in  V^a$ and $w \\in  W^b$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = y$.\n\n\\end{sproposition}\n\n\n\\begin{proof}\nWe consider the continuous embedding $i: V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b  \\to (F_0)^a \\times (F_0)^b$ and define $\\widetilde \\Omega=i^{-1}(\\Omega)$ and \nthe function $\\widetilde f= f \\circ i : \\widetilde \\Omega \\to (F_0)^c$.\n\n\\begin{sclaim}  The map $\\widetilde f$ \nis strictly differentiable at $(0,0)$ and\n$D\\widetilde{f}_{(0,0)}:  V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b \\to (F_0)^c$ is onto. \n\\end{sclaim}\n\n\nSince $f$ is $F_0$--analytic at $(0,0)$, it is strictly differentiable  \\cite[I.5.6]{Sc}, that is, there exists\nan open  neighborhood $\\Theta$ of $(0,0)$ and a positive real number $\\beta$ such that \n$$\n\\mid \\! f(x_2) -f(x_1) - Df_{(0,0)}.(x_2-x_1) \\! \\mid \\enskip \\leq \\enskip \\beta \n\\mid \\! x_2 -x_1 \\! \\mid \\qquad \\forall x_1,x_2 \\in \\Theta.\n$$\nIt is then strictly derivable as function between the \nBanach $K$--spaces $(F_0)^a \\times (F_0)^b \\to (F_0)^c$.\nOn the other hand the embedding  $i: V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b  \\to (F_0)^a \\times (F_0)^b$\nis $1$--Liftschitz so is strictly differentiable at $(0,0)$. As composite of strictly differentiable  functions,\n$\\widetilde f$ is strictly differentiable at $(0,0)$ \\cite[\\S 1.3.1]{BF}. Furthermore\nthe differential \n$D\\widetilde{f}_{(0,0)}$  is the composite of\n$$\nV[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b \\xrightarrow{\\quad i \\quad} (F_0)^a \\times (F_0)^b \\xrightarrow{\\enskip Df^a_{0} + Df^b_{0} \\enskip} (F_0)^c.\n$$\nCondition (ii) says exactly that $D\\widetilde{f}_{(0,0)}$  is surjective. The Claim is proven.\n\n\nWe apply the implicit function theorem to the function $\\widetilde f$ \\cite[\\S 1.5.2]{BF}\n(see  \\cite[\\S 4]{Sc} for concocting a proof). Lemma \\ref{lem_banach}.(4)\nshows that $V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b $ is a Banach $K$--space and so is $F_0$.\nThere exists then an open neighborhood $\\Upsilon \\subset \\widetilde \\Omega$\nof $(0,0)$ in $V[\\frac{1}{t}]^a \\times W[\\frac{1}{t}]^b$  such that $\\widetilde f_{\\mid \\Upsilon}$ is open.\nUp to shrink $\\Upsilon$ we can assume that \n$\\Upsilon \\subset V \\times W$ according to Lemma \\ref{lem_banach}.(3).\nThere exists then a real number $\\epsilon >0$ \nsuch that \n$$\n\\bigl\\{ y \\in (F_0)^c \\mid \\enskip  \\mid \\! y \\! \\mid \\leq  \\epsilon \\bigr\\} \\enskip \\subset \\enskip\n\\bigl\\{ y \\in (F_0)^c \\mid \\enskip  \\mid \\! y \\! \\mid  < 2 \\epsilon \\bigr\\} \\subset \\widetilde f(\\Upsilon).\n$$\nWe conclude that \nfor all $y \\in (F_0)^c$ with $\\mid \\! y \\! \\mid \\, \n \\leq \\epsilon$, there exist\n$v \\in  V^a$ and $w \\in  W^b$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = y$.\n\\end{proof}\n\n\n \n \n\n\\begin{scorollary}\\label{cor_analytic}\nLet $n$ be a positive integer.\nLet $\\Omega \\subset (F_0)^n \\times (F_0)^n$ be an open neighborhood of $(0,0)$ and let \n$f: \\Omega \\to (F_0)^n$ be an analytic map which satisfies \n\n\\smallskip\n\n(i) $f(0,0)=0$;\n\n\\smallskip\n\n(ii) $f(x,0)=f(0,x)=x$ over an open neighborhood $\\Upsilon$ of $0$.\n\n\n\\smallskip\n\n\\noindent Then there is a real number\n$\\epsilon  > 0$ such that for all $a $ with $\\mid \\! a \\! \\mid \\, \n \\leq \\epsilon$, there exist\n$v \\in  V^n$ and $w \\in  W^n$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = a$.\n\n\\end{scorollary}\n\n\\begin{proof} In this case we have $a=b=c=n$ and $Df^a_{0} = Df^b_{0}= \\mathrm{Id}_{ (F_0)^n}$\nso that Proposition \\ref{prop_analytic} applies.\n\\end{proof}\n\n\n\n\\subsection{Kneser-Tits' subgroups}\n\nContinuing in the previous setting, \nwe assume furthermore that $F_1$ is $t$-adically dense\nin $F_0$. Let $F \\subset F_1 \\cap F_2$ be a subfield. \nFor dealing later with Weil restriction issues it is \nconvenient to deal with  a finite field extension $E$ of $F$.\n\n\n\\begin{sproposition}\\label{prop_KT} \n Let  $H$ be a semisimple simply connected\n$E$--group scheme assumed strictly isotropic.\nWe put $G=R_{E\/F}(H)$. \nFor each overfield $L$ of $F$, we put \n$G(L)^+= H(L \\otimes_F E)^+$ where the second group \nis that defined  in \\S \\ref{subsec_PS}.\n Then  we have the decomposition\n$$\nG(F_0)^+= G(F_1)^+ \\, G(F_2)^+.\n$$\n\\end{sproposition}\n \n \n\\begin{proof} Without lost of generality we can assume that $F$ is infinite. The proof is based on an analytic argument requiring \nsome preparation.\n\nLet $P$ be a strictly proper parabolic $E$--subgroup of $H$. \nLet $U$ be its \nunipotent radical and $U_{last}$ the last part of Demazure's filtration\n\\cite[\\S 3.2]{GPS}. Let $u: \\mathbb{G}_{a,E}^d \\buildrel\\sim\\over\\lgr U_{last}$ be a $E$--group isomorphism.\nAccording to \\cite[Lemma 3.4(3)]{GPS}, $E_P(E). \\mathop{\\rm Lie}\\nolimits(U_{last})(E)$\ngenerates $\\mathop{\\rm Lie}\\nolimits(H)(E)$.\nThere exists $g_1,\\dots, g_n \\in E_P(E)$ such that \n$$\n\\mathop{\\rm Lie}\\nolimits(H)(E)= \\,  {^{g_1}\\!\\mathop{\\rm Lie}\\nolimits}(U_{last})(E) \\oplus \\,    ^{g_2}\\!\\mathop{\\rm Lie}\\nolimits(U_{last})(E) \\oplus \\dots \n\\oplus  \\,   ^{g_n}\\!\\mathop{\\rm Lie}\\nolimits(U_{last})(E) . $$\n\nWe consider the map $h:  (\\mathbb{G}_{a,E}^d)^n \\to H$, $h(x_1,\\dots, x_n)= \\, {^{g_1}u}(x_1) \\,  \\dots \\,  \n^{g_n}\\!u(x_n)$. Its differential at $0$ is \n$dh_{0,0}: E^{dn} \\cong \\mathop{\\rm Lie}\\nolimits(U_{last})(E)^n \\to \\mathop{\\rm Lie}\\nolimits(H)(E)$, $(X_1,\\dots, X_n) \\mapsto \\sum_{i=1}^n \\, \\,   ^{g_i}\\!X_i$, \nso is an isomorphism.  The map $h$ is then \\'etale at a neighborhood of $0$.\nIt follows that $h_\\sharp= R_{E\/F}(h): R_{E\/F}\\bigl((\\mathbb{G}_{a,E}^d)^n\\bigr) \\to G=R_{E\/F}(H)$\nis \\'etale also at a neighborhood of $0$ \\cite[A.5.2.(4)]{CGP}\n\n\nSince the field $F_0$ is henselian,  \nthe local inversion theorem holds \\cite[prop. 2.1.4]{GGMB}. \nWe mean that there exists an open neighborhood $\\Upsilon \\subset (F_0)^n$ such that\n the restriction \n$h_{\\sharp \\mid \\Omega}: \\Upsilon \\to G(F_0)$ is a topological open embedding.\n\nWe consider now the product morphism $q: \\Upsilon \\times \\Upsilon \\to G(F_0)$,\n$q(x,y)= h_\\sharp(x) h_\\sharp(y)$. We put  $\\Omega= q^{-1}( h_\\sharp(\\Upsilon))$, this an open\nsubset of $(F_0)^{2n}$.\nThen the restriction $q_{\\mid \\Omega}$ defines an (unique) analytical map\n$f:  \\Omega \\to \\Upsilon$ \nsuch that $q(x,y)= h_\\sharp( f(x,y) )$.\n\nBy construction we have $f(0,x)= f(x,0)=x$ for $x$ in a neighborhood of $0 \\in (F_0)^n$.\nWe apply Corollary \\ref{cor_analytic} to $f$ so that there exists $\\epsilon >0$\nsuch that  for each $a \\in \\Upsilon$ with $\\mid a \\mid  \\,  \\leq \\,  \\epsilon$, there exist\n$v \\in  V^n$ and $w \\in  W^n$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = a$.\nWe denote by $\\Upsilon_\\epsilon= \\Upsilon \\cap B(0, \\epsilon)$.\nThen $h_\\sharp^{-1}( \\Upsilon_\\epsilon)$ is an open neighborhood of \n$0$ in $(F_0)^n$.\n\n\nLet us now prove that $G(F_0)^+= G(F_1)^+ \\times G(F_2)^+$.\nSince $F_1$ is dense in $F_0$, \n$G(F_1)^+$ is dense in $G(F_0)^+$.\nIt is then enough to show that $\\Upsilon_\\epsilon  \\subset G(F_1)^+ \\times G(F_2)^+$.\nLet $g =h(a) \\in h_\\sharp( \\Upsilon_\\epsilon)$.\nThen  $a= f(v,w)$ with $(v, w) \\in  \\Omega$.\nIt follows that $g =h( a)= h_\\sharp( f(v,w)) = q(v,w) = h_\\sharp(v) \nh_\\sharp(w) \\in  G(F_1)^+ \\times G(F_2)^+$.\n\\end{proof}\n\n \nThis could be refined as follows.\n\n\\newpage\n\n\n\n \n\\begin{sproposition}\\label{prop_R_eq} Let \n$H$ be a reductive $E$-group and put $G=R_{E\/F}(H)$.\n\n\\smallskip\n\n\\noindent (1) $RG(F_1) \\, RG(F_2)$ contains an open neighborhood of $1$  in $G(F_0)$.\n\n\n\\smallskip\n\n\\noindent (2) If $RG(F_1)$ is dense in $RG(F_0)$, then  $RG(F_1) \\, RG(F_2)=RG(F_0)$;\n\n\n\\smallskip\n\n\\noindent (3) If $H$ is semisimple simply connected and $H_{F_1 \\otimes_F E}$ is \nstrictly isotropic, then we have\n$$\nG(F_1)^+ \\, RG(F_2)=G(F_0)^+ .\n$$\n\n\n\\smallskip\n\n\\noindent (4) If $H$ is semisimple simply connected and $H_{F_i \\otimes_F E}$ is strictly isotropic for $i=1,2$,\nthen $G(F_1)^+ \\, G(F_2)^+=G(F_0)^+$.\n\n\\end{sproposition}\n \n \nThe subgroups $G(F_1)^+$, $G(F_0)^+ $ are defined as in Proposition \\ref{prop_KT}.\n \n\\begin{proof}\n (1) Let $T \\subset H$ be a maximal $E$--torus and let $1 \\to S  \\to Q \\xrightarrow{s} T \\to 1$\n be a resolution of $T$  where $Q$ is a quasitrivial torus and $S$ is a torus.\n We have $Q=R_{C\/E}(\\mathbb{G}_m)$ where $C$ is an \\'etale $E$--algebra so that\n $Q$ is an open subset of the affine $E$--space $\\mathbf{W}(C)$.\n \n We use now Raghunathan's technique \\cite[\\S 1.2]{R}.\n There exists $h_1,\\dots, h_r \\in H(E)$ such that \n$\\mathop{\\rm Lie}\\nolimits(H)(E)=  \\,  ^{g_1}\\!\\mathop{\\rm Lie}\\nolimits(T)(E)\\oplus \\,   ^{g_2}\\!\\mathop{\\rm Lie}\\nolimits(T)(E) \\oplus \\,  \\dots \\oplus \\;     ^{g_r}\\!\\mathop{\\rm Lie}\\nolimits(T)(E) $.\nWe consider the map $h: (Q^n)_{E} \\to H$, $h(x_1,\\dots, x_n)= \\, ^{g_1}\\!s(x_1) \\,  \\dots \\,  \n^{g_n}\\!s(x_n)$. Its differential at $0$ is \\break \n$dh_{0}: C^r \\to \\mathop{\\rm Lie}\\nolimits(H)(E)$, $(c_1,\\dots, c_r) \\mapsto \\sum_i  \\, \\,  ^{g_i}\\!ds(c_i)$, \nand is onto (observe that $\\mathop{\\rm Lie}\\nolimits(Q)(E) \\to \\mathop{\\rm Lie}\\nolimits(T)(E)$ is surjective). \n\nWe cut  now $Q^r$ by some suitable affine $E$--subspace $\\mathbf{W}(C^n)$ of  $\\mathbf{W}(C)^r$\nsuch that the restriction $h'$ of $h$ to $X=Q^r \\cap \\mathbf{W}(C^n)$\nis such that $dh'_{0}: \\mathrm{Tan}_{X,1} \\to \\mathop{\\rm Lie}\\nolimits(H)(F)$ is an isomorphism.\nNote that $X$ and $E$ have same dimension $n$ over $E$.\n\nThe map $h'$ is then \\'etale at a neighborhood of $1$.\nIt follows that \\break $h'_\\sharp= R_{E\/F}(h'): R_{E\/F}\\bigl(X\\bigr) \\to G=R_{E\/F}(H)$\nis \\'etale also at a neighborhood of $0$ \\cite[A.5.2.(4)]{CGP}\n\nSince the field $F_0$ is henselian,  \nthe local inversion theorem holds \\cite[prop. 2.1.4]{GGMB}. \nWe mean that there exists an open neighborhood $\\Upsilon \\subset (F_0)^n$ such that\n the restriction \n$h'_{\\sharp \\mid \\Omega}: \\Upsilon \\to G(F_0)$ is a topological open embedding.\n\nWe consider now the product morphism $q: \\Upsilon \\times \\Upsilon \\to G(F_0)$,\n$q(x,y)= h'_\\sharp(x) \\, h'_\\sharp(y)$. We put  $\\Omega= q^{-1}( h'_\\sharp(\\Upsilon))$, this an open\nsubset of $(F_0)^{2n}$.\nThen the restriction $q_{\\mid \\Omega}$ defines a (unique) analytical map $f: \n \\Omega \\to \\Upsilon$ \nsuch that $q(x,y)= h'_\\sharp( f(x,y) )$.\n\nBy construction we have $f(0,x)= f(x,0)$ for $x$ in a neighborhood of $0 \\in F_0^n$.\nWe apply Corollary \\ref{cor_analytic} to $f$ so that there exists $\\epsilon >0$\nsuch that  for each $a \\in \\Upsilon$ with $\\mid a \\mid  \\,  \\leq \\epsilon$, there exist\n$v \\in  V^n$ and $w \\in  W^n$ such that\n$(v, w) \\in  \\Omega$  and $f(v, w) = a$.\nWe denote by $\\Upsilon_\\epsilon= \\Upsilon \\cap B(0, \\epsilon)$.\nThen ${h'_\\sharp}^{-1}( \\Upsilon_\\epsilon)$ is an open neighborhood of \n$0$ in $(F_0)^n$.\n\nWe claim that $\\Upsilon_\\epsilon \\subset RG(F_1) \\, RG(F_2)$.\nLet $g =h'_\\sharp(a) \\in h( \\Upsilon_\\epsilon)$.\nThen  $a= f(v,w)$ with $(v, w) \\in  \\Omega$.\nIt follows that $g= h'_\\sharp( a)= h_\\sharp( f(v,w)) = q(v,w) = h'_\\sharp(v) h'_\\sharp(w) \\in  RG(F_1) \\times RG(F_2)$.\n\n\n \\smallskip\n \n \\noindent (2)  \n If $RG(F_1)$ is furthermore dense in $RG(F_0)$, then  (1) shows that \n $RG(F_1) \\, RG(F_2)$ is a dense open subset of $RG(F_0)$.\n Thus $RG(F_1) \\, RG(F_2)=RG(F_0)$ \n \n \n \\smallskip\n \n \\noindent (3) We assume that $H$ is semisimple simply connected and \n that  $G_{F_1 \\otimes_F E_1}$ is strictly isotropic.\n According to Lemma \\ref{lem_GS}.(2), we have\n $G(F_1)^+=RG(F_1)=RH(F_1 \\otimes_F E)=H^+(F_1 \\otimes_F E)$\n and similarly for $F_0$.\n \n \nSince $H^+(F_1 \\otimes_F E)^+$ is dense in $H^+(F_0 \\otimes_F E)$, it follows that \n$RG(F_1)$ is dense in $RG(F_0)$. Assertion  \n  (2) yields then  $RG(F_1) \\, RG(F_2)=RG(F_0)$.\n  Lemma \\ref{lem_R_eq}.(2) states that $G(F_1)^+=RG(F_1)$\n  and $G(F_0)^+=RG(F_0)$. We conclude that \n  $G(F_1)^+ \\, RG(F_2)=G(F_0)^+$.\n \n\n \\smallskip\n \n \\noindent (4) We have  furthermore  $G(F_2)^+= RG(F_2)$ so that (3)\n yields  $G(F_1)^+ \\, G(F_2)^+ =G(F_0)^+$.\n\\end{proof}\n\n \n \\begin{sremark} {\\rm Note that (1) shows in particular that  \n $RG(F_0)$ is an open subgroup of $G(F_0)$.\n  }\n \\end{sremark}\n \n \n The main result of the section is the following patching statement on twisted flag varieties.\n \n\n \\begin{stheorem} \\label{thm_key} We put $F=F_1 \\cap F_2$. \n Let $H$ be a reductive $E$--group and let \n $X$ be a twisted flag $E$--variety of $H$.\n Assume that $X$ is auto-opposite, that is,\n the stabilizer $P$ of an $E_s$--point of $X$\n is conjugated to an opposite parabolic subgroup of $P$ \\cite[\\S 4.9]{BoT65}. We put $G=R_{E\/F}(H)$ and $Z=R_{E\/F}(X)$.\n If $Z(F_1) \\not = \\emptyset$\n and  $Z(F_2) \\not = \\emptyset$, then  $X(F) \\not = \\emptyset$.\n \\end{stheorem}\n\n \n \\begin{proof} Without loss of generality, we can assume that $H$ is\n semisimple simply connected and that it is absolutely $E$--simple.\n This implies that $H$ is strictly $F_i \\otimes_F E$--isotropic for $i=1,2$.\n Let $x_i \\in Z(F_i)= X( F_i \\otimes_F E)$ and denote by $P_i=\\mathop{\\rm Stab}\\nolimits_{G_{F_2}}(x_2)$\n its stabilizer, this is  parabolic $F_i \\otimes_F E$--subgroup of $H$. We denote by $U_i\/E$ its unipotent radical. \n\n Since the conjugacy class of $P_1$ is autopposite, there exists \n $h \\in H(E \\otimes_F F_0)$ such that $P_{1, E \\otimes_F F_0}$\n is opposite to $^hP_{2, E \\otimes_F F_0}$ \\cite[XXVI.5.3]{SGA3}.\nAccording to Lemma \\ref{lem_GS0}, we have \n$H(E \\otimes_F F_0)= H^+(E \\otimes_F F_0) \\,  P_2(E \\otimes_F F_0)$\nso we can assume that $h \\in H^+(E \\otimes_F F_0)$.\nAccording to Proposition \\ref{prop_R_eq}.(4), \nwe can write $h=h_1 \\, h_2$ with $h_i \\in H^+(E \\otimes_F F_i)$ for $i=1,2$.\nUp to replace $P_1$  by $^{h_1^{-1}}P_1$ and $P_2$ by  $^{h_2^{-1}}P_2$\nwe can then assume that $P_{1,F_0}$ is opposite to $P_{2,F_0}$.\n According to \\cite[XXVI.5.1]{SGA3}, we have a decomposition \n\n\\begin{equation} \\label{michel} \n H(F_0 \\otimes_F E)  =U_2(F_0 \\otimes_F E)  \\, U_1(F_0 \\otimes_F E) \\,\n P_2(F_0\\otimes_F E).\n \\end{equation}\n \n\\noindent Let $h \\in H(F_0 \\otimes_F E)$ such that $x_1=h.x_2$. The preceding decomposition permits to write\n$h= u_2 \\,  u_1 \\,  p_2$ with  $u_2 \\in U_2(F_0\\otimes_F E)$, \n $u_1 \\in  U_1(F_0 \\otimes_F E)$\nand $p_2 \\in  P_2(F_0 \\otimes_F E)$. It follows\nthat  $$\nx_1= u_1^{-1}.x_1 = (u_1^{-1}h).x_2= (u_1^{-1} \\, u_2 \\, u_1) \\, . \\, (p_2.x_2) =(u_1^{-1} \\, u_2 \\, u_1) \\, . \\, x_2\n$$\n hence $x_1 \\in H(F_0 \\otimes_F E)^+.x_2$.\n According to Proposition \\ref{prop_R_eq}.(4), we have  $G(F_0)^+=  G(F_1)^+ \\,  G(F_2)^+$\n so that $u_1^{-1} \\, u_2 \\,  u_1 =g_1 \\,  g_2$ with $g_i \\in G(F_i)^+$ for $i=1,2$.\n It follows that $g_1^{-1} \\, . \\, x_1 = g_2^{-1} \\, . \\, x_2$, this defines a point of  $X(E)=Z(F)$.\n \\end{proof}\n\n \n \n \n \\section{Relation with the original HHK method}\n \nWe recall the setting.\nLet $T$ be an excellent  complete discrete valuation ring with fraction field $K$, residue field $k$ and\nuniformizing parameter $t$. \nLet $F$ be a one-variable function field over $K$\nand let $\\goth X$ be a normal model of $F$, i.e. a normal connected\nprojective $T$-curve with function field $F$.\n We denote by $Y$ the closed fiber of $\\goth X$ and fix a separable closure $F_s$ of $F$.\n   \n For each point $P \\in Y$, let $R_P$ be the local ring of\n $\\goth X$ at $P$; its completion $\\widehat R_P$ is a domain \n with fraction field denoted by $F_P$.\n \n \n For each subset $U$ of $Y$ that is contained\n in an irreducible component of $Y$ and does not meet the other components, \nwe define\n $R_U= \\bigcap\\limits_{P \\in U} R_P \\subset F$. We denote by $\\widehat R_U$\n the $t$--adic completion of $R_U$.\n The rings $R_U$ and $\\widehat R_U$ are excellent normal domains \n and we denote by $F_U$ the fraction field of $\\widehat R_U$\n \\cite[Remark 3.2.(b)]{HHK3}.\n \n Each height one prime $\\mathfrak{p}$ in $\\widehat R_P$ that contains\n  $t$ defines a branch of $Y$ at $P$ lying \n on some irreducible component of $Y$. \n The $t$-adic completion  $\\widehat R_{\\mathfrak{p}}$ of the local ring\n $R_{\\mathfrak{p}}$ of $\\widehat R_P$ at $\\mathfrak{p}$ is a complete DVR\n whose fraction field is denoted by $F_{\\mathfrak{p}}$. \n The field $F_{\\mathfrak{p}}$ contains also $F_U$ if $U$ \n is an irreducible open subset of $Y$ such that $P \\in \\overline{U} \\setminus U$.\n We have then a diagram of fields\n \n\\[\\xymatrix{\n &   F_{\\mathfrak{p}}  & \\\\ \n F_P \\ar[ru] & & F_U  . \\ar[lu]\n}\\]\n \n \n \n \n \\begin{sexample} \\label{sexample1}{\\rm We assume that $T=k[[t]]$ and \n take $X= \\mathbb{P}^1_K$, $\\goth X= \\mathbb{P}^1_T$,\n $P= \\infty_k$, $U=\\mathbb{A}^1_k= \\mathop{\\rm Spec}\\nolimits(k[x])$.\n The ring $R_U$ contains $k[[t]][x]$ and is \n its localization with respect to the multiplicative set $S$\n of elements which are units modulo $t$.\n The  $t$-adic completion of $R_U$ is   $\\widehat R_U=\n k[x][[t]]$; we have $F_U= Frac( \\widehat R_U)=\n k(x)((t))$.\n The local ring of $\\goth X$ at $P=\\infty_k$ is $R_P=k[[t]][x^{-1}]_{(x^{-1},t)}$\n so that its completion is $\\widehat R_P= k[[t,x^{-1}]]$;\n in particular $F_P=k((t,x^{-1}))$.\n \n We take  $\\mathfrak{p}= t \\widehat R_P \\subset \\widehat R_P$\n and  the $t$-adic completion  $\\widehat R_{\\mathfrak{p}}$ \n of the local ring\n $R_{\\mathfrak{p}}$ of $\\widehat R_P$ at $\\mathfrak{p}$ is a complete DVR\n which is $k((x^{-1}))[[t]]$. In particular $F_\\goth p= k((x^{-1}))((t))$.\n }\n \\end{sexample}\n\n \\medskip\n \n \n \\begin{ssetting} \\label{setting_hhk} {\\rm\n Let $\\mathcal P$ be a non-empty finite set of closed points of $Y$ that contains\n all the closed points at which distinct irreducible components meet.\n Let $\\mathcal U$ be the set of connected components of $Y \\setminus \\mathcal P$ and let $\\mathcal B$\n be the set of branches of $Y$ at points of $\\mathcal P$.\n This yields a finite inverse system of field $F_P, F_U, F_\\goth p$ (for $P \\in \\mathcal P$;\n $U \\in \\mathcal U$, $\\goth p \\in \\mathcal B$) where $F_P, F_U \\subset F_\\goth p$ if $\\goth p$\n is a branch of $Y$ at $P$ lying in the closure of $U$.\n }\n\\end{ssetting}\n\n\n\n\\begin{slemma}  \\label{lem_dense0} We assume that  $X= \\mathbb{P}^1_K$, $\\goth X= \\mathbb{P}^1_T$,\n $P= \\infty_k$, $U=\\mathbb{A}^1_k= \\mathop{\\rm Spec}\\nolimits(k[x])$ and $\\goth p$ the branch\n of $P$. We put $F_1=F_P$, $F_2=F_U$\n and $F_0=F_\\goth p$.\n \n\n\\smallskip\n\n\\noindent (1) $F_1$ is $t$-dense in $F_0$.\n\n\n\\smallskip\n\n\\noindent (2)  We put $V= F_1 \\cap \\widehat R_\\mathfrak{p}$ and   $W= F_2 \\cap \\widehat R_\\mathfrak{p}$.\nThen $V$ and $W$ satisfy conditions \\eqref{cond_I} and \\eqref{cond_II}.\n \n\\end{slemma}\n\n\n\n \\begin{proof}\n \\noindent  (1) We are given $u_0\/v_0 \\in F_0$ with $u_0, v_0 \\in \\widehat R_\\mathfrak{p}$,  $v_0 \\not =0$.\n There exists elements $u, v \\in R_\\goth p$ very close respectively of $u_0,v_0$ with $v \\not =0$.\n Let $s_1, s_2 \\in R_p \\setminus R_p \\goth p$ such that $s_1 u \\in R_P$ and $s_2 v \\in R_\\goth p$.\n Then $u_0\/v_0$ is very close of $(s_1s_2 u) \/ (s_1s_2 v) \\in F_1$.\n \n  \n  \\smallskip\n  \n  \\noindent (2) Condition \\eqref{cond_II} is obviously fullfilled since $t \\in F_1 \\cap F_2$.\n  For establishing condition \\eqref{cond_I}, we are given an element $f$ of $\\widehat R_\\goth p$ and \n  may write it as $f= \\sum\\limits_{i=0}^\\infty x^{m_i}  \\Bigl( \\sum_{j=0}^\\infty a_{i,j} \\frac{1}{x^j} \\Bigr) t^i$\n  where the $m_i$'s are non-negative integers and $a_{i,j} \\in T$. \n  We decompose $$\n  f= f_1 + f_2=  \\sum_{i=0}^\\infty x^{m_i} \\Bigl( \\sum_{j=m_i}^\\infty a_{i,j} \\frac{1}{x^j} \\Bigr) t^i\n  \\quad +  \\quad\n   \\sum_{i=0}^\\infty x^{m_i}  \\Bigl(\\sum_{j=0}^{m_i-1} a_{i,j} \\frac{1}{x^j} \\Bigr)  t^i \n  $$\n  We observe that $f_2$ belongs to $\\widehat R_P$ so belongs to $V$.\nWe recall that $R_U$ is the localization of $T[x]$ \nwith respect to the elements which are units modulo $t$. \nWe conclude that $f_2$ belongs to $W$ as desired.\n \\end{proof}\n \n \n \n \n \\begin{stheorem} \\label{thm_main_hhk} Let $G$ be a reductive $F$--algebraic group.\n \n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if $Z(F_U) \\not = \\emptyset$ for \n each $U \\in \\mathcal U$ and  $Z(F_P) \\not = \\emptyset$ for \n each $P \\in \\mathcal P$.\n \n \\smallskip\n \n \\noindent (2) For each $U \\in \\mathcal U$ (resp.\\ each $P \\in \\mathcal P$), \n we fix an $F$--embeddings $i_U: F_s \\to F_{U,s}$  (resp.\\ $i_P: F_s \\to F_{P,s}$)\n providing identifications $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{U,s}})$\n (resp.\\, $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{P,s}})$).\n The Tits index $\\Delta_0(G)$ is the smallest subset\n  of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n  $\\Delta_0(G) \\subset \\Delta_0(G_{F_U})$ for each $U \\in \\mathcal U$ and  $\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for  each $P \\in \\mathcal P$.\n \n \\end{stheorem}\n\n The recollection for star action and Tits index is done in the beginning of the paper.\n \n \\begin{proof}\n The outline is to prove (1) under an assumption\n of autoppositeness, to prove (2) and then to prove (1) in the general case.\n \n \\smallskip\n \n \\noindent (1) We assume that $Z$ parameterizes an\n  auto-opposite conjugacy\n class of parabolic subgroups of $G$. \n  We use a Weil restriction argument as in the proof of \\cite[Thm. 4.2]{HHK2}.\nThis involves a finite morphism $f: \\goth X \\to \\mathbb{P}^1_T$ such that \n$\\mathcal P=f^{-1}(\\infty_k)$. Write $\\uF$ for the function field of  $\\mathbb{P}^1_T$, and let $d=[F:F']$. We put\n$\\uU= \\mathbb{P}^1_k \\setminus \\{\\infty \\}$, $\\uP= \\infty_k$ and\n$\\underline{\\goth p}= (U, \\mathfrak{p})$ and\n  $F_0=\\uF_{\\underline{\\goth p}}$, $F_1=\\uF_{\\uP}$ and $F_2=F_{\\uU}$.  Also patching holds for the diamond $(\\uF, F_1,F_2,F_0)$ according to \\cite[thm 3.9]{HH}\n  so in particular $K(x)= \\uF= F_1 \\cap F_2 \\subset F_0$. \n  We put $V= F_1 \\cap \\widehat R_\\mathfrak{p}$ and   $W= F_2 \\cap \\widehat R_\\mathfrak{p}$.\n  Lemma \\ref{lem_dense0} shows that \n  $F_1$ is dense in $F_0$ and that  $V$ and $W$ satisfy conditions \\eqref{cond_I} and \\eqref{cond_II}.\n\n  We consider the Weil restriction $\\uG= R_{F\/\\uF}(G)$, it is a reductive $\\uF$--group\nwhich acts on the $\\uF$--variety $\\uZ=R_{F\/\\uF}(Z)$.\nWe have $$\n\\uZ(F_1)= Z(F_1 \\otimes_{\\uF} F)  = \\prod_{P \\in \\mathcal P} Z(F_P)\n$$\naccording to \\cite[Lemma 6.2.(a)]{HH}.\nSimilarly we have\n$$\n\\uZ(F_2)= Z(F_2 \\otimes_{\\uF} F)  = \\prod_{U \\in \\mathcal U} Z(F_U)\n$$\nOur assumptions imply that $\\uZ(F_1) \\not = \\emptyset$ and \n$\\uZ(F_2) \\not = \\emptyset$. Theorem \\ref{thm_key} implies \nthat $\\uZ(\\uF) \\not = \\emptyset$. Thus $\\uZ(\\uF)= Z(F)$ is non-empty.\n\n  \\smallskip  \n  \n  \\noindent (2) Let $\\Theta$ be the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n  $\\Delta_0(G) \\subset \\Delta_0(G_{F_U})$ for each $U \\in \\mathcal U$ and  $\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for \n  each $P \\in \\mathcal P$.\n  Since $\\Theta$ is an intersection of auto-opposite subsets of $\\Delta(G_{F_s})$, it is auto-opposite.\n  \n  We observe that $\\Delta_0(G) \\subset \\Theta$ since it is \n   stable under the star action and \n  satisfies $\\Delta_0(G) \\subset \\Delta_0$ for each $U \\in \\mathcal U$ and  $\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for  each $P \\in \\mathcal P$.\n  For the converse inclusion we consider the $F$--variety $Z$ of parabolic\n  subgroups of type $\\Theta$ (which is auto-opposite).\n  For each $U \\in \\mathcal U$, we have  $\\Theta \\subset \\Delta_0(G_{F_U})$ so that\n  $Z(F_U) \\not = \\emptyset$; similarly we have \n   $\\Theta \\subset \\Delta_0(G_{F_P})$ for each $P \\in \\mathcal P$\n   so that $Z(F_P) \\not = \\emptyset$.\n  Part (1) yields that $Z(F) \\not = \\emptyset$. Thus $\\Theta \\subset \\Delta_0(G)$\n  and $\\Theta = \\Delta_0(G)$\n  \n  \\smallskip\n  \n\\noindent (3) We consider now the case of an arbitrary flag $F$--variety $Z$  for $G$. It   is associated to a subset $\\Upsilon$ of $\\Delta(G_{F_s})$\n which is invariant under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$. \n If $Z(F) \\not = \\emptyset$, we have that $Z(F_U) \\not = \\emptyset$ for each \n $U \\in \\mathcal U$ and that $Z(F_P) \\not = \\emptyset$ for each \n $P \\in \\mathcal P$. Conversely if $Z(F_U) \\not = \\emptyset$ for each \n $U \\in \\mathcal U$ and  $Z(F_P) \\not = \\emptyset$ for each \n $P \\in \\mathcal P$. \n It follows that $ \\Upsilon \\subset \\Delta_0(G_{F_U})$\n for each $U \\in \\mathcal U$.\n Part (2) of the statement yields $\\Upsilon \\subset \\Delta_0(G)$\n so that $Z(F) \\not = \\emptyset$.  \n \\end{proof}\n\n \n \n \\begin{scorollary} \\label{cor_main_hhk} Let $G$ be a reductive $F$--algebraic group.\n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if  $Z(F_P) \\not = \\emptyset$ for \n each $P \\in Y$.\n \n \\smallskip\n \n \\noindent (2) For each $P \\in Y$, \n we fix an $F$--embedding  $i_P: F_s \\to F_{P,s}$\n providing identifications \n  $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{P,s}})$.\n The Tits index $\\Delta_0(G)$ is the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n$\\Delta_0(G) \\subset \\Delta_0(G_{F_P})$ for  each $P \\in Y$.\n \n \\end{scorollary}\n\n \n \\begin{proof}\n (1)  We assume $Z(F_P) \\not = \\emptyset$ for \n each $P \\in Y$. Let $Y_1,\\dots, Y_d$ be the irreducible components of $Y$\n with respective generic points $\\eta_1, \\dots,  \\, \\eta_d$. \n According to  \\cite[prop. 5.8]{HHK2}, there exists non-empty affine subsets \n $U_i \\subset Y_i$ ($i=1,\\dots, d$) such that $Z(F_{U_i}) \\not = \\emptyset$\n for $i=1,\\dots, d$ and $U_i \\cap U_j = \\emptyset$ for $i<j$.\n We apply Theorem \\ref{thm_main_hhk} to  $\\mathcal U= \\{ U_1,\\dots, U_d\\}$, \n $\\mathcal P= Y \\setminus \\cap_i U_i$ and get that $Z(F) \\not = \\emptyset$.\n \n \\smallskip\n \n \\noindent (2) This readily follows of (1).\n  \n \\end{proof}\n\n \n\n\n \\begin{scorollary} \\label{cor_main_hhk2} Let $G$ be a reductive $F$--algebraic group\n and assume that $G$ is the generic fiber of a reductive $\\goth X$-group scheme $\\mathfrak{G}$.\n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if  $Z(F_v) \\not = \\emptyset$ for \n each discrete valuation $v$ of $F$.\n \n \\smallskip\n \n \\noindent (2) For each discrete valuation $v$ of $F$, \n we fix an $F$--embedding  $i_v: F_s \\to F_{v,s}$\n providing identifications \n  $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{v,s}})$.\n The Tits index $\\Delta_0(G)$ is the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n$\\Delta_0(G) \\subset \\Delta_0(G_{F_v})$ for  each discrete valuation $v$ on $F$.\n \n \\end{scorollary}\n\n \n \\begin{proof}\n (1) We assume that $Z(F_v) \\not = \\emptyset$ for \n each discrete valuation $v$ of $F$.\n Let $\\mathfrak{G}$ be a reductive $\\goth X$--group of generic fiber $G$. \n Then $Z$ is the generic fiber of an $\\goth X$--scheme of parabolic subgroups\n $\\goth Z$. For applying Corollary \\ref{cor_main_hhk}, we have to check\n that $Z(F_P) \\not = \\emptyset$ for each $P \\in Y$.\n If $Q$ is a point of codimension $1$ of $\\goth X$, \n it defines a discrete valuation $v_Q$ on $F$ whose completion is  $F_Q$.\n Our assumption implies then that  $Z(F_Q) \\not = \\emptyset$ in that case.\n We deal now with the case of a closed point $P$ of $\\goth X$.\n Let $D$ be an irreducible component of $Y=\\goth X_k$ containing $P$\n and let $Q$ be the generic point of $D$. Since \n $\\goth Z$ is proper over $T$, we have $\\goth Z(\\widehat R_Q)=\\goth Z(F_Q)$ which\n is not empty by the preceding case. It follows that $\\goth Z_k(k(D)) \\not = \\emptyset$.\n Again $\\goth Z_k$ is projective so that $\\goth Z_k(D) =\\goth Z_k(k(D))$ is not empty\n and in particular $\\goth Z_k(k(P))$ is not empty.\n Since $\\goth Z$ is smooth over $\\goth X$, the Hensel lemma\n shows that $\\goth Z_k(\\widehat R_{\\goth X,P}) \\to  \\goth Z_k(k(P))$ is surjective.\n Thus $\\goth Z(\\widehat R_{\\goth X,P})$ is not empty and so is $Z(F_P)$.\n \n \n \\smallskip\n \n \\noindent (2) It readily follows of (1). \n  \n \\end{proof}\n \n \n \n \n \n \\begin{scorollary} \\label{cor_main_hhk3} Let $G$ be a reductive $F$--algebraic group.\n We denote by $\\Omega^1_F$ the set of rank one valuations of $F$.