diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdqvq" "b/data_all_eng_slimpj/shuffled/split2/finalzzdqvq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdqvq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nFree probability theory was introduced by Voiculescu around 1983 motivated by \nthe isomorphism problem of von Neumann algebras of free groups. He developed a noncommutative probability theory, on a noncommutative\nprobability space, in which a new notion of freeness plays the\nrole of independence in classical probability. Around 1991, Voiculescu \\cite{V} threw a bridge connecting random matrix theory with free probability \nsince he realized that the freeness property is also present for many classes of random matrices, in the\nasymptotic regime when the size of the matrices tends to infinity. Since then, random matrices have played a key role in operator algebra whereas tools developed in operator algebras and free\nprobability theory could now be applied to random matrix problems.\\\\\nFor the reader's convenience, we recall the following basic definitions from free probability theory. For a thorough introduction to free probability theory, we refer to \\cite{VDN}.\n\\begin{itemize}\n\\item A ${\\cal C}^*$-probability space is a pair $\\left({\\cal A}, \\tau\\right)$ consisting of a unital $ {\\cal C}^*$-algebra ${\\cal A}$ and \na linear map $\\tau: {\\cal A}\\rightarrow \\mathbb{C}$ such that $\\tau(1_{\\cal A})=1$ and $\\tau(aa^*)\\geq 0$ for all $a \\in {\\cal A}$. $\\tau$ is a trace if it satisfies $\\tau(ab)=\\tau(ba)$ for every $(a,b)\\in {\\cal A}^2$. A trace is said to be faithful if $\\tau(aa^*)>0$ whenever $a\\neq 0$. \nAn element of ${\\cal A}$ is called a noncommutative random variable. \n\\item The noncommutative distribution of a family $a=(a_1,\\ldots,a_k)$ of noncommutative random variables in a ${\\cal C}^*$-probability space $\\left({\\cal A}, \\tau\\right)$ is defined as the linear functional $\\mu_a:P\\mapsto \\tau(P(a,a^*))$ defined on the set of polynomials in $2k$ noncommutative indeterminates, where $(a,a^*)$ denotes the $2k$-tuple $(a_1,\\ldots,a_k,a_1^*,\\ldots,a_k^*)$.\nFor any self-adjoint element $a_1$ in ${\\cal A}$, there exists a probability measure $\\nu_{a_1}$ on $\\mathbb{R}$ such that, for every polynomial P, we have\n$$\\mu_{a_1}(P)=\\int P(t) \\mathrm{d}\\nu_{a_1}(t).$$\nThen, we identify $\\mu_{a_1}$ and $\\nu_{a_1}$. If $\\tau$ is faithful then the support of $\\nu_{a_1}$ is the spectrum of $a_1$ and thus $\\|a_1\\| = \\sup\\{|z|, z\\in \\rm{support} (\\nu_{a_1})\\}$. \n\\item A family of noncommutative random variables $(a_i)_{i\\in I}$ in a ${\\cal C}^*$-probability space $\\left({\\cal A}, \\tau\\right)$ is free if for all $k\\in \\mathbb{N}$ and all polynomials $p_1,\\ldots,p_k$ in two noncommutative indeterminates, one has \n\\begin{equation}\\label{freeness}\n\\tau(p_1(a_{i_1},a_{i_1}^*)\\cdots p_k (a_{i_k},a_{i_k}^*))=0\n\\end{equation}\nwhenever $i_1\\neq i_2, i_2\\neq i_3, \\ldots, i_{k-1}\\neq i_k$ and $\\tau(p_l(a_{i_l},a_{i_l}^*))=0$ for $l=1,\\ldots,k$.\n\\item A family $(x_i)_{i\\in I}$ of noncommutative random variables in a ${\\cal C}^*$-probability space $\\left({\\cal A}, \\tau\\right)$ is a semicircular system if\n $x_i=x_i^*$ for all $i\\in I$, $(x_i)_{i\\in I}$\nis a free family and for any $k\\in \\mathbb{N}$, $$\\tau(x_i^k)= \\int t^k d\\mu_{sc}(t)$$\nwhere $d\\mu_{sc}(t)=\n\\frac{1}{2\\pi} \\sqrt{4-t^2}{\\1}_{[-2;2]}(t) dt$ is the semicircular standard distribution.