\n \n \\smallskip\n \n \\noindent (1) Let $Z$ be a twisted flag projective $F$--variety for $G$.\n Then $Z(F) \\not= \\emptyset$ if and only if  $Z(F_v) \\not = \\emptyset$ for \n each $v \\in \\Omega^1_F$.\n \n \\smallskip\n \n \\noindent (2) For each $v \\in \\Omega^1_F$, \n we fix an $F$--embedding  $i_v: F_s \\to F_{v,s}$\n providing identifications \n  $\\Delta(G_{F_s}) \\buildrel\\sim\\over\\lgr \\Delta(G_{F_{v,s}})$.\n The Tits index $\\Delta_0(G)$ is the smallest subset of $\\Delta(G_{F_s})$\n which is stable under the $\\star$--action of $\\mathop{\\rm Gal}\\nolimits(F_s\/F)$ and such that \n$\\Delta_0(G) \\subset \\Delta_0(G_{F_v})$ for  each $v \\in \\Omega^1_F$.\n \n \\end{scorollary}\n\n \n \\begin{proof}\n (1) We assume that $Z(F_v) \\not = \\emptyset$ for \n each $v \\in \\Omega^1_F$.\n  According to \\cite[Thm 2.5]{HHKP}, there exists a regular model $\\goth X'$ of $F$\n  such that $Z(F_P) \\not = \\emptyset$ for \n each $P \\in \\goth X' \\times_T k$.  Then Corollary \\ref{cor_main_hhk2}.(1)\n shows that $Z(F) \\not = \\emptyset$.\n \n \\smallskip\n \n \\noindent (2) This readily follows of (1).\n \\end{proof}\n \n \n \n\\section{Local-global principle for discrete valuations}\n\nLet $T$ be an excellent DVR of fraction field $K$ and residue field $k$.\nLet $X$ be a smooth, projective,\ngeometrically integral curve over $K$. Let $F=K(X)$ be the function field of $X$.\n\n \n\n \n\\begin{slemma} \\label{lem_extend} Let $\\goth X$ be a projective, flat curve over $A$\nwhich is connected and regular such that $\\goth X_K=X$.\nLet $\\goth H$ be a flat  affine $\\goth X$--group scheme of finite presentation\nand assume that  there  exists a  Zariski cover $(\\mathfrak{U}_i)_{i \\in I}$ of $\\goth X$ \n such that each $\\goth H_{U_i}$ admits a closed embedding $\\goth H_{U_i} \\subset \\GL_{n_i}$ such that \n $\\GL_{n_i}\/\\goth H_{U_i}$ is representable by an affine $U_i$--scheme,\n\nLet $\\gamma \\in H^1_{\\text{\\rm fppf}}(F,\\goth H)$ and let \n $D$ be a divisor of $\\goth X$ which contains the irreducible\ncomponents of $Y= \\goth X_k$ and such that $\\gamma$ extends \\textit\nto $X \\setminus D_K$.\n\nThen there exists a proper birational morphism $q: \\goth X' \\buildrel\\sim\\over\\lgr \\goth X$  \nsuch that $\\goth X'$ is a regular model of $X$ and \nsuch that $D'=q^*D$ is  a strict normal crossing divisor  and  \n$$\n\\gamma \\in \\mathrm{Im}\\Bigl( H^1(\\goth X' \\setminus D', \\goth H) \\to H^1(F, \\goth H) \\Bigr).\n$$\n\\end{slemma}\n\n\n\\begin{proof} Using a passage to the limit argument \\cite{Mg}, there exists an open affine subscheme \n$\\mathfrak{U}_1 \\subset \\goth X \\setminus D$\nsuch that $\\gamma$ extends to a class $\\gamma_1 \\in H^1(\\mathfrak{U}_1, \\goth H)$.\nAccording to \\cite[cor 1.8]{GP1}, $\\gamma_1$ extends to a class\n$\\gamma_2 \\in H^1(\\mathfrak{U}_2, \\goth H)$ where $\\mathfrak{U}_2$ is an open subscheme of $\\goth X  \\setminus D$\ncontaining $\\mathfrak{U}_1$ and $X \\setminus D_K$. \n\nBy purity (i.e. Theorem \\ref{thm_extend} of the appendix \\ref{app_extend}),\nwe have $H^1(\\goth X \\setminus D, \\goth H) = H^1( \\mathfrak{U}_2, \\goth H)$.\nThus $\\gamma$ extends over  $\\goth X \\setminus D$.\n\n\nAccording to Lipman's theorem we can resolve the singularities of $\\goth X$\nand transform $D$ in a  strict normal crossing divisor, see \\cite[lemma 4.7]{HHK}.\n\\end{proof}\n\n \n\\begin{sproposition} \\label{prop_patching}\n Let $G$ be a reductive $F$--group  and assume that\n $p$ does not divide the order of the automorphism group of\n the  absolute root system of $G_{ad}$.\n \n Let $Z$ be a twisted flag  variety of $G$.\nWe assume that $Z(F_v) \\not = \\emptyset$ for \nall discrete valuations of $F$ arising from models of $X$.\n \nThen there exists a regular model $\\goth X$ of $X$ and \na finite subset $\\mathcal P$ of $Y=\\goth X_k$ (containing)\nall the closed points at which distinct irreducible components meet)\n such that \n for each triple of fields $\\{F_U, F_P,F_{\\mathfrak{p}}\\}$ as in Setting \\ref{setting_hhk},  we have  $Z(F_U) \\not = \\emptyset$ and $Z(F_P) \\not = \\emptyset$.\n \\end{sproposition}\n \n\n \n \\begin{proof}\n Without loss of generality we can assume that $G$ is adjoint.\n   Let $G_0$ be the Chevalley form of $G$ and \n   let $(B_0,T_0)$ be a \n   Killing couple for  $G_0$ and let $\\Delta_0$ be\n   the associated Dynkin diagram. \n The variety $Z$ is a form of the variety $\\text{\\rm{Par}}_I(G_0)$\n  of parabolic subgroups of type $I$ where $I \\subset \\Delta_0$ is stable\n   under the star action defined by the $\\text{\\rm{Aut}}(G_0)$--torsor\n   $Q=\\mathrm{Isom}(G_0,G)$.  \n   In particular $Q$ admits a reduction $Q_I$ to the stabilizer \n   $\\text{\\rm{Aut}}_I(G_0)$ for the action  $\\text{\\rm{Aut}}(G_0)$ on $\\Delta_0$ through\nthe morphism $\\text{\\rm{Aut}}(G_0) \\to \\text{\\rm{Out}}(G_0)\\buildrel\\sim\\over\\lgr \\text{\\rm{Aut}}(\\Delta_0)$.\nFurthermore $Z$ is isomorphic to  $^{Q_I} \\! \\text{\\rm{Par}}_I(G_0)$.\n   \nWe apply  now Theorem 1.1.(b) of \\cite{CGR}\nto the $\\mathbb{Z}$--group scheme $\\text{\\rm{Aut}}_I(G_0)$. \nIt provides a finite $\\mathbb{Z}$--subgroup $S_0$ of $\\text{\\rm{Aut}}_I(G_0)$ such that the map $H^1(F, S_0) \\to H^1(F, \\text{\\rm{Aut}}(G_0))$ is onto.\nFurthermore the construction of $S_0$ is explicit in the proof,\nit is an extension of the finite constant group \n $\\text{\\rm{Aut}}_I(G_0)\/T_0$ by a finite subgroup of $T_0$.\n In particular $S_0$ is finite  free over $\\mathbb{Z}$  and our assumption on the characteristic implies that \n$S_{0,T}$ is finite \\'etale of rank prime to $p$. \n   It follows that $Q_I$ admits a reduction to an $F$--torsor \n   $E$ under $S_0$.  \n   \n   \n   The  finite flat $\\mathbb{Z}$--group scheme $S_{0,A}$ admits \n   a faitful representation $S_0 \\hookrightarrow \\GL_{N,\\mathbb{Z}}$\n   \\cite[\\S 1.4.5]{BT} and the quotient \n   $\\GL_{N,\\mathbb{Z}} \/S_0$ is representable by an affine $\\mathbb{Z}$-scheme \\cite[\\S III.2.6]{DG}.\n   \n   \n  According to Lemma \\ref{lem_extend},\n  there exists  a regular model $\\goth X$ of $X$ and \n  a strict normal crossing divisor  $D$ containing  the irreducible components $Y_k$ such that $E$ extends \n  to a $\\goth X \\setminus D$--torsor $\\mathfrak{E}$  under $S_0$.\n  We put $\\mathfrak{Q}_I= \\mathfrak{E} \\wedge^{S_0}\\text{\\rm{Aut}}_I(G_0)$ and consider\n  the $\\goth X \\setminus D$-group scheme $\\mathfrak{G}= {^{\\mathfrak{Q}_I}\\!G_0}$ of generic fiber $G$.\n  \n Let $Y_1,\\dots, Y_d$ be the irreducible components of the closed fiber $Y=\\goth X_k$\n with respective generic points $\\eta_1, \\dots,  \\, \\eta_d$. \n According to  \\cite[prop. 5.8]{HHK2}, there exists non-empty affine subsets \n $U_i \\subset Y_i$ ($i=1,\\dots, d$) such that $Z(F_{U_i}) \\not = \\emptyset$\n for $i=1,\\dots, d$ and $U_i \\cap U_j = \\emptyset$ for $i<j$.\n Up to refine  the model, we can assume that $Y_i \\setminus U_i \\subset  D_k$ for $i=1,...,d$.\n \n We put $\\mathcal P= Y \\setminus \\cup_i Y_i$ and want to establish that \n $Z(F_P) \\not = \\emptyset$ for each $P \\in \\mathcal P$.\n We are given a point $P \\in \\mathcal P$ and pick a height one prime $\\mathfrak{p}$ in $\\widehat R_P$ that contains\n  $t$. It  defines a branch of $Y$ at $P$ lying \n on some irreducible component $Y_i$ of $Y$.\n We deal now with $U=U_i$.\n\nWe consider the local ring $A=R_P$ of $\\goth X$ at $P$ and denote by $A_D$\nits localization at $D$. Since $S_{0,T}$ is finite \\'etale of degree prime\nto $p$, $H^1(A_D,S_0)$ consists in loop torsors as defined in \\cite[\\S 2.3, lemma 2.3.(2)]{Gi2},  i.e. those arising from cocycles related to tame Galois covers of $A_D$. It follows that the $A_D$--torsor\n$Q_I$ is a loop $\\text{\\rm{Aut}}_I(G)$--torsor \\cite[lemma 2.3.(3)]{Gi2},\nso that $\\mathfrak{G} \\times_{\\goth X \\setminus D} A_D$ is \nby definition a loop reductive group scheme.\n\nLet $F_{P,v}$ be the completion of the field $F_P$ for the valuation associated to the blow-up of $\\mathop{\\rm Spec}\\nolimits(A)$ at its closed point.\nOur assumption states in particular that $Z(F_{P,v}) \\not = \\emptyset$,\nthat is, $G_{F_{P,v}}$ admits a parabolic subgroup of type $I$.\nAccording to \\cite[th. 4.1 , (iii) $\\Longrightarrow$ (i)]{Gi2}, $\\mathfrak{G} \\times_{\\goth X \\setminus D} A_D$\nadmits a parabolic subgroup of type $I$. \nA fortiori $G_{F_P}$ admits a parabolic subgroup of type \n$I$ so that $Z(F_P) \\not = \\emptyset$.\n \\end{proof}\n\n\n\\begin{sremarks}{\\rm \n(a) If $p=0$, the result used \\cite[th. 4.1]{Gi2}\nadmits a simple proof, see \\cite[Ex. 4.2]{Gi2},\nby using the analogous result over Laurent polynomials\n\\cite[th. 7.1]{GP2}.\n\n\\smallskip\n\n\\noindent (b) In nice cases inspection of the proof permits \nto weaken the assumption on $p$. \nThe precise condition is that the $\\text{\\rm{Aut}}(G_0)$--torsor\n$\\mathrm{Isom}(G_0,G)$ admits a reduction to a finite\n$F$--subgroup whose degree is prime to $p$.\nFor example in type $G_2$, we need to assume only \nthat $p$ is prime to $2$.\n}\n\\end{sremarks}\n\n \n Together with Theorem \\ref{thm_main_hhk}.(1), we obtain the following consequence.\n \n \n \\begin{stheorem} \\label{thm_patching} Let $G$ be a reductive\n $F$--group and assume that $p$ does not divide the order of the automorphism group of the  absolute root system of $G_{ad}$.\n Let $Z$ be a twisted flag $F$--variety of $G$.\n Then $Z(F) \\not = \\emptyset$ if and only if $Z(F_v) \\not = \\emptyset$\n for all discrete valuations of $F$ arising from models of $X$.\n \\end{stheorem}\n\n\n \\section{Appendix: characterization of parabolic subgroups.} \\label{app_parabolic}\n Let $G$ be a reductive $F$--group \nover a field $F$. We remind the reader \nthat an algebraic $F$--subgroup $P$ of $G$ is parabolic\nif $P$ is smooth and  $G\/P$  is a projective $F$--variety.\nThe interest of the  probably known statement below\nis only in positive characteristic since in this case\nthere  exist $F$--subgroups $Q$ which are not smooth such that \n $G\/Q$ is a projective $F$--variety \\cite{W}. \n  \n\\begin{sproposition}\\label{prop_parabolic} \nLet $P$ be a $k$--subgroup of $G$\nsuch that the quotient variety $G\/P$ is projective.\nThen the following assertions are equivalent: \n\n\\smallskip\n\n(i) $P$ is an $F$--parabolic subgroup;\n\n\n\\smallskip\n\n(ii) For each $F$--field $E$, $G(E)$ acts transitively\non $(G\/P)(E)$;\n\n\\smallskip\n\n(iii) The quotient map $G \\to G\/P$ admits a rational section;\n \n\\smallskip\n\n(iv) $P$ is smooth connected;\n\n\n\\smallskip\n\n(v) $P$ is smooth.\n \n \n\\end{sproposition} \n \n \\begin{proof}\n\\noindent  $(i) \\Longrightarrow (ii)$.\nSince $(G\/P)(E)$ parameterizes the $E$--parabolic\nsubgroups of $G_E$ of same type that $P$,\n Borel-Tits' conjugacy theorem \\cite[th. 4.13.c]{BoT65}\n shows that $G(E)$ acts transitively on $(G\/P)(E)$. \n\n\n\\smallskip\n\n\\noindent  $(ii) \\Longrightarrow (iii)$.\nOur assumption rephrases by saying that\nthe map $G(E) \\to (G\/P)(E)$ is onto for each $F$--field $E$.\nApplying that to the function field $E=F(G\/P)$\nof the smooth connected $F$- variety $G\/P$\nprovides a rational section of the map $G \\to G\/P$.\n \n \n\\smallskip\n\n\\noindent  $(iii) \\Longrightarrow (iv)$.\nTo show the smoothness of $P$ we can assume\nthat $F$ is algebraically closed.\nThen the neutral component  $Q=(P_{red})^0$\nof the reduced $F$--subgroup $P_{red}$ of $P$ is smooth. Furthermore \nthe quotient $F$--variety $P\/Q$ is finite. It follows that the\nmorphism $q: G\/Q \\to G\/P$ is finite and  a fortiori projective\n\\cite[Tag 0B3I]{St}. Since the composition of projective morphisms (of qcqs schemes) is projective  \\cite[Tag 0C4P]{St}, it follows that \n$G\/Q$ is projective. The $F$--subgroup $Q$ of $G$\nis then parabolic. Our assumption is that the morphism\n$G \\to G\/P$ has a rational section\n and so \nhas a fortiori the finite morphism $q: G\/Q \\to G\/P$.\nAccording to \\cite[cor. 6.1.15]{EGA2}, $q$ is  an isomorphism. Thus $Q=P$ and we conclude that $P$ is smooth connected.\n \n\n\n\\smallskip\n\n\\noindent  $(iv) \\Longrightarrow (v)$. Obvious.\n\n\\smallskip\n\n\\noindent $(v) \\Longrightarrow (i)$.\nThis is by definition.\n\\end{proof}\n\n A variant is the following.\n\n\\begin{sproposition}\\label{prop_homog} \nLet $X$ be a smooth  projective $G$--variety.\nThen the following statements are equivalent: \n\n\\smallskip\n\n(i) $X$ is a $F$--variety of parabolic subgroups of $G$;\n\n\n\\smallskip\n\n(ii) For each $F$--field $E$, $G(E)$ acts transitively\non $X(E)$.\n\n\\end{sproposition} \n  \n \n\\begin{proof}\nThe implication $(i) \\Longrightarrow (ii)$ is again\nBorel-Tits' conjugacy theorem.\nWe assume (ii). According to \\cite[prop. 1.3]{MPW},\nwe can assume that $F$ is separably closed.\nSince $X$ is smooth, we have $X(F) \\not = \\emptyset$ \nand denote by  $P$ the stabilizer of some $F$--point $x$.\nAccording to \\cite[prop. III.3.2.1]{DG}.\ncondition (ii) implies that the orbit map $G \\to X$, $g \\mapsto g.x$\ninduces an isomorphism $f_x: G\/P \\buildrel\\sim\\over\\lgr X$.\nProposition \\ref{prop_parabolic},  \n$(ii) \\Longrightarrow (i)$, shows that \n$P$ is a $F$--parabolic subgroup.\nThus $X$ is a $F$--variety of parabolic subgroups of $G$;\n\\end{proof} \n \n \n \\begin{sremark} {\\rm\n The condition (ii) is called\n{\\it transitive action of $G$ on $X$}\nby Harbater-Hartmann-Krashen.\nIt   occurs in \\cite[th. 3.7]{HHK}. \nProjective homogeneous varieties in the result quoted above are exactly\nthe various varieties of parabolic subgroups.\n }\n \\end{sremark} \n \n \n \\section{Appendix: extending torsors} \\label{app_extend}\n \n We come back to a  purity result of Colliot-Th\\'el\\`ene and Sansuc.\n \n \\begin{stheorem}\\label{thm_extend}\n Let $X$ be a regular scheme of dimension $2$.\n Let $U$ be an open subcheme of $X$ which contains  $X^{(1)}$. \n Let $G$ be an affine $X$--group scheme. In the following  cases\n \n \\smallskip\n \n (i) $G$ is reductive,\n \n \\smallskip\n \n (ii) There exists a  Zariski cover $(U_i)_{i \\in I}$ of $X$ \n such that each $G_{U_i}$ admits a closed embedding $G_{U_i} \\subset \\GL_{n_i}$ such that \n $\\GL_{n_i}\/G_{U_i}$ is representable by an affine $U_i$--scheme,\n \n  \\smallskip\n  \nthen we have the equality $H^1_{\\text{\\rm fppf}}(X,G) \\buildrel\\sim\\over\\lgr H^1_{\\text{\\rm fppf}}(U,G)$.\n\n \\end{stheorem}\n\n\n \\begin{proof}\n  The case (i) is \\cite[th. 6.13]{CTS}. The case (ii) goes by inspection of \n  the proof. \n \\end{proof}\n \n \n \\bigskip\n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nThis article contains the completion of the project started in a previous paper \\cite{BBT1}, \nwhere we proved that there exists an ergodic conductance environment such that the \nweak (annealed) invariance principle holds for the corresponding  \ncontinuous time random walk along a subsequence but the quenched invariance principle does not hold.\nIn the present paper we give a proof of the full scaling limit for the weak invariance principle, improving the\nresult in \\cite{BBT1}.\nThe improved result is, in a sense, a quantitative form of the invariance \nprinciple. The proof consists of several lemmas. Some of them are specific to our \nmodel but some of them have the more general character and may serve as \ntechnical elements for related projects. Since this paper is a continuation of \n\\cite{BBT1}, we start by presenting basic notation and definitions from that paper.\n\nLet $d\\geq 2$ and let $ E_d$ be the set of all non oriented edges in the \n$d$-dimensional integer lattice, that is,  $E_d = \\{e = \\{x,y\\}: x,y \\in {\\mathbb Z}^d, |x-y|=1\\}$.\nLet $\\{\\mu_e\\}_{e\\in E_d}$ be a random process with non-negative values, \ndefined on some probability space $(\\Omega, \\mathcal{F}, \\Pp)$. \nThe process $\\{\\mu_e\\}_{e\\in E_d}$ represents random conductances.  \nWe write $\\mu_{xy}  = \\mu_{yx} = \\mu_{\\{x,y\\}}$ and set \n$\\mu_{xy}=0$ if $\\{x,y\\} \\notin E_d$. Set\n\\begin{align*}\n\\mu_x = \\sum_y \\mu_{xy}, \\qquad P(x,y) = \\frac{\\mu_{xy}}{\\mu_x},\n\\end{align*}\nwith the convention that $0\/0=0$ and $P(x,y)=0$ if $\\{x,y\\} \\notin E_d$. \nFor a fixed $\\omega\\in \\Omega$, let  \n$X = \\{X_t, t\\geq 0, {P}^x_\\omega, x \\in {\\mathbb Z}^d\\}$ be the \ncontinuous time random walk on ${\\mathbb Z}^d$, with transition probabilities \n$P(x,y) = P_\\omega(x,y)$, and exponential waiting times with mean $1\/\\mu_x$. \nThe corresponding expectation will be denoted ${E}_\\omega^x$. \nFor a fixed $\\omega\\in \\Omega$, the generator $\\mathcal{L}$ of $X$ is given by\n\\begin{align}\\label{e:Ldef}\n\\mathcal{L} f(x) = \\sum_y \\mu_{xy} (f(y) - f(x)).\n\\end{align}\nIn \\cite{BD} this is called the {\\em variable speed random walk} (VSRW)\namong the conductances $\\mu_e$.\nThis model, of a reversible (or symmetric) random walk in a random environment, is\noften called the Random Conductance Model. \n\nWe are interested in functional Central Limit Theorems (FCLTs)\nfor the process $X$. Given any process $X$,\nfor $\\varepsilon>0$, set $X^{\\varepsilon}_t = \\varepsilon X_{t \/\\varepsilon^2}$, $t\\geq 0$. \nLet $\\mathcal{D}_T = D([0,T], \\mathbb{R}^d)$ denote the Skorokhod space, \nand let $\\mathcal{D}_\\infty=D([0,\\infty), \\mathbb{R}^d)$.\nWrite $d_S$ for the Skorokhod metric and $\\mathcal{B}(\\mathcal{D}_T)$ for the $\\sigma$-field of \nBorel sets in the corresponding topology. \nLet $X$ be the canonical process on $\\mathcal{D}_\\infty$ or $\\mathcal{D}_T$, $P_{\\text{BM}}$ be Wiener \nmeasure on $(\\mathcal{D}_\\infty, \\mathcal{B}(\\mathcal{D}_\\infty))$  and let $E_{\\text{BM}}$ be the \ncorresponding expectation. \nWe will write $W$ for a standard Brownian motion.\nIt will be convenient to assume that $\\{\\mu_e\\}_{e\\in E_d}$ are \ndefined on a probability space $(\\Omega, \\mathcal{F}, \\bP)$, and that\n$X$ is defined on $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_\\infty, \\mathcal{B}(\\mathcal{D}_\\infty))$ \nor  $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$. \nWe also define the averaged or annealed measure ${\\bf P}$ on \n$(\\mathcal{D}_\\infty, \\mathcal{B}(\\mathcal{D}_\\infty))$ or  $(\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$ by\n\\begin{equation} \\label{e:bfPdef}\n {\\bf P}(G) = \\bE P^0_{\\omega}(G). \n\\end{equation}\n\n\\begin{definition}\\label{j1.2}\nFor a bounded function $F$ on $\\mathcal{D}_T$ and a constant matrix $\\Sigma$, let \n$\\Psi^F_\\varepsilon = {E}^0_\\omega F(X^{\\varepsilon})$ and \n$\\Psi^F_\\Sigma = E_{\\text{BM}} F(\\Sigma W)$. We will use $I$ to denote the identity matrix.\n\n\\smallskip \\noindent (i) We say that the {\\em Quenched Functional CLT} (QFCLT) holds \nfor $X$ with limit $\\Sigma W$ if for every $T>0$ and \nevery bounded continuous function $F$ on $\\mathcal{D}_T$ we \nhave $\\Psi^F_\\varepsilon \\to \\Psi^F_\\Sigma$ as $\\varepsilon\\to 0$, with $\\Pp$-probability 1.\\\\\n(ii) We say that the {\\em Weak Functional CLT} (WFCLT) \nholds for $X$ with limit $\\Sigma W$ if for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$\\Psi^F_\\varepsilon \\to \\Psi^F_\\Sigma$ as $\\varepsilon\\to 0$, in $\\Pp$-probability.\\\\\n(iii) We say that the {\\em Averaged (or Annealed) Functional CLT}\n(AFCLT) holds for $X$ with limit $\\Sigma W$ if for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$ \\bE \\Psi^F_\\varepsilon \\to \\Psi_{\\Sigma}^F$.\nThis is the same as standard weak convergence with respect to the probability measure ${\\bf P}$. \n\\end{definition}\n\nIf we take $\\Sigma$ to be non-random then, since $F$ is bounded, it is\nimmediate that QFCLT $\\Rightarrow$ WFCLT. In general for the QFCLT the matrix\n$\\Sigma$ might depend on the environment $\\mu_\\cdot({\\omega})$. However, if\nthe environment is stationary and ergodic, then $\\Sigma$ is a shift invariant\nfunction of the environment, so must be $\\bP$--a.s. constant.\nIn \\cite{DFGW} it is proved that if $\\mu_e$ is a stationary ergodic \nenvironment with $\\bE \\mu_e<\\infty$ then the WFCLT holds. In \\cite[Theorem 1.3]{BBT1} \nit is proved that for the random conductance model the AFCLT and WFCLT are equivalent.\n\n\\begin{definition}\nWe say an environment $(\\mu_e)$ on ${\\mathbb Z}^d$ is {\\em symmetric} if the law of  $(\\mu_e)$ is \ninvariant under symmetries of ${\\mathbb Z}^d$. \n\\end{definition}\n\nIf $(\\mu_e)$ is stationary, ergodic and symmetric, and the WFCLT holds with\nlimit $\\Sigma W$ then the limiting covariance matrix $\\Sigma^T \\Sigma$ must also\nbe invariant under symmetries of ${\\mathbb Z}^d$, so must be a constant \ntimes the identity.\n\nIn a previous paper \\cite{BBT1} we proved the following theorem:\n\n\\begin{theorem}\\label{T:oldmain}\nLet $d=2$ and $p<1$.\nThere exists a symmetric stationary ergodic environment $\\{\\mu_e\\}_{e\\in E_2}$\nwith $\\bE (\\mu_e^p \\vee \\mu_e^{-p})<\\infty$ \nand a sequence $\\varepsilon_n \\to 0$ such that\\\\\n(a)  the WFCLT holds for $X^{\\varepsilon_n}$ with limit $W$, \ni.e., for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$\\Psi^F_{\\varepsilon_n} \\to \\Psi^F_I$ as $n\\to \\infty$, in $\\Pp$-probability,\n\\\\\nbut \\\\\n(b) the QFCLT does not hold for  $X^{\\varepsilon_n}$ with limit $ \\Sigma W$ for any $\\Sigma$.  \n\\end{theorem}\n\nIn this paper we prove that for an environment similar to\nthat in Theorem \\ref{T:oldmain} the WFCLT holds for $X^{\\varepsilon}$ as $\\varepsilon \\to 0$,\nand not just along a subsequence.\n\n\\begin{theorem}\\label{T:main}\nLet $d=2$ and $p<1$.\nThere exists a symmetric stationary ergodic environment $\\{\\mu_e\\}_{e\\in E_2}$\nwith $\\bE (\\mu_e^p \\vee \\mu_e^{-p})<\\infty$ \nsuch that\\\\\n(a)  the WFCLT holds for $X^{\\varepsilon}$ with limit $W$, \ni.e., for every $T>0$ and every \nbounded continuous function $F$ on $\\mathcal{D}_T$ we have \n$\\Psi^F_{\\varepsilon} \\to \\Psi^F_I$ as $\\varepsilon \\to 0$, in $\\Pp$-probability,\n\\\\\nbut \\\\\n(b) the QFCLT does not hold for  $X^{\\varepsilon}$ with limit $ \\Sigma W$ for any $\\Sigma$.  \n\\end{theorem}\n\nFor more remarks on this problem see \\cite{BBT1}.\n\n\\smallskip \\noindent {\\bf Acknowledgment.}\nWe are grateful to Emmanuel Rio, Pierre Mathieu, Jean-Dominique Deuschel \nand Marek Biskup for some very useful discussions.\n\n\\section{Description of the environment}\\label{const} \n\nHere we recall the environment given in \\cite{BBT1}. We refer the reader to that\npaper for proofs of some basic properties.\n\nLet $\\Omega = (0,\\infty)^{E_2}$, and $\\mathcal{F}$ be the Borel $\\sigma$-algebra defined \nusing the usual product topology. Then every $t\\in{\\mathbb Z}^2$ defines a transformation \n$T_t (\\omega)=\\omega +t$ of $\\Omega$. Stationarity and ergodicity of the measures \ndefined below will be understood with respect to these transformations. \n\nAll constants (often denoted $c_1, c_2$, etc.) are assumed to be strictly positive and finite.\nFor a set $A \\subset {\\mathbb Z}^2$ let $E(A)\\subset E_2$ be the set of all edges with both endpoints in\n$A$. Let $E_h(A)$ and $E_v(A)$ respectively\nbe the set of horizontal and vertical edges in $E(A)$.\nWrite $x \\sim y$ if $\\{x,y\\}$ is an edge in ${\\mathbb Z}^2$. Define the exterior boundary of $A$ by\n$$ {\\partial} A =\\{ y \\in {\\mathbb Z}^2 -A: y \\sim x \\text{ for some } x \\in A \\}. $$\nLet also\n$$ {\\partial}_i A = {\\partial}({\\mathbb Z}^2 -A). $$ \nDefine balls in the $\\ell^\\infty$ norm by $\\mathcal{B}(x,r)= \\{y: ||x-y||_\\infty \\le r\\}$; of \ncourse this is just the square with center $x$ and side $2r$.\n\nLet $\\{a_n\\}_{n\\geq 0}$,  $\\{ \\beta_n\\}_{n \\ge 1}$ and $\\{b_n\\}_{n\\geq 1}$ be  \nstrictly increasing sequences of positive integers growing to infinity with $n$,\nwith \n$$ 1=a_0 < b_1 < \\beta_1 < a_1 \\ll b_2 <  \\beta_2<   a_2 \\ll b_3 \\dots $$\nWe will impose a number of conditions on these sequences in the course\nof the paper. We collect the main ones here.\nThere is some redundancy in the conditions, for easy reference.\n\n\\begin{enumerate}[(i)]\n\\item  $a_n$ is even  for all $n$. \n\\item For each $n \\ge 1$, $a_{n-1}$ divides $b_n$, \nand $b_n$ divides $\\beta_n$ and $a_n$. \n\\item  $b_1 \\geq 10^{10}$.\n\\item  $a_n\/\\sqrt{2n} \\le     b_n \\le a_n \/  \\sqrt{n} $ for all $n$, and\n$b_n \\sim a_n\/\\sqrt{n}$.\n\\item $b_{n+1} \\ge 2^n b_n$ for all $n$.\n\\item $b_n > 40 a_{n-1}$ for all $n$.\n\\item $b_n$ is large enough so that the estimates (5.1) and (6.1) of \\cite{BBT1} hold.\n\\item $100 b_n  < \\beta_n \\le  b_n n^{1\/4}  < 2 \\beta_n <   a_n\/10$ for $n$ large enough.\n\\end{enumerate}\n\nIn addition, at various points in the proof, we will assume that $a_n$ is sufficiently much\nlarger than $b_{n-1}$ so that a process $X^{(n-1)}$ defined below is such that for $a\\ge a_n$\nthe rescaled process\n$$ (a^{-1} X^{(n-1)}_{a^2 t}, t\\ge 0)$$\nis sufficiently close to Brownian motion.\nWe will mark the places in the proof where we impose these extra conditions by ($\\clubsuit$) .\n\n\n\\smallskip\\noindent\nWe begin our construction by defining a collection of squares in ${\\mathbb Z}^2$. Let\n\\begin{align*} \nB_n &= [0, a_n]^2,  \\\\\nB_n' &= [0, a_n-1]^2 \\cap {\\mathbb Z}^2 ,\\\\\n\\mathcal{S}_n(x) &= \\{ x + a_n y + B_n': \\, y \\in {\\mathbb Z}^2 \\}.\n\\end{align*} \nThus $\\mathcal{S}_n(x)$ gives a tiling of ${\\mathbb Z}^2$ by disjoint squares of side $a_n-1$\nand period $a_n$.\nWe say that the tiling $\\mathcal{S}_{n-1}(x_{n-1})$ is a refinement\nof $\\mathcal{S}_n(x_n)$ if every square $Q \\in \\mathcal{S}_n(x_n)$ is a finite\nunion of squares in $\\mathcal{S}_{n-1}(x_{n-1})$. It is clear that \n$\\mathcal{S}_{n-1}(x_{n-1})$ is a refinement of $\\mathcal{S}_n(x_n)$ if\nand only if $x_n = x_{n-1}+ a_{n-1}y$ for some $y \\in {\\mathbb Z}^2$.\n\nTake $\\mathcal{O}_1$ uniform in $B'_1$, and for $n\\geq 2$\ntake $\\mathcal{O}_n$, conditional on $(\\mathcal{O}_1, \\dots, \\mathcal{O}_{n-1})$, \nto be uniform in $B'_n \\cap ( \\mathcal{O}_{n-1} + a_{n-1}{\\mathbb Z}^2)$. We now define random tilings by letting\n\\begin{equation*}\n \\mathcal{S}_n = \\mathcal{S}_n( \\mathcal{O}_n), \\, n \\ge 1.  \n\\end{equation*}\n\nLet $\\eta_n$, $K_n$ be positive constants; we will have $\\eta_n \\ll 1 \\ll K_n$.\nWe define conductances  on $E_2$ as follows. \nRecall that $a_n$ is even, and let $a_n' = \\frac12 a_n$. Let\n$$ C_n = \\{ (x,y) \\in B_n \\cap {\\mathbb Z}^2: y \\ge x, x+y \\le a_n \\}. $$\nWe first define conductances $\\nu^{n,0}_e$ for $e \\in E(C_n)$. Let\n\\begin{align*}\nD_n^{00} &= \\big\\{ (a'_n - \\beta_n,y), a'_n - 10 b_n \\le y \\le a'_n + 10 b_n \\big\\},  \\\\\nD_n^{01} &= \\big\\{ (x, a'_n + 10 b_n),  (x, a'_n + 10 b_n + 1), (x, a'_n - 10 b_n),  (x, a'_n - 10 b_n -1),   \\\\\n\\nonumber   \n   & \\quad \\quad \\quad a'_n -\\beta_n -b_n \\le x \\le a'_n -\\beta_n + b_n \\big\\}.\n\\end{align*}\nThus the set $D^{00}_n \\cup D_n^{01}$ resembles the letter I (see Fig.~\\ref{fig1}).\n\nFor an edge $e \\in E(C_n)$ we set \n\\begin{align*} \n \\nu^{n,0}_{e}  &= \\eta_n \\quad \\text {if } e \\in E_v(D^{01}_n), \\\\\n \\nu^{n,0}_{e}  &= K_n \\quad \\text {if } e \\in E(D^{00}_n), \\\\\n  \\nu^{n,0}_{e}  &= 1 \\quad \\text {otherwise.} \n\\end{align*} \n\n\\begin{figure} \\includegraphics[width=4cm]{fig1_1}\n\\caption{The set $D^{00}_n \\cup D_n^{01}$ resembles the letter I.\nBlue edges have very low conductance. The red line represents edges with very \nhigh conductance. Drawing not to scale. \n}\n\\label{fig1}\n\\end{figure}\n\nWe then extend $\\nu^{n,0}$ by symmetry to $E(B_n)$.\nMore precisely,\nfor $z =(x,y) \\in B_n$, let $R_1 z=( y,x)$ and  $R_2z = (a_n-y,a_n-x)$, so that\n$R_1$ and $R_2$ are reflections in the lines $y=x$ and $x+y=a_n$.\nWe define $R_i$ on edges by $R_i (\\{x,y\\}) = \\{R_i x, R_i y \\}$ for $x,y \\in B_n$. \nWe then extend $\\nu^{0,n}$ to $E( B_n)$ so that\n$\\nu^{0,n}_e =  \\nu^{0,n}_{R_1 e }=\\nu^{0,n}_{R_2 e }$ for  $e \\in E(B_n)$.\nWe define the {\\em obstacle} set $D_n^0$ by setting\n$$ D_n^0 = \\bigcup_{i=0}^1 \\big( D_n^{0,i} \\cup R_1(D_n^{0,i})  \\cup  R_2(D_n^{0,i})\n \\cup R_1R_2 (D_n^{0,i} ) \\big). $$\nNote that $\\nu^{n,0}_e=1$ for every edge adjacent to the boundary of $B_n$,\nor indeed within a distance $ a_n\/4$ of this boundary.\nIf $e=(x,y)$, we will write $e-z = (x-z,y-z)$. \nNext we extend $\\nu^{n,0}$ to $E_2$ by periodicity, i.e.,\n$\\nu^{n,0}_e = \\nu^{n,0}_{e+ a_n x}$ for all $x\\in {\\mathbb Z}^2$.\nWe define the conductances $\\nu^n$ by translation by $\\mathcal{O}_n$, so that\n\\begin{equation*}\n \\nu^n_e =\\nu^{n,0}_{e-\\mathcal{O}_n}, \\, e \\in E_2.\n\\end{equation*}\nWe also define the obstacle set at scale $n$ by\n\\begin{equation}\\label{ma26.1}\n D_n = \\bigcup_{ x \\in {\\mathbb Z}^2} (a_n x + \\mathcal{O}_n + D^0_n ).\n\\end{equation}\nWe will sometimes call the set $D_n$ the set of $n$th level obstacles.\n\n\nWe define the environment $\\mu^n_e$ inductively by\n\\begin{align*}\n \\mu^n_e &= \\nu^{n}_e \\quad \\text{ if } \\nu^n_e \\neq 1, \\\\\n \\mu^n_e &= \\mu^{n-1}_e \\quad \\text{ if } \\nu^n_e=1.\n\\end{align*}\nOnce we have proved the limit exists, we will set\n\\begin{equation} \\label{e:mudef}\n \\mu_e = \\lim_n \\mu^n_e.\n\\end{equation}\n\n\n\n\\begin{lemma} \\label{L:erg} (See  \\cite[Theorem 3.1]{BBT1}).\\\\\n(a) The environments $(\\nu^n_e, e\\in E_2)$, $(\\mu^n_e, e\\in E_2)$\nare stationary, symmetric and ergodic.\\\\\n(b) The limit \\eqref{e:mudef} exists $\\bP$--a.s. \\\\\n(c) The environment $(\\mu_e, e \\in E_2)$ is  stationary, symmetric and ergodic.\n\\end{lemma}\n\n\n\nNow let \n\\begin{align}\\label{j27.4}\n\\mathcal{L}_n f(x) = \\sum_{y} \\mu^n_{xy} (f(y)-f(x)), \n\\end{align}\nand $X^{(n)}$ be the associated Markov process. Set\n\\begin{equation} \\label{e:etadef}\n  \\eta_n =   b_n^{-(1+1\/n)}, \\, n \\ge 1.\n\\end{equation}\nFrom Section 4 of \\cite{BBT1} we have:\n\n\\begin{theorem} \\label{T:eK}\nFor each $n$ there exists a constant $K_n$, depending on $\\eta_1, K_1, \\dots \\eta_{n-1}, K_{n-1}$,\nsuch that the QFCLT holds for $X^{(n)}$ with limit $W$.\n\\end{theorem}\n\nFor each $n$ the process $X^{(n)}$ has invariant measure which is counting measure \non ${\\mathbb Z}^2$. For $x \\in \\mathbb{R}^2$ and $a>0$ write $[xa]$ for the point in ${\\mathbb Z}^2$ closest to $xa$.\n(We use some procedure to break ties.) We have the following bounds on the transition\nprobabilities of $X^{(n)}$ from \\cite{BZ}. We remark that the constant $M_n$ below is\nnot effective -- i.e. the proof does not give any control on its value. \nWrite $k_t(x,y) = (2\\pi t)^{-1} \\exp( -|x-y|^2\/2t)$ for the transition density of Brownian motion\nin $\\mathbb{R}^2$, and\n$$ p^{{\\omega},n}_t(x,y) = P^x_{\\omega}( X^{(n)}_t =y )$$\nfor the transition probabilities for $X^{(n)}$.\n\n\\begin{lemma} \\label{L:hkXn} \nFor each $0< \\delta < T$  there exists $M_n=M_n(\\delta,T)$ such that for $a \\ge M_n$ \n\\begin{equation} \\label{e:GB1}\n\\frac12 k_t(x,y) \\le a^{2} p^{{\\omega},n}_{a^2t}([xa],[ya])  \\le 2 k_t(x,y) \\, \\hbox { for all }\n \\delta \\le t \\le T, |x|, |y| \\le T^2.\n\\end{equation}\n\\end{lemma} \n\n\n\n\n\\section{Preliminary results}\n\n\nSince a proof of Theorem \\ref{T:oldmain}(b) was given in \\cite{BBT1}, \nall we need to prove is part (a) of Theorem \\ref{T:main}.\nThe argument consists of several lemmas. We start with some preliminary \nresults on weak convergence of probability measures on the space of c\\`adl\\`ag functions. \nRecall the definitions of the measures $\\bP$ and $P^0_{\\omega}$.\n\nRecall that  $\\mathcal{D} := \\mathcal{D}_1 = D([0,1], \\mathbb{R}^2)$ denotes the space of c\\`adl\\`ag functions \nequipped with the Skorokhod metric ${\\rm d_S}$ defined as follows (see \\cite[p.~111]{B}). \nLet $\\Lambda$ be the family of continuous strictly increasing functions $\\lambda$ \nmapping $[0,1]$ onto itself. In particular, $\\lambda(0) =0$ and $\\lambda(1) =1$. \nIf $x(t), y(t) \\in  \\mathcal{D}$ then  \n\\begin{align*}\n{\\rm d_S}(x,y) = \\inf_{\\lambda \\in \\Lambda}\n\\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|, \\sup_{t\\in[0,1]} |y(\\lambda(t)) - x(t)| \\Big).\n\\end{align*}\nFor $x(t) \\in  \\mathcal{D}$, let $\\Osc(x, \\delta) = \\sup\\{|x(t)-x(s)|: s,t\\in[0,1], |s-t|\\le \\delta\\}$.\n\n\\begin{lemma}\\label{d21.2}\nSuppose that $\\sigma: [0,1] \\to [0,1]$ is continuous, non-decreasing and $\\sigma(0) = 0$ \n(we do not require that $\\sigma(1) = 1$).  \nSuppose that $|\\sigma(t) - t| \\le \\delta$ for all $t\\in[0,1]$.\nLet  $\\varepsilon\\geq0$, $\\delta_1>0$, $x, y \\in  \\mathcal{D}$ with\n${\\rm d_S}(x(\\,\\cdot\\,), y(\\,\\cdot\\,))\\le \\varepsilon$, and\n$\\Osc(x, \\delta) \\vee \\Osc(y, \\delta) \\le \\delta_1$. Then\n${\\rm d_S}(x(\\sigma(\\,\\cdot\\,)), y(\\sigma(\\,\\cdot\\,))) \\le \\varepsilon + 2\\delta_1$.\n\\end{lemma}\n\n\\begin{proof}\nFor any $\\varepsilon_1> \\varepsilon$ there exists $\\lambda\\in \\Lambda$ such that,\n\\begin{align*}\n\\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |y(\\lambda(t)) - x(t)| \\Big)\\le\\varepsilon_1.\n\\end{align*}\nWe have for $\\lambda$ satisfying the above condition,\n\\begin{align*}\n&\\sup_{t\\in[0,1]} |y(\\sigma(\\lambda(t))) - x(\\sigma(t))|\\\\\n&\\qquad \\le \n\\sup_{t\\in[0,1]} (|y(\\sigma(\\lambda(t))) - y(\\lambda(t))|\n+ |y(\\lambda(t)) - x(t)| + |x(t) - x(\\sigma(t))|) \\\\\n&\\qquad \\le \\Osc(y,\\delta) + \\varepsilon_1 + \\Osc(x,\\delta) \\le \\varepsilon_1 + 2 \\delta_1. \n\\end{align*}\nHence,\n\\begin{align*}\n\\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |y(\\sigma(\\lambda(t))) - x(\\sigma(t))|\n\\Big) \\le \\varepsilon_1 + 2 \\delta_1.\n\\end{align*}\nTaking infimum over all $\\varepsilon_1 > \\varepsilon$ we obtain\n${\\rm d_S}(x(\\sigma(\\,\\cdot\\,)), y(\\sigma(\\,\\cdot\\,)))\n\\le \\varepsilon + 2\\delta_1$.\n\\end{proof}\n\nLet ${d_P}$ denote the Prokhorov distance between probability measures on a probability space defined \nas follows (see \\cite[p.~238]{B}). \nRecall that \n$\\Omega = (0,\\infty)^{E_2}$ and $\\mathcal{F}$ is the Borel $\\sigma$-algebra defined \nusing the usual product topology.\nWe will use measurable spaces $(\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$ and \n $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$, for a fixed $T$ (often $T=1$).\nNote that $\\mathcal{D}_T$ and $\\Omega \\times \\mathcal{D}_T$ are metrizable, with the metrics generating the usual topologies. A ball around a set $A$ with radius $\\varepsilon$ will\nbe denoted $\\mathcal{B}(A,\\varepsilon)$ in either space. \nFor probability measures $P$ and $Q$, \n${d_P}(P,Q) $ is the infimum of $\\varepsilon>0$ such that $P(A) \\le Q(\\mathcal{B}(A,\\varepsilon)) + \\varepsilon$ and \n$Q(A) \\le P(\\mathcal{B}(A,\\varepsilon)) + \\varepsilon$ for all Borel sets $A$.\nConvergence in the metric ${d_P}$ is equivalent to the weak convergence of measures.\nBy abuse of notation we will sometimes write arguments of the function \n${d_P}(\\,\\cdot\\,,\\,\\cdot\\,)$ as processes rather than their distributions:  for example we will write\n${d_P}( \\{(1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}})$.\nWe will use ${d_P}$ for the Prokhorov distance\nbetween probability measures on $(\\Omega, \\mathcal{F}) \\times (\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$. We will write ${d_P}_\\omega$ for the metric on the space\n$(\\mathcal{D}_T, \\mathcal{B}(\\mathcal{D}_T))$.\nIt is straightforward to verify that if,  for some processes $Y$ and $Z$, \n${d_P}_\\omega(Y,Z) \\le \\varepsilon$ for $\\bP$--a.a. $\\omega$, then ${d_P}(Y,Z) \\le \\varepsilon$.\n\nWe will sometimes write $W(t)=W_t$ and similarly for other processes.\n\n\\begin{lemma}\\label{d21.1}\nThere exists a function $\\rho: (0,\\infty) \\to (0,\\infty)$ such that $\\lim_{\\delta\\downarrow 0}\n \\rho(\\delta) = 0$ and  the following holds.\nSuppose that $\\delta,\\delta'\\in (0,1)$ and $\\sigma: [0,1] \\to [0,1]$ is a non-decreasing \nstochastic process such that $t-\\sigma_t \\in [0,\\delta]$ for all $t$, with probability greater \nthan $1-\\delta'$. Suppose that $\\{W_t, t\\geq 0\\}$ has the distribution $P_{\\text{BM}}$ and \n$W^*_t = W(\\sigma_t)$ for $t\\in[0,1]$. \nThen ${d_P}(\\{W^*_t, t\\in[0,1]\\}, P_{\\text{BM}}) \\le \\rho(\\delta) + \\delta'$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $W, W^*$ and $\\sigma$ are defined on the sample space with a \nprobability measure $P$.\nIt is \neasy to see that we can choose $\\rho(\\delta)$ so that \n$\\lim_{\\delta\\downarrow 0} \\rho(\\delta) = 0$\nand $P(\\Osc(W,\\delta) \\geq\\rho(\\delta) )<\\rho(\\delta)$. \nSuppose that the event \n$F := \\{\\Osc(W,\\delta) <\\rho(\\delta)\\}\\cap \\{ \\forall t\\in[0,1]: t-\\sigma_t \\in [0,\\delta]\\}$ holds. \nThen taking $\\lambda(t) = t $,\n\\begin{align*}\n{\\rm d_S}(W, W^*) &\\le \\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |W(\\lambda(t)) - W^*(t)| \\Big) \\\\\n&= \\sup_{t\\in[0,1]} |W(t) - W(\\sigma(t))| \\le \\Osc(W, \\delta) < \\rho(\\delta).\n\\end{align*}\nWe see that if $F$ holds and $W \\in A \\subset \\mathcal{D}$ then \n$W^*(\\,\\cdot\\,)\\in \\mathcal{B}( A,\\rho(\\delta))$.\nSince $P(F^c) \\le \\rho(\\delta) + \\delta'$, we obtain\n\\begin{align*}\nP&(W \\in A) \\\\\n&\\leq P(\\{W\\in A\\} \\cap F) + P(F^c) \n\\leq P(\\{W^*\\in \\mathcal{B}( A,\\rho(\\delta))\\} \\cap F) \n+\\rho(\\delta) + \\delta'\\\\\n&\\leq P(W^*\\in \\mathcal{B}( A,\\rho(\\delta))) \n+\\rho(\\delta) + \\delta'.\n\\end{align*}\nSimilarly we have\n$P(W^*\\in A ) \\le P(W\\in \\mathcal{B}( A,\\rho(\\delta)) ) + \\rho(\\delta) + \\delta'$, and\nthe lemma follows.\n\\end{proof}\n\n\\begin{lemma}\\label{ma26.5}\nSuppose that for some processes $X, Y$ and $Z$ on the interval $[0,1]$ we have $Z= X+Y$ and $P(\\sup_{0\\leq t \\leq 1} |X_t| \\leq \\delta) \\geq 1-\\delta$. \nThen ${d_P}(\\{Z_t, t\\in[0,1]\\}, \\{Y_t, t\\in[0,1]\\}) \\le \\delta$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that the event \n$F := \\{\\sup_{0\\leq t \\leq 1} |X_t| \\leq \\delta\\}$ holds. \nThen taking $\\lambda(t) = t $,\n\\begin{align*}\n{\\rm d_S}(Z,Y) &\\le \\max\\Big( \\sup_{t\\in[0,1]} |\\lambda(t) - t|,\n\\sup_{t\\in[0,1]} |Z(\\lambda(t)) - Y(t)| \\Big) \\\\\n&= \\sup_{t\\in[0,1]} |Z(t) - Y(t)| \\le \\delta.\n\\end{align*}\nWe see that if $F$ holds and $Z \\in A \\subset \\mathcal{D}$ then \n$Y(\\,\\cdot\\,)\\in \\mathcal{B}( A,\\delta)$.\nSince $P(F^c) \\le \\delta$, we obtain\n\\begin{align*}\nP(Z \\in A) &\\leq P(\\{Z\\in A\\} \\cap F) + P(F^c) \n\\leq P(\\{Y\\in \\mathcal{B}( A,\\delta)\\} \\cap F) \n+ \\delta\\\\\n&\\leq P(Y\\in \\mathcal{B}( A,\\delta)) \n+ \\delta.\n\\end{align*}\nSimilarly we have\n$P(Y\\in A ) \\le P(Z\\in \\mathcal{B}( A,\\delta) ) +  \\delta$, and\nthe lemma follows.\n\\end{proof}\n\nRecall that the function $e\\to \\mu^n_e$ is periodic with period $a_n$.\nHence the random field $\\{\\mu^n_e\\}_{e\\in E_2}$ takes only finitely many values --\nthis is a much stronger statement than the fact that $\\mu^n_e$\ntakes only finitely many values.\n\n\nBy Theorem \\ref{T:eK} for each $n \\ge 1$,\n$$ \\lim_{a\\to \\infty} \n{d_P}( \\{  (1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) =0. $$ \nThus ($\\clubsuit$)  we can take $a_{n+1}$ so large that for every $\\omega$, $n \\ge 1$\nand $a\\geq a_{n+1}$, \n\\begin{equation} \\label{e:PdistBM}\n{d_P}_\\omega( \\{(1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n}.\n\\end{equation}\n\n\\bigskip\n\nLet $\\theta$ denote the usual shift operator for Markov processes, that is, $X^{(n)}_t \\circ \\theta_s = X^{(n)}_{t+s}$ for all $s,t\\geq 0$ (we can and do assume that $X^{(n)}$ is the canonical process on an appropriate probability space). \nRecall that\n$\\mathcal{B}(x,r) =\\{y: ||x-y||_\\infty \\le r\\}$ denote balls in the $\\ell^\\infty$ norm\nin ${\\mathbb Z}^2$ (i.e. squares), $a_n' = a_n\/2$, $B_n=[0,a_n]^2$ and $u_n =(a_n', a'_n)$. Note that $u_n$ is \nthe center of $B_n$.\nWe choose $ \\beta_n$  so that\n\\begin{align}\\label{j2.10}\n b_n n^{1\/8}\n< \\beta_n \\leq \\lfloor b_n n^{1\/4}\\rfloor  < 2 \\beta_n  < a_n\/10,\n\\end{align}\nand we assume that $n$ is large enough so that the above inequalities hold.\nLet $\\mathcal{C}_n =\\{ u_n + \\mathcal{O}_n+ a_n {\\mathbb Z}^2\\}$ be the set of centers of the squares in $\\mathcal{S}_n$, and let\n\\begin{equation} \\label{e:Hdef}\n  \\calK(r) = \\bigcup_{z \\in \\mathcal{C}_n} \\mathcal{B}(z,r).\n\\end{equation}\nNow let\n\\begin{align*}\n\\Gamma^1_n &= \\calK(2\\beta_n), \\\\\n\\Gamma^2_n &= {\\mathbb Z}^2 \\setminus \\calK(4 \\beta_n).\n\\end{align*}\nNow define stopping times as follows.\n\\begin{align*}\nS^n_0 &= T^n_0 = 0,\\\\\nU^n_k & = \\inf\\{t\\geq S^n_{k-1}: X^{(n)}_t \\in \\Gamma^2_n\\}, \\qquad k \\geq 1,\\\\\nS^n_k & = \\inf\\{t \\geq U^n_k: X^{(n)}_t \\in \\Gamma^1_n\\}, \\qquad k \\geq 1,\\\\\nV^n_1 & = \\inf \\Big\\{t \\in \\bigcup_{k\\geq 1} [U^n_k, S^n_k]: \nX^{(n)}_t \\in X^{(n)}(T^n_0) + a_{n-1} {\\mathbb Z}^2 \\Big\\}, \\\\\nT^n_k & = \\inf\\{t\\geq V^n_{k}: X^{(n)}_t \\in \\Gamma^1_n\\}, \\qquad k \\geq 1,\\\\\nV^n_k & = V^n_{1} \\circ \\theta_{T^n_{k-1}} , \\qquad k \\geq 2.\n\\end{align*}\nLet\n$$ J= \\bigcup_{k=1}^\\infty [V^n_k, T^n_k]; $$\nfor $t \\in J$  the process $X^{(n)}$ is a distance at least\n$\\beta_n$ away from any $n$th level obstacle.\nNow set for $t\\geq 0$,\n\\begin{align*}\n\\sigma^{n,1}_t &= \\int_0^t {\\bf 1}_J(s) ds = \\sum_{k=1}^\\infty \\left(T^n_{k} \\land t - V^n_{k} \\land t\\right),\\\\\n\\sigma^{n,2}_t &= t-\\sigma^{n,1}_t = \\sum_{k=0}^\\infty \\left(V^n_{k+1} \\land t - T^n_{k} \\land t\\right).\n\\end{align*}\nLet $\\widehat \\sigma^{n,j}$ denote the right continuous inverses of these processes, given by\n\\begin{equation*}\n\\widehat \\sigma^{n,j}_t = \\inf\\{s\\geq 0: \\sigma^{n,j}_s \\geq t\\}, \\, j=1,2.\n\\end{equation*}\nFinally let \n\\begin{align*}\nX^{n,1}_t &= X^{(n)}_0 + \\int_0^t {\\bf 1}_J(s) dX^{(n)}_s  \\\\\n &=  X^{(n)}_0 + \\sum_{k=0}^\\infty\n\\left(X^{(n)}(T^n_k \\land t) - X^{(n)}(V^n_{k} \\land t)\\right), \\\\\n\\widehat X^{n,1}_t &=  X^{(n)}_0 + X^{n,1}(\\widehat \\sigma^{n,1}_t), \\\\\nX^{n,2}_t &=  X^{(n)}_0 + \\int_0^t {\\bf 1}_{J^c}(s) dX^{(n)}_s  \\\\\n &=  X^{(n)}_0 + \\sum_{k=0}^\\infty\n\\left(X^{(n)}(V^n_{k+1} \\land t) - X^{(n)}(T^n_{k} \\land t)\\right), \\\\\n\\widehat X^{n,2}_t &=  X^{(n)}_0 + X^{n,2}(\\widehat \\sigma^{n,2}_t).\n\\end{align*}\n\nThe point of this construction is the following.\nFor every fixed $\\omega$, the function $e\\to\\mu_e^{n-1}$ is invariant under the shift by \n$x a_{n-1}$ for any $x\\in {\\mathbb Z}^2$, and \n$X^{(n)}(V^n_{k+1} ) = X^{(n)}(T^n_{k}) + x a_{n-1} $ for some $x\\in {\\mathbb Z}^2$. \nIt follows that for each $\\omega\\in \\Omega$, we have the following equality of  distributions: \n\\begin{equation} \\label{e:Xhatdsn}\n\\{\\widehat X^{n,1}_t, t\\geq 0\\} {\\buildrel (d) \\over {\\  =\\ }} \\{X^{(n-1)}_t, t\\geq 0\\}.\n\\end{equation}\nThe basic idea of the argument which follows is to write $X^{(n)}= X^{n,1} + X^{n.2}$.\nBy Theorem \\ref{T:eK}, or more precisely by \\eqref{e:PdistBM}, the process $X^{n,1}$ is close\nto Brownian motion, so to prove Theorem \\ref{T:main} we need to prove that $X^{n,2}$\nis small.\n\n\\bigskip\nWe state the next lemma at a level of generality greater than what we need in this article. A variant of our lemma is in the book \\cite{AF} but we could not find a statement that would match perfectly our needs.\nConsider a finite graph $G=(\\mathcal{V},E)$ and suppose that for any edge $\\overline{xy}$, $\\mu_{xy}$ is a non-negative real number. Assume that $\\sum_{y\\sim x} \\mu_{xy} >0$ for all $x$.\nFor $f: \\mathcal{V} \\to \\mathbb{R}$ set\n$$ \\sE (f,f) = \\sum_{\\{x,y\\} \\in E}  \\mu_{xy} (f(y)-f(x))^2. $$\nSuppose that $A_1, A_2 \\subset \\mathcal{V}$, $A_1 \\cap A_2 = \\emptyset$, and\nlet \n\\begin{align*}\n\\mathcal{H} &=\\{ f:\\mathcal{V} \\to \\mathbb{R} \\text{ such that } f(x)=0 \\text{ for } x\\in A_1, f(y)=1 \\text{  for } y \\in A_2\\},\\\\\n \\mathbf{r}^{-1} &= \\inf\\{ \\sE(f,f): f \\in \\mathcal{H} \\}. \n\\end{align*}\nThus $\\mathbf{r}$ is the effective resistance between $A_1$ and $A_2$.\nLet $Z$ be the continuous time Markov process on $\\mathcal{V}$ with the generator $\\mathcal{L}$  given by\n\\begin{align}\\label{ma21.1}\n\\mathcal{L} f(x) = \\sum_y \\mu_{xy} (f(y) - f(x)).\n\\end{align}\nLet $T_i = \\inf\\{t\\geq 0: Z_t \\in A_i\\}$  for $i=1,2$,\n and let $Z^{(i)}$ be $Z$ killed at time $T_i$.\n\n\\begin{lemma}\\label{L:com}\nThere exist probability measures $\\nu_1$ on $A_1$ and $\\nu_2$ on $A_2$ such that\n\\begin{align*}\nE^{\\nu_2} T_1 + E^{\\nu_1} T_2 = \\mathbf{r} |\\mathcal{V}| . \n\\end{align*}\nMoreover, for $i=1,2$, $\\nu_i$ is the capacitary measure of $A_i$ for the process $Z^{(3-i)}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $h_{12}(x) = P^x( T_1 < T_2)$. \nSet $D= \\mathcal{V}-A_1$ and recall that $Z^{(i)}$ is $Z$ killed at time $T_i$.\nLet $G_2$ be the Green operator for $Z^{(2)}$, and $g_2(x,y)$ be the density of\n$G_2$ with respect to counting measure, so that\n$$ E^x T_2 = \\sum_{y \\in \\mathcal{V}} g_2(x,y). $$\nNote that $g_2(x,y)=g_2(y,x)$.\nLet $e_{12}$ be the capacitary measure of $A_1$ for the process $Z^{(2)}$. Then\n$\\mathbf{r}^{-1} = \\sum_{z \\in A_1} e_{12}(z), $ and\n$$ h_{12}(x) =  \\sum_{z \\in A_1} e_{12}(z) g_2(z,x) . $$\nSo, if $ \\nu_1 = \\mathbf{r} e_{12} $, then\n\\begin{align*}\n\\sum_{y \\in \\mathcal{V}} h_{12}(y) &= \n   \\sum_{y \\in \\mathcal{V}}  \\sum_{x \\in A_1} e_{12}(x) g_2(x,y)  \\\\\n&= \\mathbf{r}^{-1} \\sum_{x \\in A_1} \\nu_1(x)  \\sum_{y \\in \\mathcal{V}} g_2(x,y)  \\\\\n&= \\mathbf{r}^{-1} \\sum_{x \\in A_1} \\nu_1(x) E^x T_1 = \\mathbf{r}^{-1} E^{\\nu_1} T_2.\n\\end{align*}\nSimilarly if $h_{21}(x) =\\bP^x( T_2 < T_1)$ we obtain\n$\\mathbf{r}^{-1} E^{\\nu_2} T_1 = \\sum_{y \\in \\mathcal{V}} h_{21}(y) $, and since $h_{12}+h_{21}=1$, adding these\nequalities proves the lemma.\n\\end{proof}\n\n\\section{ Estimates on the process $X^{n,2}$ } \n\nIn this section we  will prove \n\n\\begin{proposition}\\label{d22.2}\nFor every $\\delta>0$ there exists $n_1$ such that for all $n\\geq n_1$,  $u\\geq a_n^2$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{d22.3}\nP^0_\\omega\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta.\n\\end{align}\n\\end{proposition}\n\nThe proof requires a number of steps. We begin with a Harnack inequality.\n\n\\begin{lemma}\\label{L:harn}\nLet $1 \\le \\lambda \\le 10$. \nThere exist $p_1>0$ \n and $n_1 \\ge 1$ with the following properties. \\\\\n(a) Let $x \\in {\\mathbb Z}^2$, let  $B_1= \\mathcal{B}(x, \\lambda \\beta_n)$ \nand $B_2= \\mathcal{B}(x, (2\/3) \\lambda \\beta_n)$. \nLet $F$ be  the event that $X^{(n)}$ makes a closed loop around $B_2$\ninside $B_1 - B_2$\nbefore its first exit from $B_1$.\nIf $n \\ge n_1$ and $D_n \\cap B_1 = \\emptyset$ then\n$P^y_{\\omega}(F) \\ge p_1$ for all $ y \\in B_2$. \\\\\n(b) Let $h$ be harmonic in $B_1$.  \nThen \n\\begin{equation} \\label{e:harni}\n  \\max_{B_2} h \\le p_1^{-1} \\min_{B_2} h.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n(a) Using ($\\clubsuit$)  and \\eqref{e:PdistBM} we can make a Brownian approximation to\n$\\beta_n^{-1} X^{(n)}_\\cdot$ which is good enough so that this estimate holds.\\\\\n(b) Let $y \\in B_1$ be such that $h(y) = \\max_{z \\in B_2} h(z)$.\nThen by the maximum principle there exists a connected path $\\gamma$ from $y$\nto ${\\partial}_i B_1$ with $h(w) \\ge h(y)$ for all $w \\in \\gamma$.\nNow let $y'\\in B_2$.  On the event $F$ the process $X^{(n)}$ must hit $\\gamma$, and\nso we have\n$$ h(y') \\ge P^{y'}_{\\omega} (F) \\min_{\\gamma} h \\ge p_1 h(y),$$\nproving \\eqref{e:harni}. \n\\end{proof}\n\n\n\\begin{lemma}\\label{12a29lem}\nFor some $n_1$ and $c_1$, for all $n\\geq n_1$, $k\\geq 1$, and $\\omega$ such that \n$0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{z1.1}\nE^0_\\omega(U^n_k - S^n_{k-1}  \\mid \\mathcal{F}_{S^n_{k-1}} ) \\leq c_1 \\beta_n^2.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nBy the strong Markov property applied at $S^n_{k-1}$ for $k >1$, it is enough to prove \nthe Lemma for $k=1$, that is that \n$E^x_\\omega(U^n_1 ) \\le c_1 \\beta_n^2$ for all \n$x \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\n Let \n\\begin{align*}\n\\mathcal{V} &= \\mathcal{B}(u_n + \\mathcal{O}_n, 4 \\beta_n+1),\\\\\nA_1 &= \\partial_i \\mathcal{B}(u_n + \\mathcal{O}_n, (3\/2) \\beta_n),\\\\\nA_2 &= \\partial_i \\mathcal{V},\\\\\nA_3 &= \\partial_i \\mathcal{B}(u_n + \\mathcal{O}_n, 2 \\beta_n)\\\\\nT_i &= \\inf\\{t\\geq 0: X^{(n)}_t \\in A_i\\}, \\qquad i =1,2,3.\n\\end{align*}\nLet $Z$ be the continuous time Markov chain defined on $\\mathcal{V}$ by \\eqref{ma21.1}, \nrelative to the environment $\\mu^n$. Note that the transition probabilities from $x$ \nto one of its neighbors are the same for $Z$ and $X^{(n)}$ if $x $ is in the interior \nof $\\mathcal{V}$, i.e., $x\\notin \\partial_i \\mathcal{V} \\cup({\\mathbb Z}^2 \\setminus \\mathcal{V})$. \nNote also that $Z$ and  $X^{(n-1)}$ have the same transition probabilities \nin the region between $A_1$ and $A_3$.\nThe expectations and  probabilities in this proof will refer to $Z$.\nBy Lemma \\ref{L:com}, there exists a probability measure $\\nu_1$ on $A_1$ such that\n$E^{\\nu_1} T_2 \\leq \\mathbf{r} |\\mathcal{V}|$.\nWe have $|\\mathcal{V}| \\leq c_2 \\beta_n^2$.\n\nTo estimate $\\mathbf{r}$ note that by the choice of the constants $\\eta_{n-1}$ and $K_{n-1}$\nin Theorem \\ref{T:eK}, the resistance (with respect to $\\mu^{n-1}_e$) between two opposite sides of any\nsquare in $\\mathcal{S}_{n-1}$ will be 1. It follows that the resistance \nbetween two opposite sides of any square side $\\beta_n$ which is a union\nof squares in $\\mathcal{S}_{n-1}$ will also be 1. So, using Thompson's principle as\nin \\cite{BB3} we deduce that $\\mathbf{r} \\leq c_3$.\n\nSo, by Lemma \\ref{L:com} we have\n\\begin{align}\\label{ma18.1}\nE^{\\nu_1} T_2 \\leq c_4 \\beta_n^2.\n\\end{align}\n\nWe have for some $c_5$, $p_1 >0$ all $n$ and $x\\in \\mathcal{V} \\setminus \\mathcal{B}(u_n + \\mathcal{O}_n, (3\/2) \\beta_n)$,\n\\begin{align*}\nP^x_\\omega ( T_1 \\land T_2  \\leq c_5 \\beta_n^2) > p_1,\n\\end{align*}\nbecause an analogous estimate holds for Brownian motion and ($\\clubsuit$)  we have  \\eqref{e:PdistBM}. This and a standard argument based on the strong Markov property imply that for $x\\in A_3$,\n\\begin{align*}\nE^x_\\omega ( T_1 \\land T_2 ) \\leq c_6 \\beta_n^2.\n\\end{align*}\n\n\nNow for $y \\in A_1$ and $x \\in \\mathcal{V}$ set \n$$ \\nu_3^x (y) = P^x_\\omega(X^{(n)}(T_1 \\wedge T_2) = y  ).$$\n(Note that there exist $x$ with $\\sum_{y \\in A_1} \\nu^x_3(y) < 1$.)\nWe obtain for  $n\\geq n_2$ and $x\\in A_3$,\n\\begin{align}\\label{ma21.2}\nE^x_\\omega ( T_2 ) &=\nE^x_\\omega ( T_1 \\wedge T_2 ) + E^x_\\omega ((T_2 -T_1) {\\bf 1}_{ T_1 < T_2} )  \\\\\n& = E^x_\\omega ( T_1 \\wedge T_2 )  + E^{\\nu_3^x} T_2 \n\\leq  c_6 \\beta_n^2 + E^{\\nu_3^x}_{\\omega} T_2. \\nonumber\n\\end{align}\nFor $y \\in A_1$ the function $x \\to \\nu^x_3(y)$ is harmonic in $\\mathcal{V} \\setminus A_1$.\nSo we can apply the Harnack inequality Lemma \\ref{L:harn} to deduce that there exists\n$c_7$ such that\n\\begin{equation}\n \\nu^x_3(y) \\le c_7 \\nu^{x'}_3(y) \\hbox{ for all } x,x' \\in A_3, y \\in A_1.\n\\end{equation}\n\n\nThe measure $\\nu_1$ is  the hitting distribution on $A_1$ \nfor the process $Z$ starting with $\\nu_2$  (see \\cite[Chap.~3, p.~45]{AF}). So for\nany $x' \\in A_3$,\n\\begin{align*}\n\\nu_1(y) &= P^{\\nu_2}_0  ( Z_{T_1} =y) \n= \\sum_{x \\in A_3} P^{\\nu_2}_0  ( Z_{T_1} =x ) P^x_{\\omega}( Z_{T_1} =y)   \\\\\n&\\ge \\sum_{x \\in A_3} P^{\\nu_2}_0  ( Z_{T_1} =x ) P^x_{\\omega}( Z_{T_1 \\wedge T_2} =y)  \n\\ge  \\min_{x \\in A_3}   \\nu^x_3(y) \\ge c_7^{-1}  \\nu^{x'}_3(y).\n\\end{align*}\nHence for any $x \\in A_3$,\n$$  E^{\\nu_3^x}_{\\omega} T_2 \\le c_7 E^{\\nu_1}_{\\omega} T_2 \\le  c_8 \\beta_n^2,$$\nand combining this with \\eqref{ma21.2} completes the proof. \n\\end{proof}\n\nLet\n\\begin{align*}\nR_n^y =\\inf\\left\\{t\\geq 0 :  X^{(n)}_t \\in (y + a_{n-1}{\\mathbb Z}^2) \\cup \\Gamma^1_n\\right\\}.\n\\end{align*}\n\n\\begin{lemma}\nThere exist $c_1>0$ and $p_1<1$ such that for all $x,y \\in {\\mathbb Z}^2$,\n\\begin{align}\\label{ma23.1}\n&P^x_{\\omega}\\left(R_n^y \\geq c_1 b_n^2 \\right) \\le p_1,\\\\\n&P^x_{\\omega}\\left(\\sup_{0\\leq t \\leq R^y_n}|x- X^{(n)}_t| \\geq c_1 b_n \\right) \\le p_1.\n\\label{ma23.2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nRecall that the family $\\{\\mu^{n-1}_{x+ \\cdot}\\}_{x\\in {\\mathbb Z}^2}$ of translates of the \nenvironment  $\\mu^{n-1}_\\cdot$ contains only a finite number of distinct elements.\nSince each square in $\\mathcal{S}_{n-1}$ contains one point in $(y + a_{n-1}{\\mathbb Z}^2)$, \nif $b_n\/a_{n-1}$  is sufficiently large ($\\clubsuit$)  then using the transition density estimates \\eqref{e:GB1}\nas well as \\eqref{e:PdistBM}, we obtain \\eqref{ma23.1} and \\eqref{ma23.2}. \n\\end{proof}\n\n\\begin{lemma}\nFor some $n_1$ and $c_1$, for all $n\\geq n_1$, $k\\geq 1$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{ma21.5}\nE^0_\\omega(V^n_k - T^n_{k-1} \\mid \\mathcal{F}_{T^n_{k-1}}) \\leq c_1 b_n^2 n^{1\/2}.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nLet\n\\begin{align*}\n\\widehat R^n_k =\\inf\\left\\{t\\geq U^n_k :  X^{(n)}_t \\in (X^{(n)}(T^n_0) + a_{n-1}{\\mathbb Z}^2) \\cup \\Gamma^1_n\\right\\}.\n\\end{align*}\nLet $F_k = \\{ \\widehat R^n_k < S^n_k\\}$ and $G_k = \\bigcap _{j=1}^k F_j^c$.\nSince $b_n n^{1\/8} < \\beta_n$ for large $n$, we obtain from \\eqref{ma23.2} and definitions of $\\Gamma^1_n, \\Gamma^2_n, U^n_k$ and $S^n_k$ that there exists $p_2>0$ such that for $x\\in \\Gamma^2_n$,\n\\begin{align*}\nP^x_\\omega(F_k \\mid \\mathcal{F}_{U^n_k}) > p_2.\n\\end{align*}\nHence,\n\\begin{align}\\label{n5.2}\nP^x_\\omega(G_k ) < (1-p_2)^k.\n\\end{align}\nNote that if $F_k$ occurs then $V^n_1 \\leq \\widehat R^n_k$.\nWe have, using \\eqref{z1.1}, \\eqref{ma23.1} and \\eqref{n5.2},\n\\begin{align*}\nE^0_\\omega(V^n_1 - T^n_{0} ) \n&\\leq\n\\sum_{k=1}^\\infty\nE^0_\\omega((U^n_k - S^n_{k-1}) {\\bf 1}_{G_{k-1}})\n+ \n\\sum_{k=1}^\\infty\nE^0_\\omega((\\widehat R^n_k - U^n_{k}) {\\bf 1}_{G_{k-1}})\\\\\n&\\leq \\sum_{k=1}^\\infty\nc_2 \\beta_n^2 (1-p_2)^{k-1}\n+ \\sum_{k=1}^\\infty c_3 b_n^2 (1-p_2)^{k-1}  \\\\\n&\\leq c_4 \\beta_n^2 \\leq c_5 b_n^2 n^{1\/2}.\n\\end{align*}\nThis proves the lemma for $k=1$.\nThe general case is obtained by applying this estimate to the\nprocess shifted by $T^n_{k-1}$; in other words, by using the strong Markov property.\n\\end{proof}\n\n\\begin{lemma} \\label{L:sigma_1}\nFor every $\\delta>0$ there exists $n_1$ such that for all $n\\geq n_1$,  $u\\geq a_n^2$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$,\n\\begin{align}\\label{n8.5}\nP^0_\\omega\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta \\right) \\geq 1-\\delta\/2.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nFix an arbitrarily small $\\delta >0$, consider $u \\geq a_n^2$ and let $j_* = \\lceil u\/(b_n^2 n^{5\/8} )\\rceil$. Then\n\\eqref{ma21.5} implies that for some $c_1$ and $n_2$, all $n\\geq n_2$, $u \\geq a_n^2$, \n\\begin{align*}\nE^0_{\\omega} \\left( \\frac 1{j_*} \\sum_{j=1}^{j_*} V^n_j - T^n_{j-1}\\right) \n\\leq c_1 b_n^2 n^{1\/2}.\n\\end{align*}\nHence, for some $n_3$, all $n\\geq n_3$, $u \\geq a_n^2$,\n\\begin{align*\nP^0_{\\omega} \\left( \\frac 1{j_*} \\sum_{j=1}^{j_*} V^n_j - T^n_{j-1} \\ge \\delta b_n^2 n^{9\/16}  \\right) \\le \\delta\/8 ,\n\\end{align*}\nand, since $j_* \\delta b_n^2 n^{9\/16}  \\leq \\delta u$, \n\\begin{align}\\label{d27.1}\nP^0_{\\omega} \\left(  \\sum_{j=1}^{j_*} V^n_j - T^n_{j-1} \\ge \\delta u \\right) \\le \\delta\/8 . \n\\end{align}\n\nRecall $\\mathcal{K} (r)$ from \\eqref{e:Hdef}.\nLet\n\\begin{align*}\n\\widehat V^n_k & = \\inf\\{t\\geq V^n_{k}: X^{(n)}_t \\in \n{\\mathbb Z}^2 \\setminus \\mathcal{K}(b_n n ^{3\/8})\\} \\land T^n_k, \\qquad k \\geq 1,\\\\\n\\widetilde V^n_k & = \\inf\\{t\\geq \\widehat V^n_{k}: |X^{(n)}_t - X^{(n)}(\\widehat V^n_k)| \\geq (1\/2)b_n n^{3\/8} \\} , \\qquad k \\geq 1. \n\\end{align*}\nWe can use estimates for Brownian hitting \nprobabilities ($\\clubsuit$)  to see that for some $c_2, c_3$ and $n_4$,\nall $n\\geq n_4$, $k$, \n\\begin{align}\\label{j2.13}\nP^0_{\\omega}(\\widehat V^n_k < T^n_k \\mid \\mathcal{F}_{V^n_k}) \\geq c_2\n\\frac{\\log (4 \\beta_n) - \\log (2 \\beta_n)}\n{\\log (2 b_n n^{3\/8})- \\log (2 \\beta_n)} \n\\geq  c_3 \/\\log n . \n\\end{align}\nThere exist ($\\clubsuit$)  $c_4$ and $n_5$, such that for\nall $n\\geq n_5$, $k\\geq 2$, \n\\begin{align*}\nP^0_{\\omega} &(T^n_k - V^n_k \\geq c_4 b_n^2 n^{3\/4}  \\mid \\widehat V^n_k < T^n_k, \\mathcal{F}_{\\widehat V^n_k}) \\\\\n&\\ge\nP^0_{\\omega}(\\widetilde V^n_k - \\widehat V^n_k \\geq c_4 b_n^2 n^{3\/4}  \\mid \\widehat V^n_k < T^n_k, \\mathcal{F}_{\\widehat V^n_k}) \\ge 3\/4. \n\\end{align*}\nThis and \\eqref{j2.13} imply that the sequence $\\{T^n_k - V^n_k\\}_{k\\geq 2}$ is stochastically  minorized by a sequence of i.i.d.~random variables which take value $c_4 b_n^2 n^{3\/4} $ with probability $c_3 \/\\log n$ and they take value 0 otherwise.\nThis implies that for some $n_6$,\nall $n\\geq n_6$, $u \\geq a_n^2$, \n\\begin{align*\nP^0_{\\omega}\\left( \\frac 1{j_*} \\sum_{j=2}^{j_*} T^n_j - V^n_j  \\le  b_n^2 n^{3\/4}\/ \\log^2 n \\right) \\le \\delta\/4 \n\\end{align*}\nand, because  $j_* b_n^2 n^{3\/4}\/\\log^2 n \\geq u$ assuming $n_6$ is large enough,\n\\begin{align*\nP^0_{\\omega}\\left(  \\sum_{j=2}^{j_*} T^n_j - V^n_j  \\le u \\right) \\le \\delta\/4 . \n\\end{align*}\nWe combine this with \\eqref{d27.1} and the definition of $\\sigma^{n,2}_u$ to obtain\nfor some $n_7$,\nall $n\\geq n_7$, $u \\geq a_n^2$, \n\\begin{align}\\label{d27.4}\nP^0_{\\omega}(\\sigma^{n,2}_u\/u \\le \\delta) \\geq 1-3\\delta\/8.