\n\\item Let $k$ be a nonnull integer number. Denote by ${\\cal P}$ the set of polynomials in $2k $ noncommutative indeterminates.\nA sequence of families of variables $ (a_n)_{n\\geq 1} =\n(a_1(n),\\ldots, a_k(n))_{n\\geq 1}$ in ${\\cal C}^* $-probability spaces \n$\\left({\\cal A}_n, \\tau_n\\right)$ converge, when n goes to infinity, respectively in distribution if the map \n$P\\in {\\cal P} \\mapsto\n\\tau_n(\nP(a_n,a_n^*))$ converges pointwise\nand strongly in distribution if moreover the map \n$P\\in {\\cal P} \\mapsto \\Vert P(a_n,a_n^*) \\Vert$ converges pointwise.\n\\end{itemize}\n\n\nVoiculescu considered random matrices in this noncommutative probability context. Let ${\\cal A}_n$ be \nthe algebra of $n\\times n$ matrices\nwhose entries are random variables with finite moments and endow this algebra with\nthe expectation of the normalized trace defined for any $M\\in {\\cal A}_n$ by \n$\\tau_n(M) = \\mathbb{E}[\\frac{1}{n}\\Tr(M)]$. Let us \nconsider r independent $n\\times n$ so-called G.U.E matrices, that is to say random Hermitian matrices $X_n^{(v)} = [X^{(v)}_{jk}]_{j,k=1}^n$, $v=1,\\ldots,r$, for which the random variables $(X^{(v)}_{ii})$,\n$(\\sqrt{2} Re(X^{(v)}_{ij}))_{i0$ and an integer number $n_0>0$ such that, for any $x >x_0$ and any integer number $n >n_0$, we have\n\\begin{equation}\\label{condition}\\frac{1}{n^2} \\sum_{1\\leq i,j\\leq n}\\mathbb{P}\\left( \\vert X^{(v)}_{ij}\\vert >x\\right) \\leq K_v\\mathbb{P}\\left(\\vert Z^{(v)} \\vert>x\\right).\\end{equation}\n\\item $$\\sup_{1\\leq i0$ such that, for any large $n$, $[b-\\delta,c+\\delta]$ lies outside the support of the distribution of the noncommutative random variable $ P\\left(x_1,\\ldots,x_r, a_n^{(1)},\\ldots,a_n^{(t)},(a_n^{(1)})^*,\\ldots,(a_n^{(t)})^*\\right)$ in $({\\cal A},\\tau)$.\n Then, \nalmost surely, for all large $n$, there is no eigenvalue of the $n\\times n$ matrix $ P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right)$ in $[b,c].$\n\\end{theoreme}\n\\begin{remarque}\nWhen $r=t=1$, $A_n^{(1)}=(A_n^{(1)})^*$ and $P(X_1,X_2,X_2^*)=X_1+\\frac{X_2+X_2^*}{2}$, the distribution of $P(x_1,a_n^{(1)},(a_n^{(1)})^*)$ is the so-called free convolution $\\mu_{sc}\\boxplus \\mu_{A_n^{(1)}}$ where $\\mu_{A_n^{(1)}}=\\frac{1}{n} \\sum_{i=1}^n \\lambda_i(A_n^{(1)})$, denoting by \n$ \\lambda_i(A_n^{(1)})$, $i=1,\\ldots,n$, the eigenvalues of $A_n^{(1)}$.\n\\end{remarque}\n\\begin{theoreme}\\label{thprincipal} Let $({\\cal A}, \\tau)$ be a ${\\cal C}^*$-probability space\n equipped with a faithful tracial state.\nLet $x=(x_1,\\ldots,x_r)$ be a semi-circular system and $a=(a_1,\\ldots,a_t)$ be a t-tuple of noncommutative random variables which is free from $x$ in $({\\cal A},\\tau$).\\\\\nAssume moreover that $(A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*)$ converges strongly towards $a=(a_1,\\ldots,a_t, a_1^*,\\ldots,a_t^*)$ in\n $({\\cal A}, \\tau)$, that is\nfor any polynomial P in 2t noncommutative indeterminates,\\\\\n\n$\\frac{1}{n} \\Tr P\\left( A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right)$ $$ \\rightarrow_{n\\rightarrow +\\infty} \\tau \\left( P(a_1,\\ldots,a_t,a_1^*,\\ldots,a_t^*\\right)$$ and \\\\\n\n$\\left\\|P\\left( A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right)\\right\\| $ $$\\rightarrow_{n\\rightarrow +\\infty} \\left\\| P(a_1,\\ldots,a_t,a_1^*,\\ldots,a_t^*)\\right\\|_{\\cal A}.$$\n Then, almost surely, for any polynomial $P$ \\hspace*{-0.1cm}in $r+2t $ \\hspace*{-0.16cm} noncommutative variables, \n$$ \\lim_{n\\rightarrow +\\infty} \\frac{1}{n}\\Tr P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}, (A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right) $$\\begin{equation}\\label{af} =\\tau \\left( P\\left(x_1,\\ldots,x_r, a_1,\\ldots,a_t, a_1^*,\\ldots,a_t^*\\right) \\right)\\end{equation} and \n$$\\lim_{n\\rightarrow +\\infty} \\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)},(A_n^{(1)})^*,\\ldots,(A_n^{(t)})^*\\right) \\right\\|$$\\begin{equation} \\label{saf} =\\left\\| P\\left(x_1,\\ldots,x_r, a_1,\\ldots,a_t, a_1^*,\\ldots,a_t^*\\right) \\right\\|_{\\cal A}.\\end{equation}\n\\end{theoreme}\n\\begin{remarque}{ Note that it is sufficient to prove Theorem \\ref{noeigenvalue} and Theorem \\ref{thprincipal} for Hermitian matrices $A_n^{(1)},\\ldots,A_n^{(t)}$ by considering their Hermitian and anti-Hermitian parts, so that throughout the paper we assume that the $A_n^{(i)}$'s are Hermitian.}\\end{remarque}\n\n\n\\noindent We adopt the strategy from \\cite{HT} and \\cite{Schultz05} based on a linearization trick and sharp estimates on matricial Stieltjes transforms. More precisely, both proofs of Theorem \\ref{thprincipal} and Theorem \\ref{noeigenvalue} are based on the following key Lemma \\ref{inclu2}. First, note that the algebra $M_m(\\C)\\otimes {\\cal A}$ formed by the $m\\times m $ matrices with coefficients in ${\\cal A}$, inherits the structure of ${\\cal C}^*$-probability space with trace $\\frac{1}{m} \\Tr_m \\otimes \\tau$ and norm \n$$\\Vert b \\Vert = \\lim_{k\\rightarrow +\\infty} \\left( \\frac{1}{m} \\Tr_m \\otimes \\tau \\left[(b^*b)^k\\right]\\right)^{\\frac{1}{2k}}, \\; \\forall b \\in M_m(\\C)\\otimes {\\cal A} .$$\n\n\n\\begin{lemme} \\label{inclu2} Let $({\\cal A}, \\tau)$ be a ${\\cal C}^*$-probability space\n equipped with a faithful tracial state and \n $x=(x_1,\\ldots,x_r)$ be a semi-circular system in $({\\cal A}, \\tau)$. Let $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint random variables which is free from $x$ in $({\\cal A},\\tau$), such that the distribution of $a_n$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}), \\frac{1}{n}\\Tr_n)$. Then, for all $m \\in \\N$, all self-adjoint matrices $\\gamma, \\alpha_1, \\ldots,\\alpha_r, \\beta_1, \\ldots, \\beta_t$ of size $m\\times m$ and\nall $\\epsilon >0$, almost surely, for all large $n$, we have\\\\\n\n\\noindent $\nsp(\\gamma \\otimes I_n + \\sum_{v=1}^r \\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}+ \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)})$ \\begin{equation} \\label{spectre3} \\subset\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_v + \\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}) + ]-\\epsilon, \\epsilon[.\n\\end{equation}\n Here, $sp(T)$ denotes the spectrum of the operator $T$, $I_n$ the identity matrix and $1_{\\cal A}$ denotes the unit of ${\\cal A}$.\n\\end{lemme}\n\n\\begin{remarque}\\label{remarqueinversible}\nBy a density argument, it is sufficient to prove Lemma \\ref{inclu2} for invertible self-adjoint matrices $\\gamma, \\alpha_1, \\ldots,\\alpha_r, \\beta_1, \\ldots, \\beta_t$. This invertibility will be used in the proof of Lemma \\ref{inversion}.\n\\end{remarque}\n The proof of \\eqref{spectre3} requires sharp estimates of $g_n(z)-\\tilde g_n(z)$\nwhere for $z\\in \\mathbb{C}\\setminus \\mathbb{R}$, $$g_n(z) =\\mathbb{E} \\frac{1}{m} \\Tr_m \\otimes \\frac{1}{n} \\Tr_n [ (zI_m \\otimes I_n - \\gamma \\otimes I_n - \\sum_{v=1}^r \\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}(\\omega)- \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)})^{-1}]$$ and $$\\tilde g_n(z) = \\frac{1}{m} \\Tr_m \\otimes \\tau [ (zI_m \\otimes 1_{\\cal A} - \\gamma \\otimes 1_{\\cal A} - \\sum_{v=1}^r \\alpha_v \\otimes x_v - \\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)})^{-1}].