\n\\end{align}\nThis completes the proof of the lemma.\n\\end{proof}\n\n\\bigskip\n\nLet $Y^n_k = (Y^n_{k,1}, Y^n_{k,2}) = X^{(n)}(V^n_{k+1} ) - X^{(n)}(T^n_{k} )$.\nSet \n$\\bar Y^n_k = \\sup_{T^n_k \\leq t \\leq V^n_{k+1}} |X^{(n)}(t)- X^{(n)}(T^n_{k} )|$.\nFor $x\\in {\\mathbb Z}^2$, let\n $\\Pi_n(x) \\in B'_n -u_n + \\mathcal{O}_n $ be the unique point with the \nproperty that $x-\\Pi_n(x) = a_n y$ for some $y\\in {\\mathbb Z}^2$.\n\n\nWe next estimate the variance of  $X^{n,2}(V^n_{m+1}) = \\sum_{k=0}^ m Y^n_k$.\n\n\\begin{lemma} \\label{L:Ynest}\nThere exist $c_1, c_2$ and $n_1$ such that for all $n\\geq n_1$, $k\\geq 0$, $j=1,2$, and $\\omega$,\n\\begin{align}\\label{ma24.6}\nE^0_\\omega |Y^n_{k,j}| &\\leq E^0_\\omega |Y^n_k| \\leq E^0_\\omega |\\bar Y^n_k| \\leq c_1 \\beta_n, \\\\\n\\label{ma24.7}\n\\Var Y^n_{k,j} &\\le \\Var \\bar Y^n_{k} \\le c_2 \\beta_n^2 , \\qquad \\text { under } P^x_\\omega.\n\\end{align} \n\\end{lemma} \n\n\\begin{proof} Let \n\\begin{align}\\label{n8.1}\n\\mathcal{X}^{(n)}_k(t) &= X^{(n)}_t + \\Pi_n(X^{(n)}(T^n_k)) - X^{(n)}(T^n_k),\n\\qquad t\\in [T^n_k, V^n_{k+1}],\n\\end{align}\nand note that \n\\begin{align*}\nY^n_k = (Y^n_{k,1}, Y^n_{k,2}) = \\mathcal{X}^{(n)}_k(V^n_{k+1} ) - \\mathcal{X}^{(n)}_k(T^n_{k} ).\n\\end{align*}\n\nIt follows from the definition that we have $\\sup_{S^n_{k-1} \\leq t \\leq U^n_k} |X^{(n)}(t ) - X^{(n)}(S^n_{k-1} )| \\le 16\\beta_n$, a.s.\nThis, \\eqref{ma23.2} and the definition of $V^n_{k+1}$ imply that $|\\bar Y^n_k|$ is stochastically majorized by an exponential random variable with mean $c_3 \\beta_n$. This easily implies the lemma.\n\\end{proof}\n\n\nNext we will estimate the covariance of $Y^n_{k,1}$ and $Y^n_{j,1}$ for $j\\ne k$.\n\n\\begin{lemma} \\label{L:covY}\nThere exist $c_1, c_2$ and $n_1$ such that for all $n\\geq n_1$, $j < k-1$ and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, under $P^0_\\omega$,\n\\begin{align}\\label{n8.3} \n\\Cov(Y^n_{j,1},Y^n_{k,1}) & \\le c_1 e^{-c_2 (k-j)} \\beta_n^2.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nLet\n\\begin{align*}\n\\Gamma^3_n &= \\Gamma^1_n \\cap \\mathcal{B}(u_n + \\mathcal{O}_n, a_n\/2)\n= \\mathcal{B}(u_n + \\mathcal{O}_n, 2\\beta_n),\\\\\n\\Gamma^4_n &= \\partial_i \\mathcal{B}(u_n + \\mathcal{O}_n, 3\\beta_n),\\\\\n\\tau(A) &= \\inf\\{t\\geq 0: \\mathcal{X}^{(n)}_0(t) \\in A\\}.\n\\end{align*}\n\nSuppose that $x,v\\in \\Gamma^3_n$ and $y \\in \\Gamma^4_n$. \nBy the Harnack inequality proved in Lemma \\ref{L:harn},\n\\begin{align}\\label{ma23.4}\n\\frac{P_\\omega^x (\\mathcal{X}^{(n)}_0(\\tau(\\Gamma^4_n)) = y)}\n{P_\\omega^v (\\mathcal{X}^{(n)}_0(\\tau(\\Gamma^4_n)) = y)}\n\\geq c_3.\n\\end{align}\n\nLet $\\mathcal{T}^n_k$ have the same meaning as $T^n_k$ but relative to the process $\\mathcal{X}^{(n)}_k$ rather than $X^{(n)}$.\nWe obtain from \\eqref{ma23.4} and the strong Markov property applied at $\\tau(\\Gamma^4_n)$ that, \nfor any \n$x,v,y \\in \\Gamma^3_n$ we have\n\\begin{align*\n\\frac{P_\\omega^x (\\mathcal{X}^{(n)}_0(\\mathcal{T}^n_1) = y)}\n{P_\\omega^v (\\mathcal{X}^{(n)}_0(\\mathcal{T}^n_1) = y)}\n\\geq c_3.\n\\end{align*}\nRecall that $T^n_0 =0$.\nThe last estimate implies that, for  \n$x,v,y \\in \\Gamma^3_n$,\n\\begin{align*\n\\frac{P_\\omega (\\mathcal{X}^{(n)}_1(T^n_{1}) = y \\mid \\mathcal{X}^{(n)}_0(T^n_0) = x)}\n{P_\\omega (\\mathcal{X}^{(n)}_1(T^n_{1}) = y \\mid \\mathcal{X}^{(n)}_0(T^n_0) = v)}\n\\geq c_3.\n\\end{align*}\nSince the process $X^{(n)}$ is time-homogeneous, this shows that for  \n$x,v,y \\in \\Gamma^3_n$ and all $k$,\n\\begin{align}\\label{d29.2}\n\\frac{P_\\omega (\\mathcal{X}^{(n)}_{k+1}(T^n_{k+1}) = y \\mid \\mathcal{X}^{(n)}_k(T^n_k) = x)}\n{P_\\omega (\\mathcal{X}^{(n)}_{k+1}(T^n_{k+1}) = y \\mid \\mathcal{X}^{(n)}_k(T^n_k) = v)}\n\\geq c_3.\n\\end{align}\nWe now apply Lemma 6.1 of \\cite{BTW} (see Lemma 1 of \\cite{BK} for a better presentation of the same estimate) to see that \\eqref{d29.2} implies that \nthere exist constants $C_k$, $k\\geq 1$, such that for every $k$ and all \n$x,v,y \\in \\Gamma^3_n$,\n\\begin{align*\n\\frac{P_\\omega^x (\\mathcal{X}_k^{(n)}(T^n_k) = y)}\n{P_\\omega^v (\\mathcal{X}_k^{(n)}(T^n_k) = y)}\n\\geq C_k.\n\\end{align*}\nMoreover, $C_k\\in(0,1)$, $C_k$'s depend only on $c_3$, and $1-C_k \\le e^{-c_4 k}$ for some $c_4>0$ and all $k$.\nBy time homogeneity of $X^{(n)}$, for $m\\leq j<k$ and all $x,v,y,z \\in \\Gamma^3_n$,\n\\begin{align*\n\\frac{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_j^{(n)}(T^n_j) = x)}\n{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_j^{(n)}(T^n_j) = v)}\n\\geq C_{k-j},\n\\end{align*}\nand, by the strong Markov property applied at $T^n_j$, \n\\begin{align*\n\\frac{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_j^{(n)}(T^n_j) = x)}\n{P^z_\\omega (\\mathcal{X}_k^{(n)}(T^n_k) = y \\mid \\mathcal{X}_m^{(n)}(T^n_m) = v)}\n\\geq C_{k-j}.\n\\end{align*}\nThis and \\eqref{ma24.6} imply that for $j<k-1$ and $x\\in {\\mathbb Z}^2$,\n\\begin{align}\\nonumber\n|{E}^x_\\omega(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1} \\mid \\mathcal{F}_{T^n_{j+1}})|\n&=\n|{E}^x_\\omega(Y^n_{k,1}  \\mid \\mathcal{F}_{T^n_{j+1}})\n- {E}^x_\\omega Y^n_{k,1} |   \\\\\n\\nonumber\n&\\le  (1- C_{k-j-1}) \\sup_{y\\in {\\mathbb Z}^2} {E}^y_\\omega |Y^n_{k,1}| \\\\\n&\\le  e^{-c_4 (k-j-1)} c_5 \\beta_n \\leq \n c_6 e^{-c_4 (k-j)}  \\beta_n.\n\\label{ma24.8}\n\\end{align}\nHence  for $j< k-1$,\n\\begin{align*\n\\Cov(Y^n_{j,1},Y^n_{k,1})\n&= {E}^x_\\omega ((Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1})\n(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1}))\\\\\n&= {E}^x_\\omega ({E}^x_\\omega((Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1})\n(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1}) \\mid \\mathcal{F}_{T^n_{j+1}}))\\\\\n&= {E}^x_\\omega ((Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1}) {E}^x_\\omega(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1} \\mid \\mathcal{F}_{T^n_{j+1}}))\\\\\n&\\leq {E}^x_\\omega (|Y^n_{j,1} - {E}^x_\\omega Y^n_{j,1}| \\cdot |{E}^x_\\omega(Y^n_{k,1} - {E}^x_\\omega Y^n_{k,1} \\mid \\mathcal{F}_{T^n_{j+1}})|)\\\\\n& \\le 2 {E}^x_\\omega |Y^n_{j,1}| c_6 e^{-c_4 (k-j)} \\beta_n \\\\\n& \\le c_7 e^{-c_4 (k-j)} \\beta_n^2.\n\\end{align*}\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{d22.2}]\nAssume that $\\omega$ is such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$.\nWe combine \\eqref{n8.3} and \\eqref{ma24.7} to see that for some $c_{1}$ and $c_{2}$ and all  $m\\geq 1$, we have under $P^0_{\\omega}$, \n\\begin{align}\\label{d29.10}\n\\Var\\left(\\sum_{k=0}^m Y^n_{k,1}\\right) &= \n\\sum_{j=0}^m \\sum_{k=0}^m\n\\Cov(Y^n_{j,1},Y^n_{k,1})\\\\\n& \\le \\sum_{j=0}^m \\sum_{k=0}^m\nc_{1} e^{-c_{3} (k-j)} \\beta_n^2 \\leq c_{2} m \\beta_n^2. \\nonumber\n\\end{align}\n\nFor fixed $n$ and ${\\omega}$, the process \n$\\{\\mathcal{X}^{(n)}_k(T^n_k), k\\geq 1\\}$ \nis Markov with a finite state space \nand one communicating class, so it has a unique stationary distribution. We will call it $\\mathbf{p}(n)$.\nWe will argue \nthat ${E}^{\\mathbf{p}(n)}_{\\omega} Y^n_{k,1} = 0$. Since $X^{(n)}$ and $X^{(n-1)}$ \nsatisfy the quenched invariance principle and they are random walks among symmetric (in distribution) conductances, they have zero means. Recall that \n$X^{(n)} = X^{n,1} + X^{n,2}$ and $\\widehat X^{n,1}$ has the same distribution as $X^{(n-1)}$. \nIt follows that for some $c_{4}>0$ and $c_{5}<1\/4$ and all large $t$, we have \n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |\\widehat X^{n,1} _s| \\geq  c_{4}\\sqrt{t}\\right) \n=P^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{(n-1)} _s| \\geq  c_{4}\\sqrt{t}\\right) < c_{5}. \n\\end{align*}\nSince $\\widehat X^{n,1}_t =  X^{n,1}(\\widehat \\sigma^{n,1}_t)$ and $\\widehat \\sigma^{n,1}_t \\geq t$, the last estimate implies that\n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{n,1} _s| \\geq  c_{4}\\sqrt{t}\\right)  < c_{5}. \n\\end{align*}\nWe also have \nfor some $c_{6}>0$ and $c_{7}<1\/4$, and all large $t$, \n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{(n)} _s| \\geq  c_{6}\\sqrt{t}\\right)  < c_{7}. \n\\end{align*}\nSince $X^{n,2} = X^{(n)} - X^{n,1}$, we obtain\nfor some $c_{8}>0$ and $c_{9}<1\/2$ and all large $t$, \n\\begin{align*}\nP^{\\mathbf{p}(n)}_{\\omega} \\left(\\sup_{1\\le s \\le t} |X^{n,2} _s| \\geq  c_{8}\\sqrt{t}\\right)  < c_{9}. \n\\end{align*}\nThis shows that $X^{n,2}$ does not have a linear drift.\nIt is clear from the law of large numbers that $\\liminf_{t\\to\\infty} \\sigma_t^{n,2}\/t >0$, so $\\widehat X^{n,2}$ does not have a linear drift either.\nWe conclude that ${E}^{\\mathbf{p}(n)}_\\omega Y^n_{k,1} = 0$. \n\nNow suppose that $X^{(n)}_0$ does not necessarily have the distribution $\\mathbf{p}(n)$. \nThe fact that ${E}^{\\mathbf{p}(n)}_\\omega Y^n_{k,1} = 0$ and a calculation similar to that in \\eqref{ma24.8} imply that,\n\\begin{align*\n|{E}^0_\\omega  Y^n_{k,1}| \\le  c_{10} e^{-c_{11} k}  \\beta_n .\n\\end{align*}\n\nLet $c_{12} $ be the constant denoted $c_1$ in \\eqref{ma24.6}.\nThe last estimate and \\eqref{ma24.6} imply that for some $c_{13}$ and all $m\\geq 1$,\n\\begin{align} \\nonumber\n\\left| E^0_\\omega\\sum_{k=0}^{m} Y^n_{k,1}\\right| \n&\\leq \\sum_{k\\geq 0} |{E}^0_\\omega Y^n_{k,1}|\n+ \\sup_{k\\geq 1} E^0_{\\omega} |\\bar Y^n_k|   \\\\\n\\label{j1.1}\n&\\leq \\sum_{k\\geq 0} c_{10} e^{-c_{11} k}  \\beta_n +c_{12}\\beta_n\n\\leq  c_{13} \\beta_n .\n\\end{align}\nAll estimates that we derived for $Y^n_{k,1}$'s apply to $Y^n_{k,2}$'s as well, by symmetry.\n\nNote that $|X^{(n)}(U^n_{k+1} ) - X^{(n)}(T^n_{k} )|\\geq \\beta_n\/2$.\nWe have\n$V^n_{k+1}  - T^n_{k} \\geq U^n_{k+1}  - T^n_{k} $ so\nwe can assume ($\\clubsuit$)  that $b_n\/a_{n-1}$ is so large that for some $p_1>0$ and $n_{2}$, for all $n\\geq n_{2}$ and $k\\geq 1$, \n\\begin{align*\nP_\\omega^x(V^n_{k+1}  - T^n_{k} \\geq \\beta_n^2\n\\mid \\mathcal{F}_{T^n_k}) \\geq p_1.\n\\end{align*}\nLet $\\mathcal{V}_m$ be a binomial random variable with parameters \n$m$ and $p_1$. We see that $\\sigma^{n,2}(V^n_{m })= \\sum_{k=0}^m V^n_{k+1}  - T^n_{k}$ is stochastically minorized by \n$\\beta_n^2 \\mathcal{V}_m$. \n\nRecall that $u \\geq a_n^2$.\nLet $m_1$ be the smallest integer such that\n\\begin{align}\\label{c1.1}\nP^0_\\omega(V^n_{m_1 } \\leq u) < \\delta\/4.\n\\end{align}\nThen\n\\begin{align}\\label{c1.2}\nP^0_\\omega(V^n_{m_1 -1} \\leq u) \\geq \\delta\/4.\n\\end{align}\nSince $\\delta$ in \\eqref{d27.4} can be arbitrarily small, we have\nfor for some $n_{3}$ and all $n\\geq n_{3}$, \n\\begin{align}\\label{c1.3}\nP^0_\\omega(\\sigma^{n,2}_u\/u \\leq \\delta^4) \\geq 1-\\delta\/8.\n\\end{align}\nThe following estimate follows from the fact that $\\sigma^{n,2}(V^n_{m_1-1 })$ is stochastically minorized by \n$\\beta_n^2 \\mathcal{V}_{m_1-1}$, and from \\eqref{c1.2}-\\eqref{c1.3},\n\\begin{align*}\nP^0_\\omega(\\beta_n^2 \\mathcal{V}_{m_1-1}  \\leq \\delta^4 u) \n&\\geq \nP^0_\\omega(\\sigma^{n,2}(V^n_{m_1 -1}) \\leq \\delta^4 u) \\\\\n&\\geq\nP^0_\\omega(\\sigma^{n,2}_u \\leq \\delta^4 u, V^n_{m_1 -1} \\leq u) \n\\geq \\delta\/8.\n\\end{align*}\nThis implies that for some $c_{14}$, we have\n$m_1 \\leq c_{14}\\delta^3 u\/\\beta_n^2$. In other words, $u \\geq m_1 \\beta_n^2\/(c_{14} \\delta^3)$. \nNote that for a fixed $\\delta$, we have for large $n$, ($\\clubsuit$)  $u^{1\/2}\\delta\/4 -  c_{13} \\beta_n \\geq u^{1\/2} \\delta\/8$.\nThese observations, \\eqref{d29.10}, \\eqref{j1.1} and the Chebyshev inequality imply that for  $m\\le m_1$,\n\\begin{align}\\label{d29.22}\nP^0_\\omega&\\left(u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right) \\geq \\delta\/2\\right)\\\\\n&\\le\nP^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right| \\geq u^{1\/2} \\delta\/4\\right)\n+ P^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,2}\\right| \\geq u^{1\/2} \\delta\/4\\right)\n\\nonumber \\\\\n&\\le\nP^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\n- E^0_\\omega\\sum_{k=0}^{m} Y^n_{k,1}\\right| \\geq u^{1\/2} \\delta\/4\n- c_{13} \\beta_n\\right)\\nonumber \\\\\n&\\qquad + P^0_\\omega\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,2}\n- E^0_\\omega\\sum_{k=0}^{m} Y^n_{k,2}\\right| \\geq u^{1\/2} \\delta\/4\n- c_{13} \\beta_n\\right)\n\\nonumber \\\\\n&\\le  \\frac{\\Var\\left(\\sum_{k=0}^{m} Y^n_{k,1}\\right)}\n{u \\delta^2\/64} + \\frac{\\Var\\left(\\sum_{k=0}^{m} Y^n_{k,2}\\right)}\n{u \\delta^2\/64} \\nonumber \\\\\n&\\le  \\frac{2c_{2} m_1 \\beta_n^2}\n{(c_{14}^{-1}\\delta^{-3} m_1\\beta_n^2) \\delta^2\/64} \\le c_{15} \\delta.\n\\nonumber\n\\end{align}\nLet $M = \\min\\{m\\geq 1: \nu^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right)\n\\geq \\delta\\}$. By the strong Markov property applied at $M$ and \\eqref{d29.22},\n\\begin{align}\\label{d29.23}\n&P^0_\\omega\\left(\n\\sup_{1\\le m \\le m_1}\nu^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right)\n\\geq \\delta,\\  u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1}Y^n_{k,2}\\right|\\right) \\le \\delta\/2\\right)\\\\\n&\\le \nP^0_\\omega\\left( u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1-M} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1-M}Y^n_{k,2}\\right|\\right) \\geq \\delta\/2 \\mid M < m_1\\right)\n\\le c_{15} \\delta. \\nonumber\n\\end{align}\n\nRecall that $u \\geq m_1 \\beta_n^2\/(c_{14} \\delta^3)$. For a fixed $\\delta$ and large $n$, ($\\clubsuit$)  $u^{1\/2}\\delta - 2 c_{12} \\beta_n \\geq u^{1\/2} \\delta\/2$.\nIt follows from this, \\eqref{ma24.6} and \\eqref{ma24.7} that\n\\begin{align}\\label{ma25.1}\nP^0_{\\omega}\\left(\\exists k \\leq m_1: |\\bar Y^n_k| \\geq u^{1\/2}\\delta  \\right) \n&\\leq \nm_1 \\sup_{k\\leq m_1} P^0_{\\omega}\\left( |\\bar Y^n_k| \\geq u^{1\/2}\\delta  \\right) \\\\\n&\\leq \nm_1 \\sup_{k\\leq m_1} P^0_{\\omega}\\left( |\\bar Y^n_k| -E^0_{\\omega} |\\bar Y^n_k|\\geq u^{1\/2}\\delta - c_{12} \\beta_n \\right)\\nonumber \\\\\n&\\le m_1 \\frac {c_{11} \\beta_n^2}{ u  \\delta^2 \/4}  \n\\le m_1 \\frac{c_{11}  \\beta_n^2}\n{(c_{14}^{-1}\\delta^{-3} m_1\\beta_n^2) \\delta^2} \\le c_{16} \\delta.\n\\nonumber\n\\end{align}\n\nWe use \\eqref{c1.1}, \\eqref{d29.22}, \\eqref{d29.23} and \\eqref{ma25.1} to obtain\n\\begin{align*\n&P^0_\\omega \\left(\\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\geq 2\\delta\\right)\\\\\n &\\le P^0_\\omega(V^n_{m_1 } \\le u) \n+ P^0_\\omega\\left(u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1}Y^n_{k,2}\\right|\\right) \\geq \\delta\/2\\right)\\\\\n& + P^0_\\omega\\left(\n\\sup_{1\\le m \\le m_1}\nu^{-1\/2}\\left(\\left|\\sum_{k=0}^{m} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m}Y^n_{k,2}\\right|\\right)\n\\geq \\delta,\\  u^{-1\/2}\\left(\\left|\\sum_{k=0}^{m_1} Y^n_{k,1}\\right|+\\left|\\sum_{k=0}^{m_1}Y^n_{k,2}\\right|\\right) \\le \\delta\/2\\right)\\\\\n& +\nP^0_{\\omega}\\left(\\exists k \\leq m_1: |\\bar Y^n_k| \\geq u^{1\/2}\\delta  \\right) \\\\\n&\\le \\delta\/4 + c_{15} \\delta + c_{15} \\delta + c_{16}\\delta.\n\\end{align*}\nSince $\\delta>0$ is arbitrarily small, this implies that for every $\\delta>0$, some $n_{3}$ and all $n\\geq n_{3}$,\n\\begin{align*\nP^0_\\omega &\\left(\\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\geq \\delta\\right)\\le \\delta\/2.\n\\end{align*}\nThis and \\eqref{n8.5} yield the proposition.\n\\end{proof}\n\nRecall from \\eqref{e:bfPdef} the definition of the averaged measure ${\\bf P}$.\n\n\\begin{lemma}\\label{n9.1}\nFor every $\\delta>0$ there exists $n_1$ such that for all $n\\geq n_1$ and  $u\\geq a_n^2$,  \n\\begin{align}\\label{n9.2}\n{\\bf P}\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nBy Proposition \\ref{d22.2} applied to $\\delta\/2$ in place of $\\delta$, for every $\\delta>0$ there exists $n_2$ such that for all $n\\geq n_2$,  $u\\geq a_n^2$, and $\\omega$ such that $0 \\notin \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n$, \n\\begin{align}\\label{n9.3}\nP^0_\\omega\n\\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta\/2.\n\\end{align}\n\nLet $|A|$ denote the cardinality of $A\\subset {\\mathbb Z}^2$. Since $|\\Gamma^1_n| \\leq 25 \\beta_n^2 \\leq 25 a_n^2 n^{-1\/2} = 25 n^{-1\/2} |B'_n|$, the definitions of $\\mathcal{O}_n$ and $\\Gamma^1_n$ imply that ${\\bf P}(0 \\in \\Gamma^1_n \\setminus \\partial_i \\Gamma^1_n) < \\delta\/2$ for some $n_3 \\geq n_2$ and all $n \\geq n_3$. This and \\eqref{n9.3} imply \\eqref{n9.2}.\n\\end{proof}\n\nIn the following lemma and its proof, when we write the Prokhorov distance between processes such as $\\{ (1\/a)X^{(n-1)}_{ta^2}, t\\in[ 0,1]\\}$, we always assume that they are distributed according to ${\\bf P}$.\n\n\\begin{lemma}\\label{d22.1}\nThere exists a function $\\rho^*: (0,\\infty) \\to (0,\\infty)$ with \n$ \\lim_{\\delta\\downarrow 0} \\rho^*(\\delta) = 0$ \nand a sequence $\\{a_n\\}$ with the following properties,\n\\begin{align}\\label{d19.3}\n&{d_P}(\\{ (1\/a)X^{(n-1)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n},\\qquad a \\geq a_n.\n\\end{align}\nMoreover, suppose that for $\\delta<1\/2$ and all $u\\geq a_n^2$, \n\\begin{align}\\label{d19.2}\n{\\bf P} \\left(  \\sigma^{n,2}_u \/ u \\le \\delta, \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta.\n\\end{align}\nThen\n${d_P}( \\{(1\/a)X^{(n)}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n} + \\rho^*(\\delta)$, for all $a\\geq a_n$.\n\\end{lemma}\n\n\\begin{proof}\n\nFormula \\eqref{d19.3} is  special case of \\eqref{e:PdistBM}.\n\nFix some $a\\geq a_n$. We will apply \\eqref{d19.2} with $u=a^2$. \nNote that on the event in \\eqref{d19.2} we have\n\\begin{align}\\label{n9.5}\n1- \\sigma^{n,1}_{a^2}\/a^2= u\/u- \\sigma^{n,1}_u\/u \n= \\sigma^{n,2}_u \/ u \\le \\delta.\n\\end{align}\nThe function $t\\to \\sigma^{n,1}_{ta^2}\/a^2$ is Lipschitz with the constant 1 and $\\sigma^{n,1}_{ta^2}\/a^2 \\leq t$ so \\eqref{n9.5} implies for $t\\in[0,1]$,\n\\begin{align}\\label{n9.6}\nt- \\sigma^{n,1}_{ta^2}\/a^2 \\leq 1- \\sigma^{n,1}_{a^2}\/a^2 \\leq \\delta.\n\\end{align}\n\nRecall the function $\\rho(\\delta)$ from the proof of Lemma \\ref{d21.1}, \nsuch that $P_{\\text{BM}}(\\Osc(W,\\delta) \\geq\\rho(\\delta) )<\\rho(\\delta)$ and \n$\\lim_{\\delta\\downarrow 0} \\rho(\\delta) = 0$.\nBy \\eqref{n9.6}, we can apply Lemma \\ref{d21.1} with $\\sigma_t = \\sigma^{n,1}_{ta^2}\/a^2$. Recall\nthat  $W^*(t) = W(\\sigma_t)$.\nBy the definition of $\\widehat X^{n,1}$,\n\\begin{align}\n\\nonumber\n&{d_P}( \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\\n\\nonumber\n&\\le  {d_P}( \\{(1\/a) X^{n,1}_{t\/a^2}, t\\in[0,1]\\}, \\{W^*_t, t\\in[0,1]\\})\n + {d_P}( \\{W^*_t, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\ \n\\nonumber \n&\\le  {d_P}( \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}, \\{W^*_t, t\\in[0,1]\\})\n + \\rho(\\delta) + \\delta\\\\ \n&= {d_P}( \\{(1\/a)\\widehat X^{n,1}(\\sigma^{n,1}_{ta^2}), t\\in[0,1]\\}, \\{W(\\sigma^{n,1}_{ta^2}\/a^2), t\\in[0,1]\\})\n + \\rho(\\delta) + \\delta  .\n\\end{align}\n\nRecall from \\eqref{e:Xhatdsn} that for a fixed $\\omega\\in \\Omega$, the distribution of $\\{\\widehat X^{n,1}_t, t\\geq 0\\}$ \nis the same as that of $\\{X^{n-1}_t, t\\geq 0\\}$.\nIn view of Theorem \\ref{T:eK}, we can make $a_n$ so large ($\\clubsuit$)  that \n$\\Pp(\\Osc(\\widehat X^{n,1},\\delta) \\geq 2\\rho(\\delta) )< 2\\rho(\\delta)$. \nThis, Lemma \\ref{d21.2} and the definition of the Prokhorov distance imply that\n\\begin{align*}\n{d_P}( \\{(1\/a)\\widehat X^{n,1}&(\\sigma^{n,1}_{ta^2}), \\, t\\in[0,1]\\}, \\{W(\\sigma^{n,1}_{ta^2}\/a^2), t\\in[0,1]\\})  \\\\\n&\\le \n{d_P}( \\{(1\/a)\\widehat X^{n,1}_{ta^2}, t\\in[0,1]\\}, \\{W_{t}, t\\in[0,1]\\}) + 4\\rho(\\delta) \\\\\n&= {d_P}( \\{(1\/a) X^{(n-1)}_{ta^2}, t\\in[0,1]\\}, \\{W_{t}, t\\in[0,1]\\}) + 4\\rho(\\delta) \\\\\n&\\le  2^{-n}  + 4\\rho(\\delta).\n\\end{align*}\nIn the final two lines line we used  \\eqref{e:Xhatdsn} and \\eqref{d19.3}.\n\n\nCombining the estimates above, since\n$P^0_{\\omega} \\left(  \\sup_{0\\le s \\le u} u^{-1\/2} |X^{n,2}_s| \\le \\delta \\right) \\geq 1-\\delta$ and \n$X^{(n)} = X^{n,1} + X^{n,2}$, Lemma \\ref{ma26.5} shows that\n\\begin{align*}\n&{d_P}( \\{(1\/a) X^{(n)}_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\\n&\\le\n{d_P}( \\{(1\/a) X^{(n)}_{ta^2}, t\\in[0,1]\\}, \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}) \\\\\n&\\qquad + {d_P}( \\{(1\/a) X^{n,1}_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\\\\n&\\le \n\\delta + 2^{-n}  + 5 \\rho(\\delta) + \\delta .\n\\end{align*}\nWe conclude that the lemma holds if we take $\\rho^*(\\delta) =  5\\rho(\\delta) + 2\\delta $.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{T:main}]\nChoose an arbitrarily small $\\varepsilon>0$. We will show that there exists $a_*$ such that for every $a\\geq a_*$, \n\\begin{align}\\label{d23.1}\n&{d_P}( \\{(1\/a) X_{ta^2}, t\\in[0,1]\\}, P_{\\text{BM}}) \\le \\varepsilon .\n\\end{align}\n\nRecall $\\rho^*$ from Lemma \\ref{d22.1}.\nLet $n_1 $ be such that $2^{-n_1}\\le \\varepsilon\/4$ and let $\\delta>0$ be so small that \n$2^{-n_1}+ \\rho^*(\\delta) < \\varepsilon\/2$. Let $n_2$ be defined as $n_1$ in Lemma \\ref{n9.1}, \nrelative to this $\\delta$. Then, according to Lemma \\ref{d22.1}, \n\\begin{align}\\label{d22.5}\n{d_P}( \\{(1\/a)X^{n}_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le 2^{-n} + \\rho^*(\\delta) < \\varepsilon\/2,\n\\end{align}\nfor all $n\\geq n_3 := n_1\\lor n_2$ and $a\\geq a_n$.\n\nFor a set $K$ let \n$\\mathcal{B}(K,r) = \\{z: \\dist(z,K) < r\\}$ and recall the definition of $D_n$ given in \\eqref{ma26.1}. Let \n\\begin{align*}\nF_1 &= \\{0\\in \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1))\\},\\\\\nF_2 &= \\{0\\notin \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1))\\}\n\\cap\n\\{\\exists t\\in[0, a_{n+1}^2]: X^{(n)}_t \\in  D_{n+1}\\},\\\\\nG_1^k &= \\{0\\in \\mathcal{B}( D_k, b_k\/k)\\},\n\\qquad k> n+1,\\\\\nG_2^k &= \\{0\\notin \\mathcal{B}( D_k, b_k\/k)\\}\n\\cap\n\\{\\exists t\\in[0, a_{n+1}^2]: X^{(n)}_t \\in  D_k\\}, \\qquad k> n+1.  \n\\end{align*}\n\nThe area of  $\\mathcal{B}(D_{n+1}, a_{n+1}\/\\log (n+1))$ is bounded by $c_1 (a_{n+1}\/\\log (n+1))^2$ so \n\\begin{align}\\label{d23.10}\n\\Pp(F_1) \\le c_1 (a_{n+1}\/\\log (n+1))^2\/ a_{n+1}^2 = c_1 \/\\log^2 (n+1).\n\\end{align}\nWe choose $n_4 > n_3 $ such that \n$c_1 \/\\log^2 (n+1) < \\varepsilon\/8$ for $n \\geq n_4$.\n\nNote that $D_{n+1}$ is a subset of a square with side $4\\beta_{n+1} \\leq 4 a_{n+1} n^{-1\/4}$. This easily implies that\nthere exists $n_5 \\geq n_4$ such that for $n\\geq n_5$,\n\\begin{align*\nP_{\\text{BM}}\\left(\\exists t\\in[0, a_{n+1}^2]: W(t) \\in  D_{n+1} \n\\mid \n0\\notin \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1))\n\\right) \\le \\varepsilon\/16.\n\\end{align*}\nWe can assume ($\\clubsuit$)  that $a_{n+1}\/a_n$ is so large that\nfor some $n_6 \\geq n_5$ and all $n\\geq n_6$,  \n\\begin{align}\\label{d23.11}\n\\Pp(F_2) &\\le \\Pp\\left(\\exists t\\in[0, a_{n+1}^2]: X^{(n)}_t \\in  D_{n+1} \n \\mid  0\\notin \\mathcal{B}( D_{n+1}, a_{n+1}\/\\log (n+1)) \\right)   \\\\\n &\\le \\varepsilon\/8.\n\\end{align}\n\nThe area of  $\\mathcal{B}( D_k, b_k\/k)$ is bounded by $c_2 b_k^2\/k$ so \n\\begin{align}\\label{d23.12}\n\\Pp(G_1^k) \\le (c_2 b_k^2\/k)\/ a_k^2 \\le c_3 ( b_k^2\/k)\/ (k b_k^2)= c_3 \/k^2.\n\\end{align}\nWe let $n_7 >n_6$ be so large that $\\sum_{k\\geq n_7} c_3 \/k^2 < \\varepsilon\/8$.\nFor all $k>n+1\\geq n_7+1$, we make $b_k\/k$ so large ($\\clubsuit$)  that \n\\begin{align}\\label{d23.13}\n\\Pp(G_2^k) \\le \n\\Pp\\left(\\sup_{t\\in[0, a_{n+1}^2]} |X^{n}_t| \\geq b_k\/k\\right) \\le c_3\/k^2.\n\\end{align}\n\nWe combine \\eqref{d23.10}, \\eqref{d23.11}, \\eqref{d23.12} and \\eqref{d23.13} to see that for $n\\geq n_7$,\n\\begin{align}\\label{d23.14}\n\\Pp&(\\exists t\\in[0, a_{n+1}^2] \\ \\exists k\\geq n+1: X^{(n)}_t \\in  D_k)\\\\\n&\\le\n\\Pp(F_1) +  \\Pp(F_2) + \\sum_{k>n+1}\n\\Pp(G_1^k) + \\sum_{k>n+1}\n\\Pp(G_2^k) \\nonumber\\\\\n& \\le \\varepsilon\/8 + \\varepsilon\/8 + \\varepsilon\/8 + \\varepsilon\/8 = \\varepsilon\/2.\n\\nonumber\n\\end{align}\n\nLet $R_{n+1} = \\inf\\{t\\geq 0: X_t \\in \\bigcup_{k\\geq n+1} \\mathcal{D}_k\\}$. \nIt is standard to construct $X$ and $X^{(n)}$ on a common probability space so that $X_t = X_t^n$ for all $t\\in [0, R_{n+1})$. This and \\eqref{d23.14}\nimply that for $n\\geq n_7$ and all $a\\in[a_n, a_{n+1}]$ we have\n\\begin{align*\nP(\\exists t\\in[ 0,1]: (1\/a)X_{ta^2} \\ne (1\/a)X^{(n)}_{ta^2}) \\le \\varepsilon\/2.\n\\end{align*}\nWe combine this with \\eqref{d22.5} to see that for all $a\\geq a_{n_6}$, \n\\begin{align*\n{d_P}( \\{(1\/a)X_{ta^2}, t\\in[ 0,1]\\}, P_{\\text{BM}}) \\le  \\varepsilon\/2 +\\varepsilon\/2 = \\varepsilon.\n\\end{align*}\nWe conclude that \\eqref{d23.1} holds with $a_* = a_{n_7}$. \n\nThis completes the proof of AFCLT. The WFCLT then follows from Theorem 2.13 of \\cite{BBT1}.\n\\end{proof}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nOne of the fundamental tasks of the Galactic studies is to\nestimate the structure parameters\nof the major structure components.\n\\citet{Bahcall1980} fit the observations with two structure components, namely a disk and a halo. \n\\citet{Gilmore1983}  introduce a third component, namely a thick disk, \nconfirmed in the earliest Besancon Galaxy Model \\citet{Creze1983}.\nSince then, various methods and observations have been adopted to\nestimate parameters of the thin and thick disks and of\nthe halo of our Galaxy. As the quantity and quality of data available continue to improve over the years,\nthe model parameters derived have become more precise, numerically. \nIronically, those numerically more precise results do not converge (see Table~1 of \n\\citealt{Chang2011}, Table~2 of \\citealt{Lopez2014} and Sect.~5 and 6 of \\citealt{Bland2016} for a review).  \nThe scatters in density law parameters, such as scale lengths,  \nscale heights and local densities of these Galactic components, \nas reported in the literature, are rather large.\nAt least parts of the discrepancies are caused by degeneracy of \nmodel parameters, which in turn, can be traced back to  the different \ndata sets adopted in the analyses. Those differing \ndata sets either probe different sky areas \n\\citep{Bilir2006a, Du2006, Cabrera2007, Ak2007, Yaz2010, Yaz2015}, \nare of different completeness magnitudes and therefore \nrefer to different limiting distances \\citep{Karaali2007},\nor of consist of stars of different populations of different absolute magnitudes \n\\citep{Karaali2004, Bilir2006b, Juric2008, Jia2014}.  \nIt should be noted that the analysis of \\citet{Bovy2012}, using the SEGUE spectroscopic survey, \nhas given a new insight on the thin and thick disk structural parameters. This analysis provides estimate \nof their scale height and scale height as a function of metallicity and alpha abundance ratio. However, \nit relies on incomplete data (since it is spectroscopic) with relatively low range of Galactocentric radius as \nfor the thin disk is concerned. \n\nA wider and deeper sample than those employed hitherto may help break the degeneracy \ninherent in a multi-parameter analysis and yield a globally representative Galactic model. \nA single or a few fields are insufficient to break the degeneracy. \nThe resulted best-fit parameters, while sufficient for the description of  \nthe lines of sight observed, may be unrepresentative of the entire Galaxy. For the latter purpose, \nsystematic surveys of deep limiting magnitude of all or a wide sky area, such as the \nTwo Micron All Sky Survey (2MASS; \\citealt{Skrutskie2006}), the Sloan Digital Sky Survey \n(SDSS; \\citealt{York2000}), the Panoramic Survey Telescope \\& Rapid Response System \n(Pan-Starrs; \\citealt{Kaiser2002}) and the GAIA mission \\citep{Perryman2001}, \nare always preferred.\n\nSeveral authors have studied the Galactic structure with 2MASS data at low \n\\citep{Lopez2002, Yaz2015} or high latitudes \\citep{Cabrera2005, Cabrera2007, Chang2011}. \n\\citet{Polido2013} uses the model from \\citet{Ortiz1993} and rederive the parameters of this model\nbased on the 2MASS star counts over the whole sky area. \nHowever, the survey depth of 2MASS is not quite enough to reach the outer disk and the halo.\nThe survey depth of SDSS is much deeper than that of the 2MASS. Many authors \n(e.g.  \\citealt{Chen2001, Bilir2006a, Bilir2008, Jia2014, Lopez2014})\nhave previously used the SDSS data to constrain the Galactic parameters. \nThose authors have only made use of a portion of the surveyed fields, \nat intermediate or high Galactic latitudes. \n\\citet{Juric2008} obtain Galactic model parameters from the stellar \nnumber density distribution of 48 million \nstars detected by the SDSS that sample distances from 100\\,pc to 20\\,kpc and cover \n6500\\,deg$^2$ of sky. Their results are amongst those mostly quoted. \nHowever, in their analysis, they have avoided the Galactic plane. \nSo the constraints of their results on the disks, especially the\nthin disk, are weak. In their analysis, \\citet{Juric2008} have also \nadopted photometric parallaxes assuming that all stars of the same colour\nhave the same metallicity. Clearly, (disk) stars in different parts of the Galaxy have quite different \n\\citep{Ivezic2008, Xiang2015, Huang2015} metallicities, and these variations in metallicities may \nwell lead to biases in  the model parameters derived. \n\n \n In order to provide a quality input catalog for the LAMOST Spectroscopic Survey of the Galactic Anticentre\n(LSS-GAC; \\citealt{Liu2014,Liu2015, Yuan2015}), \na multi-band CCD photometric survey of the Galactic \nAnticentre with the Xuyi 1.04\/1.20m Schmidt Telescope  \n(XSTPS-GAC; \\citealt{Zhang2013,Zhang2014,Liu2014}) \nhas been carried out. The XSTPS-GAC photometric catalog contains \nmore than 100 million stars in the direction of Galactic anticentre (GAC). It provides an excellent \ndata set to study the Galactic disk, its structures and substructures. \nIn this paper, we take the effort to constrain the Galactic model \nparameters by combining photometric \ndata from the XSTPS-GAC and SDSS surveys.\nThis is the third paper of a series on the Milky Way study based on the XSTPS-GAC data.  In \n\\citet{Chen2014}, we present a three dimensional extinction map in $r$ band. The map has a spatial \nangular resolution, depending on latitude, between 3 and 9\\,arcmin and covers the entire XSTPS-GAC \nsurvey area of over 6,000 deg$^2$ for Galactic longitude 140 $< l <$220\\,deg and latitude 40 $< b <$40\\,deg. \nIn \\citet{Chen2015}, we investigate the correlation between the extinction and the $\\rm H~{\\scriptstyle I}$~ and CO emission at intermediate \nand high Galactic latitudes ($|b| >$ 10\\degr) within the footprint of the XSTPS-GAC, on small and large scales. \nIn the current work we are interested in the global, smooth structure of the Galaxy. \n\nFor the Galactic structure, in addition to the global,  smooth major components,\nmany more (sub-)structures have been discovered, including the inner bars near the Galactic centre\n\\citep{Alves2000, Hammersley2000, vanLoon2003, Nishiyama2005, Cabrera2008, Robin2012},\nflares and warps of the (outer) disk \\citep{Lopez2002, Robin2003, Momany2006, Reyle2009, Lopez2014},\nand various overdensities in the halo and the outer disk, such as the Sagittarius Stream\n\\citep{Majewski2003}, the Triangulum-Andromeda \\citep{Rocha2004, Majewski2004} and\nVirgo \\citep{Juric2008} overdensities, the Monoceros ring \\citep{Newberg2002,Rocha2003}\nand the Anti-Center Stream \\citep{Rocha2003,Crane2003, Frinchaboy2004}.\nThey show the complexity of the Milky Way. \nRecently, \\citet{Widrow2012} and \\citet{Yanny2013} have \nfound evidence for a significant Galactic North-South \nasymmetry in the stellar number density distribution,  exhibiting some \nwavelike perturbations that seem to be intrinsic to the disk.\n\\citet{Xu2015} show that in the anticentre regions\nthere is an oscillating asymmetry in the main-sequence star counts \non either sides of the Galactic plane, in support of the prediction of \n\\citet{Ibata2003}. The asymmetry oscillates in the sense that there are more \nstars in the north, then in the south, then back in the north,\nand then back in the south  at distances of about 2, 4 -- 6, 8 -- 10 and 12 -- 16\\,kpc \nfrom the Sun, respectively.\n\nThe paper is structured as follows. The data are introduced in Section~2.\nWe describe our model and the analysis method in Section~3. Section~4 \npresents the results and discussions. In Section~5 we discuss the large \nscale excess\/deficiency  of star counts that reflect the substructures in the halo and disk.  \nFinally we give a summary in Section~6.\n\n\\section{Data}\n\n\\begin{table}\n \\centering\n  \\caption{Data sets.}\n  \\begin{tabular}{lcccc}\n  \\hline\n  \\hline\n & area & field size & $N_{\\rm fields}$& $r$ ranges \\\\\n  &  (deg$^2$)    & (deg $\\times$ deg)& & (mag) \\\\\n  \\hline\nXSTPS-GAC & $\\sim$3392 & 2.5$\\times$2.5 & 574 &12--18 \\\\\nXSTPS-M31\/M33 & $\\sim$588 & 2.5$\\times$2.5 & 108 & 12--18 \\\\\nSDSS & $\\sim$6871 & 3.0$\\times$3.0 & 1592 &15--21 \\\\\n   \\hline\n\\end{tabular}\\\\\n\\end{table}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{xstpsarmap.eps}\n  \\includegraphics[width=0.48\\textwidth]{sdssdenmap.eps}\n  \\includegraphics[width=0.48\\textwidth]{allfields.eps}\n  \\caption{{\\it Upper panel}: Extinction map of the GAC and M31\/M33 areas within the footprint of \nXSTPS-GAC (\\citealt{Chen2015} map for GAC area and \\citealt{Schlegel1998} map for M31\/M33 area). \nThe selected fields for GAC area and M31\/M33 area are marked as red and blue pluses,\nrespectively. The red star symbols mark the central positions of M31 and M33, respectively. {\\it Middle panel}: \nSDSS DR12 density map of stars in a magnitude bin of $r$ = 15.5 to 16.5\\,mag at a \nresolution of 0.1\\degr. The selected fields from SDSS are marked as red pluses.   {\\it Bottom panel}: \nLocation of the 682 fields selected from the XSTPS-GAC (red) and 1592 fields selected from the SDSS (blue) in \nGalactic coordinates.}\n  \\label{data}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{selcorr.eps}\n  \\caption{Colour-magnitude distributions of stars in all selected subfields from \nthe Sample\\,C, the re-weighted Sample\\,C (see Equation~(1) \nand related discussion), and the XSTPS-GAC.\nThe bottom three panels show the grey-scaled number densities distributed in the $g-i$ vs. $r$ space\nrespectively for the XSTPS-GAC (left), the re-weighted Sample\\,C (middle), and the XSTPS-GAC (right).\nThe upper three panels show the number distribution contours in the $g-i$ vs. $r$ space\n(left) as well as number distributions \nrespectively in $r$ (middle) and $g-i$ (right) for each sample.\nThe black contours and histograms show the density of all \ntargets in the XSTPS-GAC, the red ones represent the distributions of stars in \nSample\\,C and the blue ones display the \ndistributions for the re-weighted Sample\\,C. The contours labeled with `a', `b' and `c'\nin the left-upper panel represent the contour levels of star number \nof 6\\,000,  24\\,000 and 48\\,000, respectively.\nThe re-weighted Sample\\,C perfectly reproduces the colour-magnitude\nsampling provided by the XSTPS-GAC.}\n  \\label{sel}\n\\end{figure*}\n\n\\subsection{The XSTPS-GAC Data}\n\nThe XSTPS-GAC started collecting data in the fall of 2009 and completed in the spring of 2011.\nIt was carried out in order to provide input catalogue for the LSS-GAC.\nThe survey was performed in the SDSS $g$, $r$ and $i$ bands using the \nXuyi 1.04\/1.20\\,m Schmidt Telescope equipped with a 4k$\\times$4k CCD camera, \noperated by the Near Earth Objects Research Group of the Purple Mountain Observatory.\nThe CCD offers a field of view (FoV) of 1.94\\degr $\\times$ 1.94\\degr, with a pixel scale of 1.705\\,arcsec.\nIn total, the XSTPS-GAC archives approximately 100 million stars down to a\nlimiting magnitude of about 19 in $r$ band ($\\sim$ 10$\\sigma$) \nwith an astrometric accuracy about 0.1\\,arcsec\n and a global photometric accuracy of about 2\\% \\citep{Liu2014}.\n The total survey area of XSTPS-GAC is close to 7,000\\,deg$^2$,\ncovering an area of $\\sim$ 5,400\\,deg$^2$ centered on the GAC,\nfrom RA $\\sim$ 3 to 9\\,h and Dec $\\sim~-$10\\degr to $+60\\degr$, plus\nan extension of about 900\\,deg$^2$ to the M31\/M33 area and the bridging fields \nconnecting the two areas.\n\n\\subsubsection{GAC area}\n\nIn the direction of GAC, the $r$-band extinction exceeds 1\\,mag over a significant fraction of the sky \n(see Fig.~\\ref{data}). To correct the extinction of stars in high extinguish area \nusing extinction maps integrated over lines of sight, \nsuch as \\citet{Schlegel1998}, will introduce over corrections.\nIt will make stars too bright and blue. We select a subsample, the\nso-called ``Sample\\,C'' in \\citet{Chen2014}, from XSTPS-GAC.\nExtinction of all stars in Sample\\,C were calculated by the  spectral energy distribution (SED) \nfitting to the multi-band data, including the photometric data from the optical ($g,~r,~i$ from \nXSTPS-GAC) to the near-infrared ($J,~H,~K_S$ from 2MASS and $W1, ~W2$ from the Wide-field \nInfrared Survey Explorer, WISE, \\citealt{Wright2010}). \nThe extinction of targets in the subsample, Sample\\,C, \nis highly reliable, all having minimum SED fitting \n$\\chi ^2 _{min} ~<$ 2.0 (see \\citealt{Chen2014} for more details). \nWe correct the extinction of stars in Sample\\,C using the SED fitting extinction\nand the extinction law from \\citet{Yuan2013}. \nThere are more than 13 million stars in Sample\\,C. We divide them \ninto small subfields of roughly 2.5\\degr $\\times$ 2.5\\degr. The\nwidth ($\\Delta l$) and height ($\\Delta b$) of each subfield are always exactly 2.5\\degr.\nEach subfield is not exactly 6.25\\,deg$^2$ but varies with Galactic latitude $b$. \nBecause of the heavy extinction or poor observational conditions \n(large photometric errors), some subfields have \nobviously small amount of stars, comparing to most normal \nneighboring fields and thus be excluded. \nAs a result, 574 subfields, covering about 3392\\,deg$^2$, \nare selected. The locations of these subfields are shown in the top panel of \nFig.~\\ref{data}, with the grey-scale background image \nillustrating the 4\\,kpc extinction map from \\citet{Chen2015}.\n\nFor each subfield, Sample\\,C does not contain all stars in XSTPS-GAC.\nTo connect the distribution of targets in Sample\\,C\nto the underlying distribution of all stars, it is necessary to correct for the effects of the selection \n(often referred to as selection biases). Generally, the selection effects of Sample\\,C \nare due to the following two reasons: (1) the procedure by which we cross-match the photometric \ncatalogue of the XSTPS-GAC with those of 2MASS and WISE, and (2) the $\\chi^2$ cut when we\ndefine the sample with highly reliable extinction estimates.  For the first part, we lose about 15\\,per\\,cent \nobjects, mainly due to the limiting depths of 2MASS and WISE, \nespecially at low Galactic latitudes (see Fig.~1 of \\citealt{Chen2014}). \nFor the second part, we lose more than half of the objects, because of the large photometric errors, \nhigh extinction effects, or the special targets contamination, \nsuch as blended or binaries which are not well fitted by the standard SED \nlibrary in \\citet{Chen2014}. Our model for the selection function of Sample\\,C \ncan thus be expressed as the function of the positions ($l$, $b$), colour ($g-i$) \nand magnitude ($r$) of stars, given by,\\\\\n\\begin{equation}\n  S(l,b,g-i,r) = \\frac{N_{\\rm SC}(l,b,g-i,r)}{N_{\\rm XSTPS}(l,b,g-i,r)},\n\\end{equation}\nwhere $N_{\\rm SC}(l,b,g-i,r)$ and $N_{\\rm XSTPS}(l,b,g-i,r)$ are the number of stars \nin the Sample\\,C and the XSTPS-GAC, respectively. \nThe numbers of objects are evaluated within each subfield with area of $\\sim$ 6.25\\,deg$^2$, \neach colour ($g-i$) bin ranging from 0 to 3.0\\,mag with a bin-size of 0.1\\,mag, \nand each $r$-band magnitude bin ranging from 12 to 18.5\\,mag with a bin-size of 0.1\\,mag. \n\nThe number distributions in colour $(g-i)$ and magnitude $r$ for the stars\nin all selected subfields in the Sample\\,C, the Sample\\,C\nre-weighted by the selection effect, as well as the XSTPS-GAC,\nare shown as the density grey-scales and \ndensity contours and histograms in Fig.~\\ref{sel}. \nIt is clear that our correction of selection effect leads to perfect agreement between the \ncomplete XSTPS-GAC photometric sample and the re-weighted Sample\\,C.\n\n\\subsubsection{M31\/M33 area}\n\nThe dust extinction in the M31 and M33 area is much smaller, \ncompared with the GAC area (see the top panel of Fig.~\\ref{data}). \nWe adopt the extinction map from \\citet{Schlegel1998} \nand the extinction law from \\citet{Yuan2013} \nto correct the extinction of stars in M31\/M33 area. \nSimilarly as in the GAC area, all stars in M31\/M33 area are divided into small subfields, \nwhich have width ($\\Delta l$) and height ($\\Delta b$) always of 2.5\\degr. \nWe exclude the subfields which have maximum $E(B-V)$ larger \nthan 0.15\\,mag (i.e. $A_r$ = 0.4\\,mag, according to the extinction law from \\citealt{Yuan2013}), \nto avoid the relatively large uncertainties caused \nby the high extinction in the highly extinguished regions. The subfields that cover M31 are also \nexcluded. As a result, there are 108 subfields in the\nM31\/M33 area, covering about 588\\,deg$^2$. The locations of these subfields \nare also plotted in the top panel of Fig.~\\ref{data}, with \nthe grey-scale background image illustrating the extinction map from \\citet{Schlegel1998}. \nConsidering the limiting magnitude of XSTPS-GAC ($r$ $\\sim$ 19\\,mag), \nwe claim that the data in the M31\/M33 area from XSTPS-GAC is complete in the magnitude \nrange $12 < r_0 < 18$\\,mag. \n\n\\subsection{The SDSS Data}\n\nAs the survey area of XSTPS-GAC mainly locate around the low Galactic latitudes, \nwe also use the photometric data from SDSS, for constraining better the outer disk and \nthe halo. We use the photometric data from SDSS data release 12 (DR12, \\citealt{Alam2015}). \nThe SDSS surveys mainly for high Galactic latitudes, with only a few stripes \ncrossing the Galactic plane. It complements one another with the XSTPS-GAC. We cut the \nSDSS data with Galactic latitude $|b| >$ 30\\degr, where the influence of the dust \nextinction is small. The dust extinction are corrected using the extinction \nmap from \\citet{Schlegel1998} and the extinction law from \\citet{Yuan2013}.\nThe SDSS data are divided into subfields with\nwidth ($\\Delta l$) and height ($\\Delta b$) always of 3\\degr. \nTo make sure that each subfield is fully sampled by the SDSS survey, we further\ndivide each subfield into smaller pixels (of size 0.1\\degr $\\times$ 0.1\\degr) and \nexclude the subfield which has no stars detected in at least one of the smaller pixels. \nAs a result we have obtained 1592 subfields, covering a sky area of \nabout 6871\\,deg$^2$. In the middle panel of Fig.~\\ref{data}, we show the \nspatial distributions of these subfields,\nwith grey-scale background image illustrating the number density of the SDSS data.\nTo remove the contaminations of hot white dwarfs, low-redshift quasars and \nwhite dwarf\/red dwarf unresolved binaries from the SDSS sample, we\nreject objects at distances larger than 0.3 mag from the $(r-i)_0$ vs. $(g-r)_0$ \nstellar loci \\citep{Juric2008, Chen2014}.\nThe 95 per\\,cent completeness limits of the SDSS images are $u$, $g$, $r$, $i$ and $z$ $=$\n 22.0, 22.2, 22.2, 21.3 and 20.5\\,mag, respectively \\citep{Abazajian2004}. Thus \nthe SDSS data is complete in the magnitude range of $15 < r_0 < 21$\\,mag. \n\nA brief summary of the data selection in the current work is given in Table~1. In total, \nthere are 2274 subfields, covering nearly 11,000\\,deg$^2$,\nwhich is more than a quarter of the whole sky area. \nThe positions of all the subfields, from both the XSTPS-GAC and the\nSDSS, are plotted in the bottom panel of \nFig.~\\ref{data}. They cover the whole range of Galactic latitudes.\nGenerally,  the XSTPS-GAC provides nice constraints of the Galactic disk(s), \nespecially for the thin disk, while the SDSS provides us a good opportunity to \nrefine the structure of Galactic halo, as well as the outer disk.   \n \n\\section{The Method}\n\n\\subsection{The Galactic model}\n\nWe adopt a three-components model for the smooth stellar distribution of the\nMilky Way. It comprises two exponential disks (the thin disk and the thick disk) \nand a two-axial power-law ellipsoid halo \\citep{Bahcall1980, Gilmore1983}. \nThus the overall stellar density $n(R,Z)$ at a location $(R,Z)$ can be decomposed\nby the sum of the thin disk, the thick disk and the halo,\n\\begin{equation}\n  n(R,Z)=D_1(R,Z)+D_2(R,Z)+H(R,Z),\n\\end{equation}\nwhere $R$ is the Galactocentric distance in the Galactic plane, $Z$ is the\ndistance from the Galactic mid-plane.\n$D_1$ and $D_2$ are stellar densities of the thin disk and the thick disk,\n\\begin{equation}\n  D_i(R,Z)=f_{i}\\,n_{0}\\exp\\left[-\\,{(R-R_\\odot)\\over L_{i}}-\\,{(|Z|-Z_\\odot)\\over H_{i}}\\right],\n\\end{equation}\nwhere the suffix $i=1$ and $2$ stands for the thin disk and thick disk, respectively. \n$R_\\odot$ is the radial distance of the Sun to the Galactic centre on the plane, \n$Z_\\odot$ is the vertical distance of the Sun from the plane, \n$n_0$ is the local stellar number density of the thin disk at ($R_\\odot$, $Z_\\odot$),\n$f_i$ is the density ratio to the thin disk ($f_1$=1),\n$L_{i}$ is the scale-length and $H_{i}$ is the scale-height. \nWe adopt $R_\\odot=8$\\,kpc  \\citep{Reid1993} and\n$Z_\\odot = 25$\\,pc \\citep{Juric2008} in the current work.\n$H$ is the stellar density of the halo,\n\\begin{equation}\n H(R,Z)=f_{h}\\,n_{0}\\left[R^2+(Z\/\\kappa)^2\\over R_\\odot^2+(Z_\\odot\/\\kappa)^2\\right]^{-p\/2},\n\\end{equation}\nwhere $\\kappa$ is the axis ratio, $p$ is the power index and $f_h$ is the halo normalization \nrelative to the thin disk.\n\n\\subsection{Halo fit}\n\n\\begin{table}\n \\centering\n  \\caption{The parameter space and results of the halo fit}\n  \\begin{tabular}{lcccc}\n  \\hline\n  \\hline\nParameters & Range & Grid size & Best value  & Uncertainty \\\\\n  \\hline\n$\\kappa$  & 0.1--1.0 & 0.01  & 0.65 & 0.05 \\\\\n$p$  & 2.3--3.3 & 0.01 & 2.79 & 0.17 \\\\\n   \\hline\n\\end{tabular}\\\\\n\\end{table}\n\nWe fit the component of the halo first.\nThe metallicity distribution of the halo stars can be described as a single \nGaussian component, with a median halo metallicity of $\\mu_{\\rm H}$=$-$1.46\\,dex and spatially\ninvariant of $\\sigma_{\\rm H}$=0.30\\,dex \\citep{Ivezic2008}. \nWe assume the metallicity of all halo stars as  [Fe\/H]$=-1.46$\\,dex and adopt the\nphotometric parallax relation from \\citet{Ivezic2008},\n\\begin{equation}\n\\begin{split}\nM_{r}  = &  4.50-1.11{\\rm [Fe\/H]}-0.18{\\rm [Fe\/H]}^2 \\\\\n             &  -5.06+14.32(g-i)_0-12.97(g-i)_0^2 \\\\\n             & + 6.127(g-i)_0^3 - 1.267(g-i)_0^4+0.0967 (g-i)_0 ^5.\n\\end{split}\n\\end{equation}\nThe distances of the halo stars can thus be calculated from the standard relation, \n\\begin{equation}\n  d=10^{0.2(r_0-M_r)+1}.\n\\end{equation}\n\nStar in a blue colour bin $0.5 \\le g-i < 0.6$ are selected. They do not suffer \nfrom the giant star contamination and probe larger distances to constrain the halo.\nWe calculate their distance using Equations~(5) and (6). The distances of the disk stars will be\nunderestimated because they are more metal-rich. To exclude the contamination of the disk stars, \nwe use stars with absolute distance to the Galactic plane \n$|Z| >$ 4\\,kpc. For each subfield, we divide all halo stars\ninto suitable numbers of logarithmic distance bins and then count the \nnumber for each bin. This number can be modelled as,\n\\begin{equation}\nN_{\\rm H}(d)=H(d)\\Delta V(d),\n\\end{equation}\nwhere $H(d)$ is the halo stellar density given by Equation~(4) and  \n$\\Delta V(d)$ is the volume, given by,\n\\begin{equation}\n\\Delta V(d) =\\frac{\\omega}{3}(\\frac{\\pi}{180})^2(d^3_2-d^3_1),  \n\\end{equation}\nwhere $\\omega$ denotes the  area of the field (unit in deg$^2$), $d_1$ and $d_2$ are\nthe lower distance limit and upper distance limit of the bin, respectively. \n\nWe fit the halo model parameters $p$ and $\\kappa$ to the data.\nAs we explicitly exclude the disk, we cannot fit for \nthe halo-to-thin disk normalization $f_h$.\nA maximum likelihood technique is adopted to explore the best \nvalues of those halo model parameters.\nIn Table~2, we list the searching parameter space and the grid size. \nFor each set of parameters, a reduced likelihood is computed between the simulated \ndata (star counts in bins of distances) and the observations,\ngiven by \\citet{Bienayme1987} and \\citet{Robin2014},\n\\begin{equation}\n  Lr=\\sum_{i=1}^{N} q_i \\times (1-R_i+{\\rm ln}(R_i)),\n\\end{equation}\nwhere $Lr$ is the reduced likelihood for a binomial statistics, \n$i$ is the index of each distance bin, $f_i$ and $q_i$ are the \nnumber of stars in the $i$th bin for the model \nand the data, respectively and $R_i=f_i\/q_i$. \nThe uncertainties of the halo parameters are estimated similarly as those in \\citet{Chang2011}.\nWe calculate the likelihood for 1000 times using the observed data  and the \nsimulations of the best-fit model adding with the Poisson noises. \nThe resulted likelihood range defines the confidence level and thus the uncertainties.  \n\n\\subsection{Disk fit}\n\nThe metallicity distribution of the disk is more complicated than that of the halo. \nThus we fit the disk model parameters through a different way.\nWe compare the $r$-band differential star counts in different colour bins and compare \nthem to the simulations to search for the best disk model parameters\n($n_0, ~L_1,~H_1,~f_2,~L_2$ and $H_2$), as well \nas the halo-to-thin disk normalization $f_h$.\n\nTowards a subfield of galactic coordinates ($l,~b$) \nand solid angle $\\omega$, the $r$-band differential star counts $N_{\\rm sim} (r^k_0)$ \n($k$ is the index of each magnitude bin) in a given colour bin $(g-i)^j_{0}$ \n($j$ is the index of each colour bin) can be simulated as follows:\n\\begin{enumerate}\n\\item The line of sight is divided into many small distance bins. For a given distance \nbin with centre distance of $d_i$ ($i$ is the index of each distance bin),\nthe $r$-band apparent magnitude of a star is given by \n\\begin{equation}\n  r_0(d_i) = M_r ((g-i)^j_0, {\\rm [Fe\/H]}|l,b,d_i)+\\mu,\n\\end{equation}\nwhere $\\mu$ is the distance modulus [$\\mu = 5{\\rm log}_{10}(d_i)-5$] and $M_r$\nis the $r$-band absolute magnitude of the star given by Equation (5). The metallicities \nof halo stars are again assumed to be $-$1.46\\,dex and those of disk stars are \ngiven as a function of positions, which is fitted using the metallicities of \nmain sequence turn off stars from LSS-GAC \\citep{Xiang2015},\n\\begin{equation}\n {\\rm  [Fe\/H]} = -0.61+0.51 \\cdot {\\rm exp}{(-|Z|\/1.57)}.\n\\end{equation}\n\n\\item The number of stars in each distance bin can be calculated by,\n\\begin{equation}\n  N(d_i)=n(R,Z|l,b,d_i)V(d_i), \n\\end{equation}\nwhere $V(d_i)$ is the volume given by Equation~(8) and $n(R,Z|l,b,d_i)$\nis the stellar number density given by Equation~(2, 3 and 4). The halo model parameters,\n$\\kappa$ and $p$, resulted from the halo fit are adopted and settled to be not changeable here.\n\n\\item Combining all distance bins, \nwe can obtain the modeled $r$-band star counts $N(r^k_0)$, by\n\\begin{equation}\n  N(r^k_0) = \\Sigma N(d_i) ~  {\\rm where} ~ r^k_0 - \\frac{\\rm rbin}{2} < r_0(d_i) < r^k_0 + \\frac{\\rm rbin}{2},\n\\end{equation}\nwhere rbin is the bin size of $r$-band magnitude (we adopt rbin=1\\,mag in the current work).\n$N(r^k_0)$ is the underlying star counts. When comparing to the observations, we \nneed to apply the selection function, by\n\\begin{equation}\n  N_{\\rm sim}(r^k_0) = N(r^k_0)S(l,b,g-i,r) C,\n\\end{equation}\nwhere $S(l,b,g-i,r)$ is the selection function, calculated by Equation~(1) for \nXSTPS-GAC subfields in GAC area and equals to one for \nXSTPS-GAC subfields in M31\/M33 area and all the SDSS subfields. Besides,\n \\begin{equation}\n   C =\n\\begin{cases}\n1       & {\\rm for~} d_{\\rm min} < d_i < d_{\\rm max}; \\\\\n0\t\t&  {\\rm otherwise};  \\\\\n\\end{cases}\n\\end{equation}\n\\begin{eqnarray}\n  d_{\\rm min} &=&  10^{0.2(r_{\\rm min}-A_r(d_i)-M_r((g-i)_0,{\\rm [Fe\/H]}))+1},\\\\\n  d_{\\rm max}&=&10^{0.2(r_{\\rm max}-A_r(d_i)-M_r((g-i)_0,{\\rm [Fe\/H]}))+1},\n\\end{eqnarray}\nwhere $r_{\\rm min}$ and $r_{\\rm max}$ are the \nmagnitude limits of each subfield.\nWe adopt $r_{\\rm min} = 12$ and  $r_{\\rm max} = 18$ for all XSTPS-GAC subfields,\nand $r_{\\rm min} = 15$ and  $r_{\\rm max} = 21$ for all SDSS subfields. \n$A_r(d_i)$ is the extinction in $r$-band at distance of $d_i$. We adopt the 3D extinction map from\n\\citet{Chen2014} for XSTPS-GAC subfields in GAC area and \n2D extinction map from \\citet{Schlegel1998} for\nXSTPS-GAC subfields in M31\/M33 area and all SDSS subfields. As the size of each subfield is quite\nlarge ($\\sim$ 4\\,deg$^2$), the extinction $A_r(d)$ varies within a subfield. We thus adopt the maximum \nvalues to make sure that our data are complete.\n\\end{enumerate}\n\nThe photometric parallax relation of Equation~(5) is only valid for the single \nstars. A large fraction of stars in the Milky Way are\nactually binaries (e.g. \\citealt{Yuan2015b}). In the current work we adopt the binary fraction \nresulted from \\citet{Yuan2015b} and assume that 40\\,per\\,cent of\nthe stars are binaries. The absolute magnitudes $M_r$ of the binaries are calculated as the same way\nas in \\citet{Yuan2015b}. \n\nWe also consider the effects of photometric errors, \nthe dispersion of disk star metallicities and the errors due to\nthe photometric parallax relation of \\citet{Ivezic2008}. The $r$-band photometric errors\nof most stars in the XSTPS-GAC and the SDSS are  smaller than 0.05\\,mag \n(\\citealt{Chen2014} for the XSTPS-GAC and \\citealt{Sesar2006} for the SDSS).\nWhen we fit the metallicities of disk stars as a function of positions [Equation~(11)], we \nfind a dispersion of the residuals of about 0.05\\,dex. \nAccording to Equation~(5), this dispersion \nwould introduce an offset of about 0.05\\,mag for \nthe absolute magnitude when [Fe\/H] = $-$0.2\\,dex. \nAs a result, the effect of the photometric errors and the\ndisk stars metallicities dispersions would \nintroduce a distance errors of smaller than 5\\,per\\,cent.\nCombining with the systematic error of the photometric parallax \nrelation, which is claimed to be smaller than 10\\,per\\,cent \\citep{Ivezic2008},\nwe assume a total error of distance of 15\\,per\\,cent. This distance error is added \nwhen we model the $r$-band\nmagnitude of stars in a given distance bin [Equation~(10)].\n\nWe select three different colour bins for the disk fit. \nTwo of them correspond to G-type stars with \n$0.5 \\le (g-i)_0 < 0.6$\\,mag and $0.6 \\le  (g-i)_0 < 0.7$\\,mag,\nand the other one corresponds to late K-type stars with $1.5 \\le (g-i)_0 < 1.6$\\,mag. \nThe giant and sub-giant contaminations for the first two G-type star bins are very small. \nFor the late K-type stars, we exclude stars with $r$-band magnitude $r_0 < 15$\\,mag \nto avoid the giant contaminations. For each colour bin, we count the differential\n$r$-band star counts with a binsize of $\\Delta r=1$\\,mag and then \ncompare them to the simulations to search for the \nbest disk model parameters, i.e $n_0,~L_1,~H_1,~f_2,~L_2,$ and $H_2$ and\nthe halo-to-thin disk normalization $f_h$. \nSimilarly as in \\citet{Robin2014}, \nan ABC-MCMC algorithm is implemented using the reduced likelihood calculated by Equation~(9) \nin the Metropolis-Hastings algorithm acceptance ratio \n\\citep{Metropolis1953, Hastings1970}. \nWe note that the 68\\,per\\,cent probability intervals of the \nmarginalised probability distribution functions (PDFs) of each parameter, given\nby the accepted values after post-burn period in the MCMC chain are only the fitting uncertainties \nwhich do not include systematic uncertainties.\nA detailed analysis of errors of the scale parameters will be given in Sect.~4.2.\n\nThe stellar flare is becoming significant at $R \\ge$15\\,kpc \\citep{Lopez2014} \nwhile the limiting magnitude we adopt for XSTPS-GAC is $r=18$\\,mag, which corresponds\nto $R ~\\sim$ 13\\,kpc for early G-type dwarfs.  On the other hand, the disk warp is \na second order effect on the star counts and the XSTPS-GAC centre around the \nGAC, with $l$ around 180\\degr. The effect of the disk warp is thus\nnegligible \\citep{Lopez2002}.  So in the current work we ignore the influences of the disk warps and flares. \nIn order to minimise the effects coming from other irregular structures (overdensities)\nof the Galactic disk and halo (e.g., Virgo overdensity, etc. ), we iterate our fitting \nprocedure to automatically and gradually \nremove pixels contaminated by unidentified irregular structures, similarly as in \\citet{Juric2008}. \nThe model is initially fitted using all the data points.\nThe resulted best-fit model is then used to define the outlying data, \nwhich have ratios of residuals (data minus the best-fit model) to the best-fit model\nhigher than a given value, i.e. $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod} > a_1$. \nThe model is then refitted with the outliers excluded. \nThe newly derived best-fit model is again compared to all the data points. \nNew outliers with $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod} > a_2$\nare excluded for the next fit. We repeat this procedure with a sequence of values\n$a_i$ = 0.5, 0.4, and 0.3.  The iteration, which gradually reject about 1, 5 and 15\\,per\\,cent of\nthe irregular data points of smaller and smaller significance, will make our model-fitting algorithm to \nconverge toward a robust solution which describes the smooth background best. \n\n\\section{The Results and Discussion}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{halogrid.eps}\n  \\caption{Reduced likelihood surface of the halo parameters $p$ and $\\kappa$ \n  space (see Table~2). The best-fitted values and uncertainties are marked \n  as a red plus with error bars. \n  The red contour ellipse shows the likelihood ranges used for estimating \n  the uncertainties.}\n  \\label{halog}\n\\end{figure}\n\n\n\\begin{table*}\n \\centering\n  \\caption{The best-fit values of the disk fit}\n  \\begin{tabular}{lcccccccc}\n  \\hline\n  \\hline\n  Bin & $n_1$ & $L_1$ & $H_1$ & $f_2$ & $L_2$ & $H_2$ & $f_H$ & $Lr$  \\\\\n         & $10^{-3}$stars\\,$pc^{-3}$ & pc & pc & per\\,cent & pc & pc & per\\,cent &  \\\\\n\\hline\nJoint fit\\\\\n\\hline\n $0.5 \\le (g-i)_0 < 0.6$    &  1.25 & 2343 & 322 & 11 & 3638 & 794 & 0.16  & $-$86769   \\\\\n    $0.6 \\le (g-i)_0 < 0.7$         &  1.20 & & & & & & &   \\\\\n       $1.5 \\le (g-i)_0 < 1.6$           &  0.54 & & & & &  & & \\\\ \n        \\hline    \n    Individual fit\\\\\n    \\hline\n $0.5 \\le (g-i)_0 < 0.6$ & 1.31 & 1737 & 321  & 14   \n                               & 3581& 731  & 0.16   & $-$43699   \\\\     \n $0.6 \\le (g-i)_0< 0.7$ & 1.65  & 2350  & 284  & 7  \n                               & 3699  & 798  & 0.12   & $-$35774  \\\\        \n $1.5 \\le (g-i)_0 < 1.6$ & 0.41  & 2780  & 359  & 8 \n                               & 2926  & 1014  & 0.50 & $-$4028   \\\\\n \\hline\n stddev & & 429 & 31 & 3 & 360 & 124 & 0.02 &  \\\\\n \\hline \n\\end{tabular}\n\\end{table*}     \n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.95\\textwidth]{cross2d.eps}\n  \\caption{\n  Two-dimensional marginalized PDFs for the disk model parameters,\n$L_1,~H_1,~f_2,~L_2,$ and $H_2$\nand the halo-to-thin disk normalization $f_h$, obtained from the MCMC analysis. \nHistograms on top of each column show the one-dimensional marginalized PDFs \nof each parameter labeled at the bottom of the column. \nRed pluses and lines indicate the best solutions. \nThe dash lines give the 16th and 84th percentiles, \nwhich denotes only the fitting uncertainties.}\n  \\label{cross2d}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.68\\textwidth]{rntestmodel.eps}\n  \\caption{Star count (per deg$^2$) for the colour bin $0.5 \\le (g-i)_0 <0.6$\\,mag and magnitude bins,\n   $r_0$ = 15 (left) and 16\\,mag (right), of both the XSTPS-GAC\n  (red pluses) and the SDSS (blue pluses) data as a function of the Galactic latitude\nfor example subfields with Galactic longitude 177\\degr  $<l<$ 183\\degr.\nThe black dots are the model predictions.}\n  \\label{rnmodel}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.7\\textwidth]{datamode.eps}\n  \\caption{Difference map for the all subfields \n  $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod}$. \n  The populations in excess in the data are most \n  likely irregular structures of the Milky Way.\nThe squares denote different regions with significant excesses of the\nstar counts.}\n  \\label{resd}\n\\end{figure*}\n\n\\subsection{Fitting results}\n\nThe best-fit halo model parameters obtained from the halo fit are listed in Table~2 and the\nreduced likelihood $L_r$ surface for the fit is shown in Fig.~\\ref{halog}.\nThe $L_r$ contour dramatically changes along $\\kappa$, \nbut relatively mildly along $p$. The best-fit halo model parameters are \n$\\kappa=0.65\\pm 0.05$ and $p=2.79\\pm 0.17$. They are not \nsurprisingly in good agreement with those values from \\citet{Juric2008}, \nwhich have $\\kappa=0.64$ and $p=2.8$, since the data used for the halo fit in the current work\nare mainly from the SDSS subfields. In addition, the photometric parallax relation that\n\\citet{Juric2008} used for the blue stars was corrected for low-metallicities \n([Fe\/H] $\\sim$ $-$1.5\\,dex), which is of minor difference from \nthe photometric parallax we used for the halo stars.\n\nThe best-fit values for the disk fit are listed in Table~3.\nWe performed the disk fit using first jointly all three colour bins, then separately for each colour bin.\nFor the joint fit case, all the model parameters, excepted for \nthe local density ($n_0$), are settled to be the same for stars in every bin,\nas we expect a universal density profile for all stellar populations \\citep{Juric2008}. \nAll the parameters resulted from the joint fit appear to be very well constrained.\nWe explore the correlations between different parameter pairs in Fig.~\\ref{cross2d}. \nThe Figure shows the marginalized one- and \ntwo-dimensional PDFs of the model parameters. \nThe correlations between different parameters are rather weak in general. \nWe identify small correlations \nbetween several pairs of the parameters, such as ($H_1$, $H_2$), ($H_1$, $f_2$) \nand ($l_1$, $f_h$), and a strong degeneracy between the scale height and\nthe local normalised densities ratio of thick disk ($H_2$, $f_2$).\n\nFor the separate fits, we fit all parameters independently for each color bin. \nThe individual fits can be served as a consistency check of our method.\nFrom Table~3, we can declare that those best-fit solutions \nare generally consistent. The thin disk scale height varies around\n$H_1 ~ \\sim$ 320 pc, by $\\pm 40$\\,pc, which is \nconsistent with the result from the  joint fit.\nThe variations of the disk scale lengths are relatively large, with \n$L_1~\\sim$ 1.7 $-$ 2.8\\,kpc and $L_2~\\sim$ 2.9 -- 3.7\\,kpc for the\nthin disk and the thick disk, respectively. \nThe large variations could be due to the limited ranges of Galactocentric distances \non the Galactic plane for stars from both the XSTPS-GAC (for the limited depth) and \nthe SDSS (for the poor sky coverage in low Galactic latitudes). \nData with deeper depth and better sky coverage in the low Galactic latitudes\n (such as Pan-Starrs) may help to improve the situation.\nThe thick disk normalization $f_2$ and scale height $H_2$\nappear also weakly constrained, with $f_2$ ranging from 7 to 14\\,per\\,cent and\n$H_2$ from 730 to 1000\\,pc. This is mainly due to the strong\ncorrelation between these two parameters (see \nthe  $f_2$ vs. $H_2$ panel in Fig.~\\ref{cross2d}).  The halo normalisation $f_h$\nis well constrained to $\\sim$ 0.15\\,per\\,cent for the two G-type star bins. While\nthe value of $f_h$ is abnormally large for the late K-type star bin [1.5$ \\le (g-i)_0 < $1.6]. \nThis is mainly due to the fact that late K dwarfs in the halo are cut by our limiting magnitude ($r_0<$21\\,mag).\n\nWe plot in Fig.~\\ref{rnmodel} the star counts in the colour bin $0.5 \\le (g-i)_0 < 0.6$\\,mag\nand magnitude bins, $r_0=$15 and 16\\,mag of both the XSTPS-GAC \nand the SDSS data as a function of the Galactic latitude for example subfields \nwith Galactic longitude 177\\degr $ < l < $ 183\\degr. \nThe best-fit model is in good agreement with the observations,\nwith some small deviations.\nIn Fig.~\\ref{resd}, we show the differences between the observed star counts, integrated from stars\nin all the three colour bins and $r_0$ from 12 to 21\\,mag, and \nthe model predictions as a function of position on the sky. \nDifferent colours in the Figure indicate the values of the\nratio $(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod}$. For most \nof the fields, we do not see obvious deviation,\nwith residual smaller than 10\\,per\\,cent, i.e. \n$|(N_{\\rm obs}-N_{\\rm mod})\/N_{\\rm mod}| < 0.1$.\n\n\n\\subsection{Systematics}\n\nThe dispersions of the resultant parameters from both the jointly fit and individual fit \nare also listed in Table.~3, which could be used to denote the \nsystematic errors of the corresponding parameters. \nThe typical errors of the parameters are about 10\\,per\\,cent.  \nSome of the parameters, i.e. the thin disk scale length $H_1$ and the thick disk \nlocal normalised densities ratio $f_2$, have relatively larger uncertainties, which are mainly due\nto the limits of our data and the degeneracies between different parameters. \nOther dominant sources of the errors are (in order of decreasing importance):\n1) the systematic distance determination uncertainties, \n2) the misidentification of binaries as single stars, \n3) the value of distance error, i.e., finite width of the photometric parallax relation,  \n4) the contamination of non-dwarf stars, and \n5) the effects from disk warp and flare. Finally, the structure parameters for different \nstellar populations (colour bins) would be intrinsically different, which may also contribute to \nthe dispersions.\n\nDue to the absolute calibration errors of the photometric parallax relation, the distances of stars \ncould be systematically over- or under- estimated. To check this effects, we\nredo the fit by changing the distance scale by \udbff\udc0e15\\,per\\,cent, i.e., \nusing the parallax relations 0.3\\,mag brighter and 0.3\\,mag fainter than Equation~(5).\nThe relative differences between \nthe original resultant parameters and those derived after \nchanging the distance scale by 15\\,per\\,cent are about 10 -- 15\\,per\\,cent.\n\nComparing to the single stars of the same colour, the binaries are \nbrighter, i.e. of smaller absolute magnitudes. Thus if one model  \nthe Galaxy with no or less fraction of binaries, the resultant \nmodel would be more `compact' than it truly is. \nIn the current work we select a binary fraction $f_b$=40\\,per\\,cent, \nwhich is an average binary fraction \nfor field FGK stars \\citep{Yuan2015b}. To check the possible effects \nof the binary fraction, we have tried two extreme \nvalues, the lowest value, $f_b$=0, assuming that all stars are \nsingle stars and the highest value, $f_b$=1, assuming that all stars\nare binaries, to redo the fit. \nThe relative differences between \nthe original resultant parameters and those derived after \nchanging the binary fraction to the extreme values are about 10\\,per\\,cent. \n\nWhen simulating the star counts for each colour bin, we assumed an error of distances\nof 15\\,per\\,cent, which includes the effect of photometric \nuncertainties, dispersion of metallicities of disk stars and  \nuncertainties of the \\citet{Ivezic2008} photometric parallax \nrelation. We redo the fit by changing the \ndistance dispersion to 0 and 30\\,per\\,cent, respectively. \nThe results show that changing the uncertainties of \ndistance do not introduce any significant bias in the derived Galactic model parameters.     \n\nThe effects for the model parameters caused by the non-dwarf (i.e. subgiants and giants) contaminations\nare similar as the binaries. From the Galaxy stellar population synthesis models, BESANCON\n\\citep{Robin2003} and TRILEGAL \\citep{Girardi2005}, we find that the fraction of giants in the three \nchosen colour bins is no more than 1\\,per\\,cent and the fraction \nof sub-giants is less than 5\\,per\\,cent. Thus the systematics \ncaused by the giant and sub-giant contaminations are likely to be negligible. \n\nIn the current work we have ignored the effects of disk flare, as we assume that the stellar flare\nis becoming significant at further distances \\citep{Lopez2014}. While \\citet{Derriere2001} and \nAmores et al. (2016, submitted) find that the flare starts at about 9 to 10\\,kpc. Having a shorter\nstart of the flare could have an impact on the thin ansd thick disk scale lengths that we determine, \nwhich would make the values to be underestimated. \n   \n\\subsection{Comparisons with other work}\n\nAs stated in Section~4.1, our results of the halo model parameters are very similar\nas those derived from \\citet{Juric2008}, because of the similar method (stellar \nnumber density fits) and data (the SDSS) adopted in both work. \nThe main differences in the halo fit between those from \\citet{Juric2008}\nand our work are that they adopt the $\\chi^2$ fitting while we use the reduced\nlikelihood $L_r$; and they calculate the number densities \nfor the entire sample in $(R,~Z)$ space while we do that separately for each line of sight. \nThe result in our work confirms those from \\citet{Juric2008}.\n\nWhen constraining the disk model parameters,\nour method and data are both quite different from those in \\citet{Juric2008}\nbut the results are in agreement at a level of about 10\\,per\\,cent.\nHowever, \\citet{Juric2008} admit that their result suffers large uncertainties\nfrom the uncertainty in calibration of the photometric parallax relation and the poor sky coverages \nfor the low Galactic latitudes, which is not the case in our work.\nGenerally, our derived value of thin disk scale height, 322\\,pc, is in the range of values, \n150--360\\,pc, which resulted from the recent work by \\citet{Bilir2008, Juric2008, \nYaz2010, Chang2011, Polido2013, Jia2014} and \\citet{Lopez2014}. \nSpecially, this value is in good agreement with the canonical value of 325\\,pc \n\\citep{Gilmore1984, Yoshii1987, Reid1993, Larsen1996}. Our derived value of the\nthin disk scale length, 2.3\\,kpc, is \nconsistent with those results found by \\citet{Ojha1996, Robin2000, Chen2001, Siegel2002, Karaali2007, \nJuric2008, Robin2012, Polido2013, Lopez2014} and \\citet{Yaz2015}, which ranges between 2 and 3\\,kpc. \nWe find a local thick disk normalization of 11 per\\,cent.  In the different colour bins, \nthis value varies between 7 and 14\\,per\\,cent, probably because of the degeneracy with the \nscale height. It is well in agreement with the values in the literature, which ranges between 7 and 13\\,per\\,cent \\citep{Chen2001, Siegel2002, Cabrera2005, Juric2008, Chang2011, Jia2014}. \nThe range of values for the thick disk scale height and scale length from the recent literature \nare respectively 600 -- 1000\\,pc and 3 -- 5\\,kpc \\citep{Bilir2008, Juric2008, \nYaz2010, Chang2011, Polido2013, Jia2014, Lopez2014, Robin2014}. The results deduced here, \nwhich have thick disk scale height of $\\sim$800\\,pc and scale length of 3.6\\,kpc, are both in the\nmiddle of the ranges of those values reported in the literature. Notice that there is a significant difference \nwith the \\citet{Robin2014} result for the thick disc. Their thick disk is modelled with 2 episodes, \none of which has very similar parameters as the present result (their old thick disk). But their young thick \ndisk is more compact, with smaller scale height and scale length. The difference can be due to the different \nshapes used (they use secant squared density laws and they include the flare) while in the present study \nthe thick disk is a simple exponential vertically. \n\n\\section{Substructures}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{cmdplot_l181.eps}\n  \\includegraphics[width=0.48\\textwidth]{rhist_l181.eps}\n  \\caption{{\\it Upper panel:} Differences of star counts in the $r_0$ vs. $(g-i)_0$ diagram for XSTPS-GAC data \n  in 2.5\\degr $\\times$ 2.5\\degr patches of sky, i.e. ($l,b$) = (181.25\\degr, $-$8.75\\degr) minus ($l,b$) = (181.25\\degr, \n  8.75\\degr). The red curve which fits the white main-sequence pattern of the newly identified substructure\n  is an isochrone with [Fe\/H]=$-$0.2\\,dex, age of 3.981\\,Gyr and distance of 1.3\\,kpc. The green curve that fits the black \n   main-sequence pattern of the ``north near structure'' is an isochrone with [Fe\/H]=$-$0.5\\,dex, age of 2.818\\,Gyr and distance of 3.7\\,kpc. The blue curve that fits the white main-sequence pattern of the ``south middle structure'' \n  is an isochrone with [Fe\/H]=$-$0.5\\,dex, age of 1.995\\,Gyr and distance of 6.8\\,kpc. \n{\\it Bottom panels:} The $r$-band star counts in the colour bin $0.5< (g-i)_0<0.6$\\,mag for the south patch\n  ($l,b$) = (181.25\\degr, $-$8.75\\degr), north patch ($l,b$) = (181.25\\degr, 8.75\\degr) and \n  the subtraction of the two.}\n  \\label{l181}\n\\end{figure}\n\nOne of the main purposes of the Galactic smooth structure modelling  is to define the \nirregular structure of the Milky Way. Assuming that our model now \nreproduces the populations of the halo and the disk, \nwe can investigate if any further structures are missing from our modelling. \nAs seen in Fig.~\\ref{resd}, most of the significant deviations are \noverdensities, i.e. $N_{obs} > N_{mod}$, except for \na few subfields such as those located at ($l,b$)=(170\\degr, 0\\degr). The extinction in\nthese fields are large \\citep{Chen2014}. It is very difficult to distinguish that whether it is a real  `hole'  \nor it is caused by the selection effects or extinction correction errors. \nFor the overdensities, we find three large scale structures,\nwhich are located at different positions on the sky and appear at\ndifferent magnitudes. We describe them as follows.\n\nThe first large region where star counts are in excess is located at \n240\\degr\\ $<l<$ 330\\degr~and  60\\degr\\ $<b<$ 90\\degr.\nIt is a large and diffuse structure, which corresponds \nto the Virgo overdensity found by \\citet{Juric2008}. \nFor stars with $0.5 ~<~(g-i)_0~<~0.7$\\,mag, the excess of star counts \noccurs at $r_0 \\sim$20\\,mag.\nSo the distance of the structure is between 9 -- 11\\,kpc, which is \nconsistent with the work of \\citet{Juric2008}. A plausible explanation of the Virgo \noverdensity is that it is a result of a merger event involving the Milky Way and \na smaller, low metallicity dwarf galaxy \\citep{Juric2008}.   \n\nThe second significant structure is located at\n170\\degr\\ $<l<$ 200\\degr~and  10\\degr\\  $<b<$ 30\\degr. \nThe excess of star counts occurs at $r_0~\\sim$ 16\\,mag \nfor stars with colour $0.5 ~<~(g-i)_0~<~0.7$\\,mag, \ncorresponding to a distance between 1.5--2\\,kpc. \nThis feature is consistent with the so-called \n``north near structure'' found by \\citet{Xu2015}. \n\\citet{Juric2008} also report an overdensity at\n$R$ = \udbff\udc09 9.5\\,kpc and $Z$ =\udbff\udc09 0.8\\,kpc, which\nmay be connected with this substructure.\n\\citet{Xu2015} report that this substructure \nrepresents one of the locations of peaks in the oscillations\n of the disk mid-plane, observed at \nabout $\\pm$15\\degr Galactic latitude, toward the Galactic anticentre. \n\nThe third star counts excess structure is located \nat 150\\degr\\ $<l<$ 210\\degr~and  $-$15\\degr\\  $<b<$ $-5$\\degr. \nWe have not found any corresponding substructures to this \nfeature from the literature.  \nWe need to check first that whether this\nsubstructure is real or caused by the selection bias or \nhigh extinction effects. To test its existence in a more direct way, \nwe examine the Hess diagrams constructed for a subfield located in\nthe area of the feature and for a control subfield which is outside the region.\nWe choose the control subfield with the same galactic longitude but opposite \nlatitude as the selected subfield that includes the overdensity.\nTo avoid the effect of the selection bias of Sample\\,C, \nwe select data from the complete XSTPS-GAC photometric sample in the regions \n($l,b$) = (181.25\\degr, $-$8.75\\degr, south field) and\n(181.25\\degr, 8.75\\degr, north field). Each region is of size 2.5\\degr $\\times$ 2.5\\degr.\nWe correct the extinction of stars using the extinction map from \\citet{Chen2015}\ntogether with the extinction law from \\citet{Yuan2013}.\nThe average extinction for the south and north fields\nare respectively $A_r$ = 0.66 and 0.27\\,mag. In Fig.~\\ref{l181}, \nwe show the differences between Hess diagrams\nand the star counts for stars with colour 0.5 $< (g-i)_0 < $ 0.6\\,mag for the two regions. \nWe see a white-black-white main-sequence pattern in the \ndifference Hess diagram. The fainter white sequence which indicating that \nthere are more stars on the south side\n and the middle black sequence indicating that \nthere are more stars on the north side respectively correspond \nto the ``south middle structure'' and ``north near structure'' described  in \\citet{Xu2015}.  \nThe bright white sequence corresponds to the newly found substructure. \nThe $r$-band extinction in the these two fields are rather small. Even if we assume an error of \n20\\,per\\,cent in extinction, the uncertainties of colour $g-i$ and magnitude $r$ in the two \nfields are only $\\delta _{g-i}$ = 0.09 and $\\delta _r$ = 0.13 for the south field and \n$\\delta _{g-i}$ = 0.04 and $\\delta _r$ = 0.05 for the north field, respectively. \nThe observed feature will not be erased by adjusting the reddening values. \nSo we believe that the feature is real. The effect of selection function and inaccuracies \nin the reddening correction cannot cause the apparent overdensities.\n\nThe excess of star counts occurs at around $r_0~\\sim 15$\\,mag for stars in the \ncolour ranges of 0.5 $< (g-i)_0 < $ 0.6\\,mag, indicating a distance between 1 -- 1.5\\,kpc. \nWe select a series of isochrones from \\citet{An2009}. The isochrone with [Fe\/H] = $-$0.2\\,dex \nand age of 3.981\\,Gyr, shifted by a distance of 1.3\\,kpc, fit well with the maximum overdensities of\nthe bright white sequence (see Fig.~\\ref{l181} ). The young age and high \nmetallicity are consistent with those of the field thin disk stars. Thus we believe that \nthis feature is unlikely a substructure originated from the outer disk. The distance to the \nGalactic plane of this feature is about $Z~\\sim~-$0.3\\,kpc and that for the ``north near structure''\nis about $Z~\\sim~$0.5\\,kpc. These two features, which show the significant\nNorth-South asymmetry in the star number count distributions, are consistent with the \nvertical oscillations in the stellar density in the solar neighbourhood discovered\nby \\citet[see their Fig.~18]{Yanny2013}. \n\n\\section{Summary}\n\nBased on the data from the XSTPS-GAC and the SDSS, we have modelled the global \nsmooth structure of the Milky Way. We adopt a three-component stellar distribution model.\nIt comprises two double exponential disks, the thin disk and the thick disk, and a\ntwo-axial power-law ellipsoid halo. The stellar number density of halo stars in the colour bin\n$0.5 < (g-i)_0 < 0.6$\\,mag and the $r$-band differential\nstar counts in three colour bins, $0.5 < (g-i)_0 < 0.6$\\,mag,\n $0.6 < (g-i)_0 < 0.7$\\,mag and  $1.5 < (g-i)_0 < 1.6$\\,mag, are\nused to determine the Galactic model parameters. \nThe best-fit values are listed in Table~2 and 3. In summary, the scale height and length\nof the thin disk are $H_1$=322\\,pc and $L_1$=2343\\,pc,\nand those of the thick disk are $H_2$=794\\,pc \nand $L_2$=3638\\,pc. The local stellar density ratio of thick-to-thin disk \nis $f_2$=11\\,per\\,cent, and that of \nhalo-to-thin disk is $f_h$=0.16\\,per\\,cent.  \nThe axis ratio and power-law index of the halo are $\\kappa=0.65$ \nand $p=2.79$. Our results are all well constrained and in good \nagreement with the previous works.\n\n\nBy subtracting the observations from our best-fit model, we find three\nlarge overdensities. Two of them have been previously identified, \nincluding the Virgo overdensity in the Halo \\citep{Juric2008}, which located \nat 240\\degr\\  $<l<$ 330\\degr\\ and  60\\degr\\  $<b<$ 90\\degr\\ with a distance between 9 -- 11\\,kpc, \nand the so-called ``north near structure'' in the disk \\citep{Xu2015}, which located \nat 170\\degr\\ $<l<$ 200\\degr\\ and 10\\degr\\ $<b<$ 30\\degr\\ with a distance between 1.5 -- 2\\,kpc. \nThe third structure, located at 150\\degr\\  $<l<$ 210\\degr\\ and   $-$15\\degr\\  $<b<$ $-5$\\degr\\ \nwith a distance between 1 -- 1.5\\,kpc, is a new identification. \nThrough the Hess diagram examination, we conclude that \nit could not be a artifact caused  by extinction correction or selection effects. \nThis feature, together with the ``north near structure'' confirms the earlier \ndiscovery of  \\citet{Widrow2012} and \\citet{Yanny2013} of a significant \nGalactic North-South asymmetry in the stellar number density distribution.\n\n\\section*{Acknowledgements}\n\nWe want to thank the referee, Prof. Gerry Gilmore, for his insightful comments.\nThis work is partially supported by National Key Basic Research Program of China\n2014CB845700,  China Postdoctoral Science Foundation 2016M590014 and \nNational Natural Science Foundation of China 11443006 and U1531244.\nThe LAMOST FELLOWSHIP is supported by Special Funding for Advanced Users,\nbudgeted and administrated by Center for Astronomical Mega-Science, Chinese\nAcademy of Sciences (CAMS). This research has made use of the Chinese \nVirtual Observatory (China-VO) resources and services.\n\nThis work has made use of data products from the Guoshoujing Telescope (the\nLarge Sky Area Multi-Object Fibre Spectroscopic Telescope, LAMOST). LAMOST\nis a National Major Scientific Project built by the Chinese Academy of\nSciences. Funding for the project has been provided by the National\nDevelopment and Reform Commission. LAMOST is operated and managed by the\nNational Astronomical Observatories, Chinese Academy of Sciences.\n\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the\nParticipating Institutions, the National Science Foundation, and the U.S. Department\nof Energy Office of Science. The SDSS-III web site is http:\/\/www.sdss3.org\/.\nSDSS-III is managed by the Astrophysical Research Consortium for the\nParticipating Institutions of the SDSS-III Collaboration including the University\nof Arizona, the Brazilian Participation Group, Brookhaven National Laboratory,\nCarnegie Mellon University, University of Florida, the French Participation Group,\nthe German Participation Group, Harvard University, the Instituto de Astrofisica de\nCanarias, the Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins\nUniversity, Lawrence Berkeley National Laboratory, Max Planck Institute for\nAstrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico\nState University, New York University, Ohio State University, Pennsylvania State\nUniversity, University of Portsmouth, Princeton University, the Spanish Participation\nGroup, University of Tokyo, University of Utah, Vanderbilt University, University of\nVirginia, University of Washington, and Yale University.\n\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\n\nThe key contribution in advancing sequence-to-sequence (encoder-decoder) models accounted to neural attention (NA) networks has rendered the latter a core part of any deep learning toolkit. Indeed, their obtained state-of-the-art results in MT \\cite{Bahdanau2014,Luong2015}, abstractive document summarization (ADS) \\cite{Manning2017}, speech recognition (SR) \\cite{Chorowski2015,Chan2015} and question answering (QA) \\cite{Sukhbaatar2015}, amongst others, vouch for their broad applicability. NA network functionality consists in enabling the output layer of sequence-to-sequence models to access a variable-length memory \\cite{weston2014,Graves2014,Graves2016} of stored source sequence encodings. Specifically, NA networks infer a time-varying representation of the memory content, which encapsulates the affinity of the stored source sequence encodings with the (elements in the) decoded target sequence \\cite{Bahdanau2014}. This is usually referred to as the inferred \\emph{context vector representation}, $\\boldsymbol{c}_{t}$, of the stored source sequence encodings, and is used to drive output generation (decoding) at time $t\\in{1,\\dots,T}$. Consequentially, application of NA mechanisms enables encoder-decoder models to perform inference over long temporal horizons; this is not feasible without NA, due to the model having to encode the entire source information into a single fixed vector $\\boldsymbol{h}$.\n \nThis major breakthrough has rendered NA a focal point in recent research developments, mainly revolving around the concept of \\emph{Soft-Attention} (SA) \\cite{Xu2015}. Despite their wide success, SA mechanisms also suffer from inherent weaknesses, which are actively scrutinized by the research community. One major weakness is that SA essentially implements a soft-selection mechanism over the source sequence encodings. This procedure relies on the assumption of full independence between the selections at each decoding step, and over all the available encodings. Unfortunately, under this simplistic selection rationale, the model is discouraged from exploring the dynamics in the memory space to their full potential.\n\nStructural attention models have been recently proposed as a means of overcoming these issues \\cite{wang2017translating,logeswaran2016sentence}. They consist in establishing an SA mechanism which infers a joint attention distribution over consecutive decoding steps. This gives rise to a structural dependencies prior, meant to allow for the model to attend to partial segmentations e.g., pertaining to phrases or other coherent syntactic structures in natural language. However, while this approach may be helpful, for instance, in reducing grammatical and syntax errors in translation outputs, it requires relying on dynamic programming to infer the attention distributions. Unfortunately, such algorithms generally impose exponential-scale complexity. On the other hand, adoption of simplistic relaxations for the sake of computational tractability, e.g. of first-order Markovian dynamics assumptions, or of a maximum length $L$ for each attended segment \\cite{wang2017translating,logeswaran2016sentence}, is clearly limiting.\n\nThis lack of the existing NA paradigm is the driving force of our work. We posit that enabling the attention mechanism to capture structural ambiguity due to higher-order dynamics in the source element selection process will allow for improved inferential capabilities. To effect this goal, for the first time in the literature, we draw inspiration from quantum statistical concepts.\nQuantum statistical systems replace the concept of Categorical distributions with the concept of \\emph{Density matrices} \\cite{sato2009quantum,warmuth2010bayesian,sordoni2013modeling,chatzis2012quantum}. A density matrix is a square matrix, the order of which corresponds to the number of available system selections; these are referred to as \"pure states\" in the literature of quantum statistics. Its diagonal elements represent the probabilities of the underlying pure states (which sum to one). In NA terms, this corresponds to a traditional soft selection over the source sequence elements. On the other hand, its non-diagonal elements correspond to the possibility that the system becomes inherently incapable of discerning between individual (pure) states. \nThis quantum-statistical modeling rationale does not reinstate the induced higher-order dependencies modeling as introduction of additional latent variables, nor stochastically as joint probabilities \\cite{sordoni2013modeling}, similar to related works on robotics \\cite{chatzis2012quantum} and information retrieval tasks \\cite{sordoni2013modeling}.\n\nTo exhibit the efficacy of our novel approach, we implement it into the context of encoder-decoder models with attention. We dub our approach \\emph{attention density matrix (ADM) seq2seq networks}. We perform an extensive experimental evaluation of our approach using several benchmark MT datasets. As we show, our method compares favorably to alternative NA mechanisms. \n\nThe remainder of this paper is organized as follows. Section \\ref{methodological_background} covers the methodological background. We introduce our novel quantum statistics-inspired regard towards NA in Section \\ref{proposed_approach}. In Section \\ref{experimental_eval}, we perform the experimental evaluation of our approach. We further discuss and conclude this work in Section \\ref{discuss}.\n\n\\section{SA for Encoder-Decoder Networks} \\label{methodological_background}\n\nEncoder-decoder networks with attention \\cite{Bahdanau2014,Luong2015} are formulated as follows: A sequence of source tokens $\\mathbf{x} = {x}_{i},\\dots,{x}_{T_x}$ are fed to an encoder function ${f}_{enc}$ that generates a sequence of corresponding representations (encodings) $\\mathbf{h} = {\\boldsymbol{h}}_{1},\\dots,{\\boldsymbol{h}}_{T_x}$. Typical encoder-decoder models adopt a bidirectional RNN encoder. This yields encodings obtained as the concatenation of the states of the component backward and forward RNNs; that is $\\boldsymbol{h}_{i} = [\\overrightarrow{\\boldsymbol{h}}_{i};\\overleftarrow{\\boldsymbol{h}}_{i}] \\in \\mathbb{R}^{D_h}$. On the other hand, the decoder receives at each step, $i$, the current state of the decoder RNN, ${\\boldsymbol{s}}_{i}$, and a context vector, ${\\boldsymbol{c}}_{i}$, to obtain a distribution over the next element in the target sequence, $ \\mathbf{y} = {y}_{1}, \\dots, {y}_{{T}_{y}}$.\n\n\nMore formally, the distribution of the generated (decoded) sequence yields:\n\\begin{equation}\nP(\\mathbf{y}|\\mathbf{x}) = \\prod_{i=1}^{T_y}P({y}_{i}|\\mathbf{y}_{<i}, \\mathbf{x}), \\;\\;\\; P({y}_{i}|\\mathbf{y}_{<i}, \\mathbf{x}) = \\mathrm{softmax}(\\mathrm{Dense}({\\boldsymbol{s}}_{i}, {\\boldsymbol{c}}_{i})) \\label{decoding}\n\\end{equation}\n\n\\noindent In these expressions, ${y}_{i}$ is the target sequence element generated by the decoder at step $i$, and $\\boldsymbol{s}_i$ is the state of the decoder RNN, $\\mathrm{f_{dec}}$, at step $i$:\n\\begin{equation}\n\\boldsymbol{s}_i=\\mathrm{f_{dec}}(\\boldsymbol{s}_{i-1}, y_{i-1}) \\in \\mathbb{R}^{D_s},\n\\end{equation}\nTypically, the used RNN is a Long Short-Term Memory (LSTM) \\cite{Hochreiter1997} or a Gated Recurrent Unit (GRU) network \\cite{Cho2014}. On the other hand, the \\emph{context vector}, $\\boldsymbol{c}_i$, is calculated as a weighted average of the encoder outputs pertaining to the source sequence, $\\boldsymbol{h}$ (encodings):\n\\begin{equation}\n\\boldsymbol{c}_{i}=\\sum_{j=1}^{T_{x}}a_{ij}\\boldsymbol{h}_{j} \\; \\; \\; \\mathrm{s.t.:} \\, \\boldsymbol{h}_{j} = \\mathrm{f_{enc}}(\\boldsymbol{h}_{j-1}, \\boldsymbol{x}_{j}) \\label{base_context_vector}\n\\end{equation}\n\nIn computing the context vector, $\\boldsymbol{c}_{i}$, the used weights $a_{ij}$ assigned to each encoding, $\\boldsymbol{h}_{j}$, define a Categorical distribution over the source sequence elements. Essentially, these weights imply establishment of a soft-selection mechanism; the probability of each source element being selected (attended to) is considered analogous to the affinity of the corresponding encoding with the current state of the decoder.\nSpecifically, we have \n\\begin{equation} \\label{pure_states_computation}\na_{ij} = \\mathrm{softmax}(\\alpha(\\boldsymbol{s}_{i}, \\boldsymbol{h}_{j})), \\; \\; \\mathrm{s.t.:} \\sum_{j}{a_{ij}}=1\n\\end{equation} \nHere, ${\\alpha}(\\cdot)$ is an \\emph{alignment network} \\cite{Bahdanau2014}. In the most usual case, the alignment network takes the form of an additive variant that reads \n\\begin{equation} \\label{soft-attention-scores}\n\\alpha(\\boldsymbol{s}_{i}, \\boldsymbol{h}_{j}) = \\boldsymbol{u}^{\\top}_{\\alpha}\\mathrm{tanh}(\\boldsymbol{W}_{\\alpha}\\boldsymbol{s}_{i} + \\boldsymbol{V}_{\\alpha}\\boldsymbol{h}_j + \\boldsymbol{b}_{\\alpha}),\n\\end{equation}\nwhere $\\boldsymbol{W}_{\\alpha}$ and $\\boldsymbol{V}_{\\alpha}$ are trainable weight matrices, while $\\boldsymbol{u}_{\\alpha}$ and $\\boldsymbol{b}_{\\alpha}$ are trainable parameter vectors of the same size. \n\n\\section{Proposed Approach} \\label{proposed_approach}\n\nThe formulation of our approach draws inspiration from the concept of the \\emph{density matrix}. A density matrix $\\boldsymbol{\\Phi}$ is symmetric, $\\boldsymbol{\\Phi} = \\boldsymbol{\\Phi}^{\\top}$, positive-semidefinite, $\\boldsymbol{\\Phi} \\succeq 0$, and of trace one. It constitutes a natural extension of Categorical probability distributions in the context of quantum statistics, whereby a modeled system may become inherently incapable of assigning selection probabilities to individual (\"pure\") states; assumingly due to underlying higher-order dynamics. \n\nIndeed, a Categorical distribution can be represented via a density matrix of diagonal form, referring to Gleason's theorem \\cite{gleason1957measures}. On the other hand, the non-diagonal elements of a density matrix correspond to the probabilities of the so-called \"mixed states\". These essentially reflect a situation where the quantum-statistical system \"lives\" in a mixture of states, as opposed to selecting to live in a specific \"pure\" state.\n\nDrawing from this inspiration, in this work we suggest that we infer attention dynamics in the form of a \\emph{pseudo}-density matrix, $\\boldsymbol{\\Psi}$. Under our proposed approach, the diagonal elements of $\\boldsymbol{\\Psi}$ constitute \\textit{attention scores} that correspond to the system selecting to attend to one of  \nthe available source sequence elements, in a fashion similar to the selection principles of SA. On the other hand, the non-diagonal elements of $\\boldsymbol{\\Psi}$ constitute scores tied to the possibility that selection may not be feasible for the model at the level of individual source sequence elements. On the contrary, the model may only become capable of performing selection at the level of pairs of elements, while lacking the capacity to discern which specifically of the pair elements it must attend to. \n\nLet us consider an NA mechanism performing attention over $N$ source sequence elements. We succinctly represent the above-prescribed concepts and assumptions by introducing the notion of the Attention Density Matrix (ADM). Considering the decoding step $i$, we define the pseudo-density matrix:\n\\begin{equation}\n\\boldsymbol{\\Psi}_i\\triangleq\\left[\\begin{array}{cccc}\n\\alpha_{i\\,1} & \\sigma_{i\\,(1,2)} & \\dots & \\sigma_{i\\,(1,N)}\\\\\n\\sigma_{{i\\,(2,1)}} & \\alpha_{i2} & \\dots & \\sigma_{{i\\,(2,N)}}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\sigma_{{i\\,(N, 1)}} & \\sigma_{i\\,({N, 2)}} & \\dots & \\alpha_{iN}\n\\end{array}\\right] \\label{attention_density_matrix}\n\\end{equation}\nWe dub this matrix the ADM of the introduced NA model at time $i$. Its diagonal elements $\\alpha_{ij}$ correspond to the attention scores of attending to source element $j$ at decoding step $i$, defined in the classical sense. On the other hand, the non-diagonal elements, $\\sigma_{i\\,(j,k)} \\in \\mathbb{R}$, correspond to the score pertaining to the system's inability to discern between source elements $j$ and $k$ at decoding step $i$; instead, it selects to attend to their pair, while being oblivious as to which specific element of the pair is the most appropriate to attend to. In this way, we seek to elevate the system's \\emph{uncertainty} as an additional model feature. Apparently, since the ADM $\\boldsymbol{\\Psi}_i$ is symmetric, it holds $\\sigma_{i\\,(j,k)}=\\sigma_{i\\,(k,j)},\\;\\forall\\,j\\neq k$. \n\nNote that the introduced ADM may reduce to the Categorical attention distribution employed in conventional SA, by setting its non-diagonal elements equal to zero. In addition, we emphasize that we elect not to enforce a positive-semidefiniteness constraint on the ADMs, $\\boldsymbol{\\Psi}_i$, nor do we express their elements as probabilities. This is reasonable, since our approach does not model an actual quantum-statistical system; we draw inspiration from it, in an effort to effect computationally efficient and elegant modeling of attention dynamics.\n\nLet us now proceed to derivation of the context vector representations under our ADM-based NA mechanism. To this end, we begin from the classical definition of context vectors adopted under the SA scheme. From definition (\\ref{base_context_vector}), it follows that the expression of SA context vectors can be rewritten in the following form:\n\\begin{equation}\n\\boldsymbol{c}_{i}=\\sum_{j=1}^{N}a_{ij}\\boldsymbol{h}_{j} = \\boldsymbol{H} \\boldsymbol{\\alpha}_{i}\\\\\n\\end{equation} \\label{base_context_vector_rewritten}\nwhere $\\boldsymbol{H}=[\\boldsymbol{h}_{j}]_{j=1}^{N}$ is the columnwise concatenation of the vectors set $\\{\\boldsymbol{h}_{j}\\}_{j=1}^{N}$. \n\nOn this basis, we seek to devise an effective means of generalizing the expression (7) in the context of our quantum-statistical formulation. To this end, we perform row-wise averaging of the ADM-induced, state-related quantities. This is an efficient formulation; we defer a supporting discussion of this aspect in Section \\ref{discuss}. \n\nSpecifically, our quantum statistics-inspired view towards NA materializes by adopting the same functional form for the derived context vectors, as in (7). This brings to the fore a natural way to generalize attention weight formulation under our approach, with the same output dimensionality. We postulate\n\\begin{equation}\n\\boldsymbol{c}_{i}=\\boldsymbol{H}\\boldsymbol{\\omega}_{i},\n\\; \\; \\; \\mathrm{s.t.:} \\, \\boldsymbol{\\omega}_{i}=\\mathrm{softmax}(\\;\\mean\\limits_{1\\leq j\\leq N}[\\boldsymbol{\\Psi}_{i}]_{jk}\\;)\n\\label{quantum_context_vector}\n\\end{equation}\nwhere $[\\boldsymbol{\\Psi}_{i}]_{jk}$ is the $(j,k)$th element of matrix $\\boldsymbol{\\Psi}_{i}$.\n\n\n\nThe key difference from (7) is that we replace the attention weights of conventional SA with the probabilities of the row-wise mean score of state mixtures, defined in (\\ref{attention_density_matrix}). The rationale of this statistical function is to roughly describe mean \\emph{temporal variation} during source element selection. In essence, we establish a mean-pooling mechanism, which amortizes the variance among scores pertaining to each source sequence element. This is in line with recent work related to enhancing sentence embeddings by leveraging mean-pooling operations across multiple NLP tasks, e.g. \\cite{chen2018enhancing}.\n\n\n\n\n\n\n\n\nHaving provided the definition of the introduced ADM concept, as well as the corresponding derivations pertaining to context vector inference under our approach, we now elaborate on how we can compute the ADM elements. We begin with the diagonal elements, $\\alpha_{ij}$, which correspond to classical soft-selection over the source sequence elements. In this work, we obtain these quantities by simply adopting the \\emph{score} functions employed in conventional SA. Indeed, this is a natural selection, since the diagonal elements of the ADM have the same physical interpretation as the scores in conventional SA.\n\nLet us now turn to the non-diagonal elements, $\\sigma_{i\\,(j,k)}$ of the postulated ADM, $\\boldsymbol{\\Psi}_i$, at time $i$. To introduce an effective way of computing them, we draw inspiration from their physical interpretation. As we have already discussed, a non-diagonal ADM element, $\\sigma_{i\\,(j,k)}$, assigns a \nscore to the scenario that the postulated NA model cannot effectively discern between the $j$th and the $k$th source sequence encoding, $\\{\\boldsymbol{h}_j, \\boldsymbol{h}_k\\}\\; \\forall{j\\neq k}$. Therefore, to obtain these  \nscores, we need to introduce a transformation that can effectively reflect two distinct aspects: (i) the affinity between pairs of source sequence encodings, $\\{\\boldsymbol{h}_j, \\boldsymbol{h}_k\\}\\; \\forall{j\\neq k}$; and (ii) the affinity of the current decoder state, $\\boldsymbol{s}_i$, with the latent dynamics shared by the encodings $\\boldsymbol{h}_j$ and  $\\boldsymbol{h}_k$. \n\nTo effect this goal, at each decoding step, $i$, we first compute a set of $D$-dimensional score vectors between each of the available source sequence encodings.\nWe define the 3-dimensional tensor $\\boldsymbol{L}=[l_{(j,k)}]_{j,k}$, where: \n\\begin{equation} \\label{order-1-quantum}\nl_{(j,k)}=\\mathrm{tanh}(\\boldsymbol{h}_{j}+\\boldsymbol{h}_{k}),\n\\end{equation}\nThis way, we yield a high-dimensional representation of the correlations between all pairs of the available source sequence encodings.\nThis enables us to further extract rich information regarding the latent dynamics shared between all possible pairs of encodings, and in respect to the current decoder state. Specifically, the sought higher-order interaction dynamics matrix is computed through Eq. (\\ref{luong:quantum}) using the tensor $\\boldsymbol{L}$ in (\\ref{order-1-quantum}). We have\n\\begin{equation} \\label{luong:quantum}\n\\boldsymbol{M}_{i}= w_{s}\\boldsymbol{L}\\boldsymbol{s}_{i},\n\\end{equation}\nwhere $w_{s}\\in\\mathbb{R}$ is a trainable weight scalar. Because of the multiplicative style it employs, resembling the attention mechanism in \\cite{Luong2015}, we refer to it as the \\textit{Multiplicative Quantum-Theoretical (MQT)} scheme. We make this distinction as it is possible to follow a different scheme, by replacing Eqs. (\\ref{order-1-quantum} - \\ref{luong:quantum}) with:\n\n\\begin{equation} \\label{bahdanau:quantum}\n\\boldsymbol{M}_{i}= \\mathrm{tanh}(\\boldsymbol{L} + \\boldsymbol{s}_{i}) \\boldsymbol{w}_{s},\n\\end{equation}\nwhere $\\boldsymbol{w}_{s}$ is a trainable weights vector.\n\\noindent We dub this alternative formulation as the \\textit{Additive Quantum-Theoretical (AQT)} scheme, as it resembles \\cite{Bahdanau2014}.\n\nEventually, we use the non-diagonal elements of the so-derived matrix $\\boldsymbol{M}_{i}$ as the non-diagonal elements of the sought ADM, $\\boldsymbol{\\Psi}_{i}$: $\\sigma_{i(j,k)}=[\\boldsymbol{M}_i]_{(j,k)}$. \n\n\\noindent\\textbf{Relation to recent work.} As demonstrated above, our approach enables capturing higher-order interactions in the\ncomputation of context vectors. Recent efforts\ntoward generalizing neural attention, by deriving more complex attention\ndistributions, contrast our proposed approach. We refer again to \\cite{structuredAttention}, which recently\nintroduced \\emph{structured attention}, whereby the attention probabilities are considered to be\ninterdependent over consecutive time-steps; for instance, they postulate\nthe first-order Markov dynamics assumption: \n\\begin{equation}\np(\\{\\boldsymbol{z}_{t}\\}_{t=1}^{T};\\pazocal{D})=p(\\boldsymbol{z}_{1};\\pazocal{D})\\prod_{t=1}^{T-1}p(\\boldsymbol{z}_{t+1}|\\boldsymbol{z}_{t};\\pazocal{D}),\n\\label{eq:Structured}\n\\end{equation}\nwhere $\\{z_{t}^{i}\\}_{i=1}^{N}$,\n$z_{t}^{i}\\in\\{0,1\\}$, with $z_{t}^{i}=1$ being a set of binary latent variables indicating the source element the model attends to at time $t$. To perform training and inference, the computationally complex forward-backward algorithm \\cite{rabiner1989tutorial} has to be used. This also discourages the method's use in scale, contrary to our approach which imposes minimal additional computational costs. This is the case as all the extra computations of our model, described in Eqs. (10)-(12), constitute single additional forward propagation computations; this renders our model comparable to classical SA in terms of computational costs.\n\n\\noindent\\textbf{Inference Algorithm.} To perform target decoding by means of a \\emph{seq2seq} model with ADM, we resort to \\emph{Beam sampling} \\cite{russell2016artificial}. In our experiments, \\emph{Beam} \\emph{width} is set to ten across all MT tasks. \n\n\\noindent\\textbf{Training Algorithm.} To perform training of a \\emph{seq2seq} model with ADM, we rely on minimization of an appropriate loss function, $\\pazocal{L}(\\theta)$, where $\\theta$ is the set of model parameters, in a similar fashion to conventional models. For example, to address an MT task, one may use the categorical cross-entropy between the generated decoding probabilities (\\ref{decoding}) and the available groundtruth.\n\n\n\n\n\n\\section{Experimental Evaluation} \\label{experimental_eval}\n\n\\begin{table*}[t]\n\\caption{Translation results on the En$\\leftrightarrow$Vi and En$\\leftrightarrow$Ro pairs.}\n\\small\n\\centering{}%\n\\begin{tabu}{|c|c|c|c|c|[1.5pt]c|c|c|c|}\n\\hline \n\\multirow{3}{*}{Method} & \\multicolumn{8}{c|}{BLEU}\\tabularnewline\n& \\multicolumn{2}{c}{En$\\rightarrow$Vi} & \\multicolumn{2}{c}{Vi$\\rightarrow$En} & \\multicolumn{2}{c}{En$\\rightarrow$Ro} & \\multicolumn{2}{c|}{Ro$\\rightarrow$En}\n\\tabularnewline\n\\cline{2-9} \n & dev & test & dev & test & dev & test & dev & test\\tabularnewline\n\\hline\n\\multirow{1}{*}{Baseline} & \\multirow{1}{*}{22.90} & \\multirow{1}{*}{24.11} & \\multirow{1}{*}{19.14} & \\multirow{1}{*}{20.76} & \\multirow{1}{*}{13.65} & \\multirow{1}{*}{15.31} & \\multirow{1}{*}{17.15} & \\multirow{1}{*}{16.74}\n\\tabularnewline\n\\cline{2-9}\n\\multirow{1}{*}{\\makecell{Structured Attention}} & \\multirow{1}{*}{16.81} & \\multirow{1}{*}{17.00} & \\multirow{1}{*}{17.19} & \\multirow{1}{*}{18.08} & \\multirow{1}{*}{7.04} & \\multirow{1}{*}{7.03} & \\multirow{1}{*}{11.02} & \\multirow{1}{*}{11.08}\n\\tabularnewline\n\\cline{2-9}\n\\multirow{1}{*}{\\textbf{MQT}} & \\multirow{1}{*}{23.41} & \\multirow{1}{*}{25.34} & \\multirow{1}{*}{20.65} & \\multirow{1}{*}{22.23} & \\multirow{1}{*}{15.01} & \\multirow{1}{*}{16.95} & \\multirow{1}{*}{19.03} & \\multirow{1}{*}{19.09}\n\\tabularnewline\n\\cline{2-9}\n\\multirow{1}{*}{AQT} & \\multirow{1}{*}{22.71} & \\multirow{1}{*}{24.34} & \\multirow{1}{*}{19.95} & \\multirow{1}{*}{21.33} & \\multirow{1}{*}{3.98}\n& \\multirow{1}{*}{4.56} & \\multirow{1}{*}{5.08} & \\multirow{1}{*}{5.62}\n\\tabularnewline\n\\hline\n\\end{tabu}\n\\label{translation:results}\n\\end{table*}\n\n\nOur experiments make use of publicly available corpora, namely IWSLT'15 English to and from Vietnamese (IWSLT En$\\leftrightarrow$Vi) and \nWMT'16 English to and from Romanian (WMT En$\\leftrightarrow$Ro). \nWe benchmark our models against word-based vocabularies and present our results in terms of the BLEU score \\cite{blue}. Note that, in this paper, we employ relatively small datasets to better demonstrate the applicability and generalization strength of our approach. To allow seamless handling of rare words, we utilize \\emph{byte pair encoding} (BPE) \\cite{sennrich2015neural} for En$\\leftrightarrow$Ro. \n\n\n\n\n\n\n\n\n\\subsection{Datasets}\n\n\\noindent\\textbf{IWSLT En$\\leftrightarrow$Vi.} One benchmark we consider is the IWSLT En$\\leftrightarrow$Vi machine translation task.\nThe dataset includes $\\sim$133K training sentence pairs, from translated TED talks,\nprovided by the IWSLT 2015 Evaluation Campaign \\cite{cettolo2015iwslt}. Following the same preprocessing steps in \\cite{Luong2015,raffel2017online}, we use the TED tst2012 ($\\sim$1.5K sentences) as a validation set for hyperparameter tuning and TED tst2013 ($\\sim$1.2K sentences) as a test set. The Vietnamese and English vocabulary sizes are $\\sim$7.7K and $\\sim$17.2K respectively.\n\n\n\\noindent\\textbf{WMT En$\\leftrightarrow$Ro.} A second dataset our evaluation addresses is the WMT En$\\leftrightarrow$Ro machine translation task. The dataset consists of $\\sim$400K parallel sentences. The shared vocabulary sizes (obtained from BPE with 32K operations) total $\\sim$31K words.  We use newsdev2016 as our development set, and newstest2016 as our test set (both with 1.9K sentences).\n\n\n\n\\subsection{Experimental setup} \n\n\\begin{table}[t]\n\\small\n\\caption{Vi$\\rightarrow$En, dev set - Examples (a) 281 and (b) 1086.}\n\\begin{tabularx}{\\textwidth}{|X|} \\hline\n\\textbf{Reference Translation}\\tabularnewline \\hline \n(a) And for me that means spending time thinking and talking about the poor, the disadvantaged, those who will never get to TED. \\\\\n(b) Now the transition point happened when these communities got so close that, in fact, they got together and decided to write down the whole recipe for the community together on one string of DNA.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Baseline}\\tabularnewline \\hline\n(a) For me, it means \\textcolor{orange}{to be} spending time to think about the poor, who have a situation, about who don't have a chance to \\textbf{[verb]} TED. \\\\\n(b) The time that transitions \\textbf{[verb]} when communities are closer together, in  \\textcolor{blue}{reality}, they're together, form a  \\textcolor{red}{collective tool} for the whole  \\textcolor{red}{chromosome} on a single  \\textcolor{blue}{array} of DNA.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Structured Attention}\\tabularnewline \\hline\n(a) For me, that means for time to think, \\textcolor{orange}{to think} about the poor, to talk about people who are not \\textcolor{orange}{having to} go to TED. \\\\\n(b) The time \\textcolor{red}{$<$unk$>$} occurs when the communities are close to each other, in fact\\textcolor{orange}{, in fact,} they're together to form a shared structure for the entire \\textcolor{red}{body}.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - MQT}\\tabularnewline\n\\hline \n(a) For me, it means to spend time to think, to talk about the poor people, who have a \\textcolor{blue}{difficult situation}, about who there is \\textcolor{blue}{no opportunity} to go to TED. \\\\\n(b) A time of \\textcolor{red}{communication} happens when communities are close to each other, in fact, they together form a \\textcolor{blue}{formula} that is shared for the whole \\textcolor{blue}{colony} on a single \\textcolor{blue}{array} of DNA.\n\\tabularnewline \\hline\n\\end{tabularx}\n\\label{vi-en:translations}\n\\end{table}\n\nWe perform dropout regularization of the trained models, with a dropout rate equal to 0.2. We minimize $\\pazocal{L}(\\phi)$ by employing the Adam \\cite{adam} optimizer with its default settings for En$\\leftrightarrow$Vi and simple stochastic gradient descent (SGD) for En$\\leftrightarrow$Ro\\footnote{En$\\leftrightarrow$Vi models are trained for $\\sim$12 epochs; En$\\leftrightarrow$Ro for$\\sim$12 epochs for structured attention models and $\\sim$4 for the rest.}. We preserve homogeneity throughout the trained architectures as follows.\nBoth the encoders and the decoders of all the evaluated models are presented with 256-dimensional \\emph{trainable} word embeddings. The maximum inference length is set to 50. We utilize 2-layer BiLSTM encoders, and 2-layer LSTM decoders; all comprise 256-dimensional hidden states on each layer. For the remainder of hyper-parameters, we adopt the default settings used in the code\\footnote{https:\/\/github.com\/harvardnlp\/struct-attn.} provided by the authors in \\cite{structuredAttention} for structured attention models and the code in \\cite{luong17} for the rest. Except specified otherwise, the default settings used by the latter for En$\\leftrightarrow$Vi also apply to En$\\leftrightarrow$Ro.\n\n\n\n\\subsection{Results} \n\\begin{table}[t]\n\\caption{Ro$\\rightarrow$En, dev set - Examples (a) 5 and (b) 182.}\n\\begin{tabularx}{\\textwidth} {|X|} \\hline \n\\textbf{Reference Translation}\\tabularnewline \\hline \n(a) Dirceu is the most senior member of the ruling Workers' Party to be taken into custody in connection with the scheme.\n\\\\\n(b) With one voice the lobbyists talked about a hoped-for ability in Turnbull to make the public argument, to cut the political deal and get tough things done.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Baseline}\\tabularnewline \\hline\n(a) He is the oldest member of the \\textcolor{orange}{Dutch People's Party} on \\textcolor{orange}{Human Rights} in custody for \\textcolor{orange}{the} \\textcolor{blue}{links} with this scheme.\n\\\\\n(b) The representatives of \\textcolor{blue}{lobbyists} have spoken about their hope in the ability of \\textcolor{orange}{Turngl} to \\textcolor{blue}{satisfy} the public interest, to reach a political agreement and to do things well.\n\\tabularnewline \\hline \\hline\n\\textbf{Generated Translation - Structured Attention}\\tabularnewline \\hline\n(a) \\textcolor{orange}{It} is the oldest member of the \\textcolor{orange}{Mandi} \\textcolor{orange}{of the Massi} in \\textcolor{orange}{the} government \\textcolor{red}{in the government}.\n\\\\\n(b) The representatives of the \\textcolor{blue}{interest groups} have spoken \\textcolor{red}{in mind} about their hope to \\textcolor{blue}{meet} the public interest, to achieve a political and good thing.\n\\tabularnewline \\hline\n\\textbf{Generated Translation - MQT}\\tabularnewline \\hline \n(a) \\textcolor{orange}{Dirre} is the oldest member of the \\textcolor{orange}{People's Party} in government \\textcolor{blue}{held} in custody for \\textcolor{blue}{ties} with this scheme.\n\\\\\n(b) The representatives of \\textcolor{blue}{interest groups} have spoken \\textcolor{orange}{to} \\textcolor{blue}{unison} about their hope in \\textcolor{orange}{Turkey's} ability to \\textcolor{blue}{meet} the public interest, to reach a political agreement and to do things well.\n\\tabularnewline \\hline\n\\end{tabularx}\n\\label{ro-en:translations}\n\\end{table}\n\nTable \\ref{translation:results} shows superior performance for our \\emph{multiplicative} approach. In addition, note that despite their extended training requirements, structured attention models demonstrate an inability to properly capture long-temporal information, both score and output-wise, as presented in Tables \\ref{vi-en:translations} and \\ref{ro-en:translations}. These showcase some characteristic examples of generated translations for a hands-on inspection of model outputs. We annotate deviations from the reference translation with orange and red, for minor\nand major deviations\nrespectively. Synonyms are highlighted with blue. We also indicate missing tokens, such as verbs, articles and adjectives, by adding the \\textbf{[$<$token$>$]} identifier.\n\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{discuss}\n\nIn this section, we want to further explore how and why our proposed approach can enable\nbetter utilization of infrequent words through second-order interactions. Furthermore, we outline technical augmentations that we consider as future research directives.\n\n\\subsection{Model uncertainty} \n\n\\begin{table*}[t]\n\\caption{Rare word mean reference frequency deviation}\n\\small\n\\centering{}%\n\\begin{tabu}{|c|c|c|c|c|}\n\\hline \n\\multirow{3}{*}{Language Pair} &  & \\multicolumn{3}{c|}{Deviation (\\%)}\\tabularnewline\n\\cline{3-5} \n & \\makecell{Mean reference \\\\ frequency (\\%)} & Baseline & \\makecell{Structured \\\\ Attention} & MQT\\tabularnewline\n\\hline\n\\multirow{1}{*}{En$\\rightarrow$Vi} & \\multirow{1}{*}{3.59} & \\multirow{1}{*}{-4.82} & \\multirow{1}{*}{11.57} & \\multirow{1}{*}{\\textit{\\textbf{0.28}}}\n\\tabularnewline\n\\cline{2-5}\n\\multirow{1}{*}{\\makecell{Vi$\\rightarrow$En}} & \\multirow{1}{*}{8.91} & \\multirow{1}{*}{-10.77} & \\multirow{1}{*}{-20.85} & \\multirow{1}{*}{\\textit{\\textbf{-8.68}}}\n\\tabularnewline\n\\cline{2-5}\n\\multirow{1}{*}{En$\\rightarrow$Ro} & \\multirow{1}{*}{7.00} & \\multirow{1}{*}{\\textbf{4.59}} & \\multirow{1}{*}{\\textit{14.61}} & \\multirow{1}{*}{-7.87}\n\\tabularnewline\n\\cline{2-5}\n\\multirow{1}{*}{Ro$\\rightarrow$En} & \\multirow{1}{*}{6.75} & \\multirow{1}{*}{34.23} & \\multirow{1}{*}{37.47} & \\multirow{1}{*}{\\textbf{\\textit{23.17}}}\n\\tabularnewline\n\\hline\n\\end{tabu}\n\\label{rare-words-table}\n\\end{table*}\n\n\nA first case study is inspired from the interesting work of \\cite{ott2018analyzing}. Therein, the major claim is that if model and data distributions match, then samples drawn from both should also match. As a broader extension to our evaluation, we present deviation from unigram word distributions (across 1 - 30\\% frequency groups). Table \\ref{rare-words-table} reveals how well our evaluated models estimate said frequencies. Words are split into groups based on their appearance in their respective training datasets. We present this against development set frequencies. Note that evaluation criteria include not only \\textit{deviation} but also \\textit{least over-representation} (favored to any magnitude of under-representation); these are presented in bold and italics, respectively. This is because swapping a frequent for a rare word would not be as harmful as the reverse, in terms of translation quality\nIn 3 out of 4 cases, our approach achieves close representation of target frequencies.\n\n\n\n\\subsection{High rank matrix approximation}\nThe capacity of most natural language models is crippled by their inability to cope in highly context-dependent settings \\cite{yang2017breaking}. Admittedly, this shortcoming is due to their limited capacity to capture complex hierarchical dependencies. To address this issue, we need to devise computationally efficient ways of capturing higher-order dynamics. A first step towards this goal is offered by our approach. However, our approach is limited to second-order interactions; in addition, to allow for computational efficiency, we have resorted to a mean-pooling solution in the computation of the final context vector.\nA more general solution would be to resort to spectral decomposition, which would require performing eigen\/tensor decomposition.\nHowever, differentiating this operation during back-propagation may lead to numerical instability issues \\cite{dang2018eigendecomposition}, rendering it non-differentiable. \nWe aim to examine solutions to these issues in our future work.\n\\section{Conclusions}\n\nIn this work, we introduced a novel regard towards formulating attention layers in \\emph{seq2seq}-type models. Our work was inspired from the quest of a more expressive way of computing dependencies between input and output sequences. Specifically, our aim was to enable capturing of second-order dependencies between the source sequence encodings and the generated output sequences. \n\nTo effect this goal, for the first time in the literature, we leveraged concepts from the field of quantum statistics. We cast the operation of the attention layer into the computation of the \\emph{Attention Density Matrix}, which expresses how pairs of source sequence elements correlated with each other, and jointly with the generated output sequence. Our formulation of the ADM was based on \\emph{Density Matrix} theory; it is an attempt to encapsulate the core concepts of the Density Matrix in the context of attention networks, without adhering though to the exact definition and properties of density matrices.\n\nWe exhibited the merits of our approach on \\emph{seq2seq} architectures addressing competitive MT tasks.\nWe have showed that the unique modeling capacity of our approach translates into better handling of (rare) words in the model outputs. Hence, this finding offers a quite plausible explanation of the obtained improvement in the achieved BLEU scores.\nFinally, we emphasize inference using our method entails minor computational overhead compared to conventional SA, with only a single extra forward-propagation computation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, entanglement has been regarded as a quantum\nresource for many novel tasks such as quantum computation, quantum\ncryptography, quantum teleportation and so on \\cite{Nielsen}.\nThese quantum-information tasks cannot be carried out by classical\nresources and they rely on the entangled states. Although the\nmixed entangled states are directly used in some\nquantum-information tasks \\cite{Murao}, most of them require the\npure entangled states of bipartite or multipartite system to be\nthe crucial elements. However in a lab, it turned out that the\npure entangled states always become mixed by the decoherence due\nto the coupling with the environment. A central topic in quantum\ninformation theory is thus how to extract pure entangled states\nfrom mixed states \\cite{Horodecki1}.\n\n\nAn entangled state $\\rho$ is distillable if one can asymptotically\nor explicitly extract some pure entangled state from infinitely\nmany copies of $\\rho$ by using only local operations and classical\ncommunication (LOCC). It has been proved that the entangled\n2-qubit states are always distillable\n\\cite{Bennett1,Bennett2,Horodecki2}. Nevertheless there exist\nbound entangled (BE) states which are not distillable under LOCC\n\\cite{Horodecki4}. Concretely, a bipartite entangled state $\\rho$\nin the Hilbert space $H_A\\otimes H_B$ is BE if it has positive\npartial transpose (PPT) with respect to system $A$ (or $B$),\nnamely $\\rho^{T_A}(\\mbox{or}\\ \\rho^{T_B})\\geq0$. Such states are\ncalled PPT BE states and usually it cannot be used for\nquantum-information tasks under LOCC \\cite{Murao,Eggeling}.\n\n\nA more formidable challenge is that whether a bipartite state\n$\\rho_{AB}$ having non-positive partial transpose (NPT) with\nrespect to system $A$ (or $B$) is always distillable. This class\nof states are always entangled due to the celebrated\nPeres-Horodecki criterion \\cite{Peres}. It was pointed out by\n\\cite{Horodecki5} that any NPT state can be converted into some\nNPT Werner state under LOCC. Much efforts have been devoted to\ndistilling this kind of states and there has been a common belief\nthat NPT BE Werner states indeed exists\n\\cite{DiVincenzo,Vianna,Pankowski,Hiroshima,Bandyopadhyay,Kraus,Clarisse}.\nIn addition, it has been proved that the NPT states in $2\\times N$\nspace are distillable \\cite{Horodecki2,Kraus2} and the rank two\nNPT states of bipartite systems are also distillable\n\\cite{Horodecki6}. However, the situation becomes more complex\nwhen we distill the entangled state whose subsystems have higher\ndimensions or that has a higher rank.\n\nIn this paper we show that the rank three bipartite entangled\nstates are distillable under LOCC. We give the concrete method of\ndistilling this class of states. It helps infer the analytical\ncalculation of distillable entanglement \\cite{Rains,Devetak}. A\nrank three state is entangled if and only if (iff) it is NPT,\nnamely there is no PPT BE state of rank three \\cite{Lewenstein}.\nSo we also obtain that there are no rank-three NPT BE states and\nall of them can be used for quantum-information tasks. It is\nsimilar to the case of rank two states and we conclude: a rank two\nor three state is distillable iff it is entangled. This conclusion\ndoes not hold for the bipartite entangled states with higher\nranks, e.g., there have been the rank four PPT BE states\nconstructed by the unextendible product bases (UPB) \\cite{Mor}.\n\nMoreover, we will investigate the NPT states of rank four and find\nout some families of states that are distillable. This helps\ndistill the NPT states which have more complex structure. In\naddition, we will show that locally converting the Werner state\ninto the rank three entangled state is difficult, so our result is\nindependent of the expectant fact that there exists NPT BE Werner\nstate.\n\nThe rest of this paper is organized as follows. In Sec. II we\nprove our main result on rank three states and then we use it to\ndistill the rank four NPT states. We also discuss the relationship\nbetween the result in this paper and the Werner state. We conclude\nin Sec. III.\n\n\n\\section{distillation of rank three and four bipartite states}\n\nThroughout this paper we will use the following notations. The\nrank of a bipartite state $\\rho_{AB}$ is referred to as\n$r(\\rho_{AB})$, and the reduced density operator of it as\n$\\rho_A=\\mbox{Tr}_B\\rho_{AB},\\rho_B=\\mbox{Tr}_A\\rho_{AB}$. The\nrange of the density operator $\\rho_{AB}$ is referred to as\n$R(\\rho_{AB})$. Another useful tool is the so-called invertible\nlocal operator (ILO) (or the local filter) \\cite{Dur}, namely the\nnonsingular matrix. Physically, it can be probabilistically\nrealized through the positive operator valued measure (POVM)\n\\cite{Nielsen}, so we can use it when distilling the NPT states.\n\n\nWe first consider the NPT states of rank three. Before proving our\nmain theorem, we recall a useful lemma that was proved in\n\\cite{Horodecki6}.\n\n\\textit{Lemma 1}. If\n$r(\\rho_{AB})<\\mbox{max}[r(\\rho_{A}),r(\\rho_{B})]$, then the\nbipartite state $\\rho_{AB}$ is distillable.\n\\hspace*{\\fill}$\\blacksquare$\n\nThe lemma has been used to show that there is no rank two BE state\n\\cite{Horodecki6}. It was proven by using the reduction criterion\n\\cite{Horodecki5}, i.e., a state is distillable when the reduction\ncriterion is violated (See Eq. (6) in \\cite{Horodecki6}). It\nfollows from lemma 1 that any rank three state in $M\\times N$\nspace with $\\mbox{max}[M,N]>3$ is distillable. Since an NPT state\nin $2\\times2$ or $2\\times3$ space is also distillable\n\\cite{Bennett1,Bennett2,Horodecki2}, it suffices to consider the\nrank three NPT states $\\rho_{AB}$ in $3\\times3$ space. Moreover,\nwe can perform some ILO on the subsystem $B$ such that\n$\\rho_B=\\frac13I$. Then only the state having the following form\ndoes not violate the reduction criterion ( up to local unitary\ntransformations )\n\\begin{equation}\n\\sigma_{AB}\\equiv\\frac13|\\psi_0\\rangle\\langle\\psi_0|+\\frac13|\\psi_1\\rangle\\langle\\psi_1|\n+\\frac13|\\psi_2\\rangle\\langle\\psi_2|,\\sigma_B=\\frac13I\n\\end{equation}\nwhere the three eigenvectors satisfy\n$\\langle\\psi_i|\\psi_j\\rangle=\\delta_{ij}$ and\n\\begin{eqnarray}\n|\\psi_0\\rangle&=&\\cos\\theta|00\\rangle+\\sin\\theta|11\\rangle,\\\\\n|\\psi_1\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}b_{ij}|ij\\rangle,\\\\\n|\\psi_2\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}c_{ij}|ij\\rangle.\n\\end{eqnarray}\nNotice that there is always at least a Schmidt rank two state by\nlinear combination of the eigenvectors. In addition, any spectral\ndecomposition of the state $\\sigma_{AB}$ have the form in Eq. (1)\n(in which the state $|\\psi_0\\rangle$ has a more general form,\ne.g., $|\\psi_0\\rangle=\\sum\\nolimits^{2}_{i,j=0}a_{ij}|ij\\rangle$).\n\nIn what follows we will concentrate on the NPT state $\\sigma_{AB}$\nin Eq. (1) because any rank three NPT state in $3\\times3$ space\ncan be locally converted into $\\sigma_{AB}$, otherwise it is\ndistillable in terms of the reduction criterion. There is a simple\nsituation we can treat easily as follows.\n\n\\textit{Lemma 2}. The state $\\sigma_{AB}$ is distillable when\nthere is a product state in its range.\n\n\\textit{Proof.} Without loss of generality, we consider the state\n$\\sigma_{AB}$ with $\\theta=0$. Then its coefficients\n$b_{i0},c_{i0},i=0,1,2$ equal zero because of the condition\n$\\sigma_B=\\frac13I$. We project the state $\\sigma_{AB}$ by using\nthe local projector\n$I_A\\otimes(|1\\rangle\\langle1|+|2\\rangle\\langle2|)_B$ and obtain\nthe resulting state\n$\\frac12|\\psi_1\\rangle\\langle\\psi_1|+\\frac12|\\psi_2\\rangle\\langle\\psi_2|$.\nIt's a rank two NPT state and hence distillable. It implies the\nstate $\\sigma_{AB}$ is also distillable.\n\\hspace*{\\fill}$\\blacksquare$\n\nLemma 2 has given a criterion that tells whether a rank three NPT\nstate is distillable. We will generalize it to the case of rank\nfour states later. It is also useful for the distillation of\ngeneral rank three NPT state as shown below. Let us consider the\nstate $\\sigma_{AB}$ whose range has no product state. We take the\nprojector $P_{AB}$ onto the $2\\times3$ subspace spanned by\n$\\{|00\\rangle,|01\\rangle,|02\\rangle,|10\\rangle,|11\\rangle,|12\\rangle\\}$\nand obtain the state\n\\begin{equation}\n\\sigma^1_{AB}=|\\psi^1_0\\rangle\\langle\\psi^1_0|+|\\psi^1_1\\rangle\\langle\\psi^1_1|\n+|\\psi^1_2\\rangle\\langle\\psi^1_2|,\n\\end{equation}\nwhich is not normalized for convenience. The resulting states\n$|\\psi^1_i\\rangle$ equal $P_{AB}|\\psi_i\\rangle$, respectively. We\nwill follow this notation below, e.g.,\n$|\\psi^2_i\\rangle=V_A\\otimes V_B|\\psi^1_i\\rangle$, etc.\n\nThe state $\\sigma^1_{AB}$ is distillable if it is entangled since\nit is in $2\\times2$ or $2\\times3$ space. Let us consider the case\nin which $\\sigma^1_{AB}$ is separable. First, the state\n$\\sigma^1_{AB}$ is in $2\\times2$ space iff\n$b_{i2}=c_{i2}=0,i=0,1$. In this case, the condition\n$\\sigma_B=\\frac13I$ leads to\n$b_{2i}b^*_{22}+c_{2i}c^*_{22}=0,i=0,1$ and\n$|b_{22}|^2+|c_{22}|^2=1$. When $b_{22}c_{22}=0$, either the state\n$|\\psi_1\\rangle$ or $|\\psi_2\\rangle$ becomes a product state and\nhence $\\sigma_{AB}$ is distillable in terms of lemma 2; When\n$b_{22}c_{22}\\neq0$, we can remove the coefficients\n$b_{2i},c_{2i},i=0,1$ by using linear combination of the\neigenvectors $|\\psi_i\\rangle,i=0,1,2$. It is then easy to see that\n$R(\\sigma_{AB})$ contains a product state and thus $\\sigma_{AB}$\nis distillable.\n\n\nSecond, we investigate the state $\\sigma^1_{AB}$ in $2\\times3$\nspace. Notice the rank of $\\sigma^1_{AB}$ remains three, otherwise\nthere will be a product state in $R(\\sigma_{AB})$ and it is\ndistillable. We can always write a rank three separable state\n$\\rho$ in $2\\times3$ space as the sum of three product states\n\\cite{Wootters,Werner}. To prove it, suppose the state has the\nform\n\\begin{equation}\n\\rho=\\sum\\nolimits^{d-1}_{i=0}|\\phi_i\\rangle|\\omega_i\\rangle\\langle\\phi_i|\\langle\\omega_i|,d>3.\n\\end{equation}\nWithout loss of generality we choose the first three product\nstates as a set of linearly independent vectors, so any other\nproduct state can be written as\n$|\\phi_j\\rangle|\\omega_j\\rangle=\\sum\\nolimits^2_{i=0}k_{ij}|\\phi_i\\rangle|\\omega_i\\rangle,j=3,...$\nNotice the vectors $|\\omega_i\\rangle,i=0,1,2$, and two vectors in\n$|\\phi_i\\rangle,i=0,1,2$ are linearly independent, respectively.\nSo the product state $|\\phi_j\\rangle|\\omega_j\\rangle,j>3$ equals\neither one of the first three product states, or\n$|\\phi_j\\rangle|\\omega_j\\rangle=\\sum\\nolimits^1_{i=0}k_{ij}|\\phi_i\\rangle|\\omega_i\\rangle$\nin which $|\\phi_0\\rangle$ is proportional to $|\\phi_1\\rangle$. In\nthis case it is easy to write the state $\\rho$ as the sum of three\nproduct states.\n\nUsing the above conclusion, we can express the state $\\sigma_{AB}$\nby means of eigenvectors\n$|\\psi_i\\rangle=(a_{i0}|0\\rangle+a_{i1}|1\\rangle)|\\phi_{i1}\\rangle+|2\\rangle|\\phi_{i2}\\rangle,i=0,1,2.$\nMoreover, the vectors $|\\phi_{i1}\\rangle$'s are linearly\nindependent, while $|\\phi_{i2}\\rangle$'s linearly dependent. We\nperform some ILO's on the state $\\sigma_{AB}$ and remove two\ncoefficients $a_{00}$ and $a_{11}$. The resulting state\n$\\sigma^2_{AB}$ still has the form in Eq. (1), otherwise it is\ndistillable.\n\nFor the state $\\sigma^2_{AB}$ when the condition $a_{20}a_{21}=0$\nis satisfied, we find that $R(\\sigma^2_{AB})$ contains a product\nstate because of the orthogonal conditions\n$\\langle\\psi^2_i|\\psi^2_j\\rangle=\\delta_{ij}$. So the state\n$\\sigma_{AB}$ is distillable. Let us move to investigate the state\n$\\sigma^2_{AB}$ satisfying the condition $a_{20}a_{21}\\neq0$. By\nperforming ILO's on $\\sigma^2_{AB}$ we greatly simplify its form\nsuch that\n\\begin{equation}\n\\sigma^3_{AB}=|\\psi^3_0\\rangle\\langle\\psi^3_0|+|\\psi^3_1\\rangle\\langle\\psi^3_1|\n+|\\psi^3_2\\rangle\\langle\\psi^3_2|,\n\\end{equation}\nwhere\n\\begin{eqnarray}\n|\\psi^3_0\\rangle&=&|00\\rangle+|2\\rangle|\\psi\\rangle,\\\\\n|\\psi^3_1\\rangle&=&|11\\rangle+|2\\rangle|\\phi\\rangle,\\\\\n|\\psi^3_2\\rangle&=&(|0\\rangle+|1\\rangle)|2\\rangle+\n|2\\rangle(\\alpha|\\psi\\rangle+\\beta|\\phi\\rangle),\\\\\n|\\psi\\rangle&=&x_0|0\\rangle+x_1|1\\rangle+x_2|2\\rangle,\\\\\n|\\phi\\rangle&=&y_0|0\\rangle+y_1|1\\rangle+y_2|2\\rangle.\n\\end{eqnarray}\nNotice the state is not normalized and the condition\n$\\sigma^3_B=\\frac13I$ is also not required. We project\n$\\sigma^3_{AB}$ by the projector\n$[|0\\rangle(\\langle0|+a\\langle1|)+|2\\rangle\\langle2|]_A\\otimes\nI_B,a\\in R$ and obtain the state $\\sigma^4_{AB}$ in $2\\times3$\nspace. It is entangled and thus distillable when its partial\ntranspose is not positive \\cite{Peres}. Nevertheless, there may be\nsome cases in which the coefficients $x_i,y_i,i=0,1,2$ make that\n$(\\sigma^4_{AB})^{T_A}\\geq0$. We are going to find out such\ncoefficients by calculating several average values\n$\\mbox{Tr}[(\\sigma^4_{AB})^{T_A}|\\omega_i\\rangle\\langle\\omega_i|]$,\nwhere\n$|\\omega_0\\rangle=|00\\rangle+b|22\\rangle,|\\omega_1\\rangle=|00\\rangle+b|21\\rangle\n,|\\omega_2\\rangle=|01\\rangle+b|20\\rangle,|\\omega_3\\rangle=|02\\rangle+b|20\\rangle,b\\in\nC.$ To keep the average value always positive, we find that it is\nnecessary that $x_1=x_2=0$. However, this means the state\n$|\\psi^3_0\\rangle$ is of product form and hence the state\n$\\sigma^3_{AB}$ is distillable. As it can be converted into the\nstate $\\sigma_{AB}$ by ILO's, the latter is also distillable. Now\nwe reach our main theorem in this paper.\n\n\\textit{Theorem.} The rank three NPT states are distillable under\nLOCC. \\hspace*{\\fill}$\\blacksquare$\n\nSo the rank three entangled states can be used for\nquantum-information tasks. In fact, we have proposed the method of\ndistilling $\\sigma_{AB}$ in the proof of the theorem. First, when\nthe given state contains a product state in its range, it can be\nprojected onto a rank two entangled state. According to the\nreduction criterion, we can distill it by the procedure similar to\nthe famous BBPSSW protocol \\cite{Bennett2,Horodecki5}. It is also\nthe method of distilling the rank three entangled states that\ncannot be converted into $\\sigma_{AB}$. Second, when the given\nstate $\\rho$ contains no product state in $R(\\rho)$, we project it\nby the projector $(|0\\rangle\\langle0|+|1\\rangle\\langle1|)_A\\otimes\nI_B$. The resulting state is entangled and thus distillable;\notherwise, we should project the initial state $\\rho$ by the\nprojector\n$[|0\\rangle(\\langle0|+a\\langle1|)+|2\\rangle\\langle2|]_A\\otimes\nI_B$ after performing some ILOs on $\\rho$. There will be a\nsuitable parameter $a$ making the resulting state entangled and\nthus distillable.\n\nThe rank three entangled states are a quite special class of\nstates. As there have been PPT BE states of any higher rank (e.g.,\nrank four PPT BE states constructed by UPB \\cite{Mor}), we indeed\nhave found out the lowest rank space in which there is no BE\nstate. It also implies that when a state can be locally projected\ninto some rank three NPT state, then it is distillable. This\ncauses new methods of distilling quantum states having more\ncomplex structure. We will show it in terms of distilling the rank\nfour NPT states below. One may also find other way to distill the\nentangled states based on the theorem. For example, the tensor\nproduct of the rank three entangled states are also entangled for\ncertain.\n\nOn the other hand, the analytical calculation of distillable\nentanglement is also an important issue in quantum information\ntheory. The problem is very difficult and there have been some\noptimal bounds on distillable entanglement \\cite{Rains,Devetak}.\nSpecially, the bound is saturated if we can find a way to distill\nthe state and get the same amount of pure entanglement as the\nbound. In this case we get the analytical result of distillable\nentanglement. As it is possible to find out whether the bound on\nrank three NPT state is saturated by using our method of\ndistilling it, we indeed provide new ways to calculate the\ndistillable entanglement.\n\nThird, our result is also independent of the expectant fact that\nthere exist NPT Werner states $\\rho_w$. We do not know whether an\nNPT state $\\rho$ is distillable, even it can be converted into\nsome Werner state which is proved to be not distillable under\nLOCC. One can easily exemplify it by locally taking some rank\nthree NPT state into $\\rho_w$, while the latter is expected to be\nnot distillable. Conversely, it is difficult to convert the Werner\nstate into the state $\\sigma_{AB}$, so we still do not know\nwhether the latter is distillable. To see it, we have the Werner\nstate in a $N\\times N$ space as follows \\cite{Werner}\n\\begin{eqnarray}\n\\rho_w&=&(a+b)\\sum\\nolimits^{N-1}_{i,j=0}|ij\\rangle\\langle\nij|\\nonumber\\\\\n&-&2b\\sum\\nolimits^{N-1}_{i<j=0}\\frac{|ij\\rangle-|ji\\rangle}{\\sqrt2}\\frac{\\langle\nij|-\\langle ji|}{\\sqrt2},\n\\end{eqnarray}\nwhere $a>0,b<0$ are two parameters satisfying $a+b\\geq0$. The most\ngeneral local transformation on a quantum state $\\rho$ has the\nform $\\Lambda(\\rho)=\\sum\\nolimits_i A_i\\otimes B_i\\rho\nA^{\\dag}_i\\otimes B^{\\dag}_i$ \\cite{Vedral}. Because the resulting\nstate is entangled, there must be at least one pair of Kraus\noperators $A_i,B_i$ that have at least rank two, respectively. In\nthis case, the state $\\Lambda(\\rho)$ will have the rank not less\nthan four when $a+b>0$, which means a rank three NPT state cannot\nbe output by this local channel. The only exception happens when\n$a+b=0$, but it is difficult to judge whether the state\n$\\Lambda(\\rho)$ is of rank three and entangled.\n\nLet us investigate further the problem of distilling rank four\nstates by using the theorem in this paper. Different from the case\nof rank three state, it is well-known that there indeed exist PPT\nBE states of rank four even in the $3\\times3$ space. It is easy to\nshow that the NPT BE states of rank four possibly exist only in\nthree kinds of spaces, $4\\times4,3\\times4,3\\times3$ in terms of\nlemma 1. One will meet lots of difficulties when applying the\ntechnique in this paper to distill the rank four NPT state $\\rho$,\ne.g., the resulting state from $\\rho$ by projection can be\n$3\\times3$ and it may be PPT BE. Besides, the Peres-Horodecki\ncriterion is no more a sufficient condition for the separability\nof state in $2\\times4$ space, etc. Nevertheless, we still can\nobtain some useful results on this problem when the target state\nhas a special form.\n\n\\textit{Lemma 3}. For a rank four NPT state in $4\\times4$ or\n$3\\times4$ space, it is distillable when there is a product state\nin its range.\n\n\\textit{Proof.} By employing similar deduction for the state\n$\\sigma_{AB}$, only the state having the following form does not\nviolate the reduction criterion\n\\begin{equation}\n\\rho_{AB}=\\frac14\\sum\\nolimits^3_{i=0}|\\psi_i\\rangle\\langle\\psi_i|,\\rho_B=\\frac14I,\n\\end{equation}\nwhere the four eigenvectors satisfy\n$\\langle\\psi_i|\\psi_j\\rangle=\\delta_{ij}$. Up to the local unitary\ntransformations we have $|\\psi_0\\rangle=|00\\rangle$. Next, we\nproject the state $\\rho_{AB}$ by the projector\n$I_A\\otimes(|1\\rangle\\langle1|+|2\\rangle\\langle2|+|3\\rangle\\langle3|)_B$\nand obtain the NPT state\n$\\rho^{\\prime}_{AB}=\\frac13\\sum\\nolimits^3_{i=1}|\\psi_i\\rangle\\langle\\psi_i|$\nin $2\\times3$, or $3\\times3$, or $4\\times3$ space. By means of the\nBBPSSW and Horodeckis' protocol, our theorem and the reduction\ncriterion, respectively, the state $\\rho^{\\prime}_{AB}$ and hence\n$\\rho_{AB}$ is always distillable. \\hspace*{\\fill}$\\blacksquare$\n\nSo we have generalized lemma 1 to the case of rank four NPT\nstates. Moreover, we hope that it always holds for the NPT states\nwhose rank equal to its maximal dimension of subsystems. However,\nit does not hold when the rank of a state is larger, e.g, the PPT\nBE state in $3\\times3$ space constructed in \\cite{Horodecki4}\ncontains infinitely many product states in its range, but its rank\nequals eight. It is also unclear that whether the rank four NPT\nstates $\\rho$ in this space are distillable. Solving this problem\nis more difficult since we cannot rely on the reduction criterion.\nHowever, $\\rho$ is distillable when we can project it onto a rank\nthree NPT state in terms of our theorem.\n\nFor example, the following $3\\times3$ rank four NPT state is\ndistillable\n\\begin{eqnarray}\n\\rho_{AB}&=&\\lambda_0|00\\rangle\\langle00|+\\lambda_1|01\\rangle\\langle01|+\n\\lambda_2|\\psi_2\\rangle\\langle\\psi_2|+\\lambda_3|\\psi_3\\rangle\\langle\\psi_3|,\\nonumber\\\\\n|\\psi_2\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}c_{ij}|ij\\rangle,\\nonumber\\\\\n|\\psi_3\\rangle&=&\\sum\\nolimits^{2}_{i,j=0}d_{ij}|ij\\rangle,\\lambda_0,\\lambda_1,\\lambda_2,\\lambda_3>0.\n\\end{eqnarray}\nTo prove it, we project the state $\\rho_{AB}$ by the projector\n$(|1\\rangle\\langle1|+|2\\rangle\\langle2|)_A\\otimes I_B$. When the\nresulting state\n$\\rho^{1}_{AB}=\\lambda_2|\\psi^1_2\\rangle\\langle\\psi^1_2|+\\lambda_3|\\psi^1_3\\rangle\\langle\\psi^1_3|$\nis entangled, it is also distillable. On the other hand when\n$\\rho^{1}_{AB}$ is separable, we can write it as the sum of two\nproduct states since it is in a space not larger than $2\\times3.$\nBesides, the rank of $\\rho^{1}_{AB}$ must be two because of\n$r(\\rho_A)=3$. By performing some ILOs on the state $\\rho_{AB}$\nand linear combination of $|\\psi_2\\rangle$ and $|\\psi_3\\rangle$,\nwe can convert them into\n$|\\psi^2_2\\rangle=|0\\rangle|\\phi_0\\rangle+|1\\rangle|\\phi_1\\rangle$\nand\n$|\\psi^2_3\\rangle=|0\\rangle|\\omega_0\\rangle+|2\\rangle|\\omega_2\\rangle$,\nand keep the other two terms $|00\\rangle$ and $|01\\rangle$\nunchanged.\n\nWhen either of the states $|\\psi^2_2\\rangle$ and\n$|\\psi^2_3\\rangle$ is of product form, we easily project the state\n$\\rho^2_{AB}$ onto a $2\\times3$ subspace, the resulting state is\nstill entangled and distillable. On the other hand when both the\nstates $|\\psi^2_2\\rangle$ and $|\\psi^2_3\\rangle$ are entangled, we\nproject the state $\\rho^2_{AB}$ by the projector\n$[|0\\rangle(\\langle0|+a\\langle1|)+|2\\rangle\\langle2|]_A\\otimes\nI_B$. The obtained state $\\rho^3_{AB}$ is $2\\times3$ and its rank\nis four by choosing suitable parameter $a$. This state is\nseparable iff it has the decomposition\n$\\rho^3_{AB}=|\\psi\\rangle_A\\langle\\psi|\\otimes|\\omega_2\\rangle_B\\langle\\omega_2|\n+|0\\rangle_A\\langle0|\\otimes\\rho^3_B$ with $r(\\rho^3_B)=3.$\nHowever it is impossible, since it requires a $4\\times4$\ncoefficient unitary matrix $[a_{ij}]$ in which $a_{i3}=0,i=1,2,3,$\nand $a_{0i},i=0,1,2$ cannot be zero simultaneously. Hence the\nstate $\\rho^3_{AB}$ is entangled and thus distillable. This also\ncompletes the proof showing that the state $\\rho_{AB}$ in Eq. (15)\nis distillable.\n\n\nAs above we have given several families of states that can be\ndistilled by means of the fact that the rank three NPT states are\ndistillable. The main difficulty in entanglement distillation is\nthe great amount of parameters that cannot be removed during the\nfiltering process. For example, it is unknown that whether the\nrank four NPT states are distillable. All in all, more efforts are\nrequired to distill other classes of rank four NPT states.\n\n\n\n\n\n\n\\section{conclusions}\n\nWe have proved that the bipartite rank three NPT states and some\nfamilies of rank four NPT states are distillable. So they are\nindeed available resource for quantum-information tasks. An open\nproblem is that whether all rank four NPT states are distillable.\nOur result also gives an insight into the relationship between the\nlow rank states and the Werner states.\n\n\n\nThe work was partly supported by the NNSF of China Grant\nNo.90503009, No.10775116, and 973 Program Grant No.2005CB724508.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}