$$\n More precisely we are going to establish that \n there exists a polynomial $Q$ with nonnegative coefficients such that, for $z \\in \\C \\setminus \\mathbb{R}$, \n\\begin{equation} \\label{estimdiffeqno}\n\\left|g_n(z)-\\tilde g_n(z)+{\\tilde{E}_n(z)}\\right|\\leq \\frac{Q(\\vert \\Im z\\vert^{-1})}{n\\sqrt{n}},\n\\end{equation}\nwhere $\\tilde{E}_n$ is the Stieltjes transform of a compactly supported distribution $\\nabla_n$ on $\\mathbb{R}$ whose support is\nincluded in the spectrum of $\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r \\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$ and such that $\\nabla_n(1)=0$.\\\\\n But this required sharp estimate makes necessary a technical piece of work and a fit use of free operator-valued subordination maps (see Section \\ref{free}). In particular, we need an explicit development of the Stieltjes transform up to the order $\\frac{1}{n\\sqrt{n}}$ but the stability under perturbation argument used in \\cite{CamilleM} does not provide this development from the approximate matricial subordination equation. Therefore we use a strategy based on an invertibility property of matricial subordination maps related to semi-circular system (see Lemma \\ref{inversion}).\\\\\n \n Theorem \\ref{thprincipal} can be deduced from Lemma \\ref{inclu2} by following the proofs in \\cite{HT} and \\cite{Schultz05}. \nGiven a noncommutative polynomial $P$, choosing in Lemma \\ref{inclu2} the $\\gamma$, $(\\alpha_v)_{v=1,\\ldots,r}$, $(\\beta_u)_{u=1,\\ldots,t}$ corresponding to a self-adjoint linearization of $P$ as defined in \\cite{A} allows to deduce Theorem \\ref{noeigenvalue}. \\\\\n\n\nIn Section \\ref{troncation}, we explain why, using a truncation and Gaussian convolution procedure, it is sufficient to prove Theorem \\ref{thprincipal} and Theorem \\ref{noeigenvalue}\n when we assume that the $X_{ij}^{(v)}$'s satisfy:\n\\begin{itemize} \n\\item[(H)] the variables $\\sqrt{2}\\Re X_{ij}^{(v)}$, $\\sqrt{2}\\Im X_{ij}^{(v)}$, $1\\leq i0$ and a nonnegative random variable $Y$ with finite fourth moment for which there exists $x_0>0$ and an integer number $n_0>0$ such that, for any $x >x_0$ and any integer number $n >n_0$, we have \\begin{equation}\\label{majquatreZ}\\frac{1}{n^2} \\sum_{1\\leq i\\leq n,1\\leq j\\leq n}\\mathbb{P}\\left( \\vert Z_{ij}\\vert >x\\right) \\leq K \\mathbb{P}\\left( Y >x\\right).\\end{equation}\nThen, setting $\\sigma^*=\\{\\sup_{1\\leq it\\right) dt \\geq \\sum_{l=1}^{+\\infty} u_l^3 (u_{l+1}-u_l) \\mathbb{P} \\left( \\frac{Y}{\\sigma^*}>u_{l+1}\\right), $$\n it readily follows that for any $\\delta>0$, choosing $u_l= \\delta 2^{(l-1)\/2}$, we have\n $$\\sum_{l=1}^\\infty 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\delta 2^{l\/2}\\right) <\\infty.$$\n In particular, there exists an increasing sequence $(N_k)_{k\\geq 1}$ of integer numbers such that \n for any $k \\geq 1$, $$\\sum_{l=N_k+1}^\\infty 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\frac{1}{2^{\\frac{k}{8}}} 2^{l\/2}\\right) \\leq \\frac{1}{2^k}.$$\n Set for any $l \\in ]0,N_{1}]$, $\\epsilon_l={1}$ and for any $l \\in ]N_k,N_{k+1}]$, $\\epsilon_l=\\frac{1}{2^{\\frac{k}{8}}}$. Then, \n $$\\sum_{l=N_1 +1}^\\infty 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\epsilon_l 2^{l\/2}\\right)= \\sum_{k=1}^{+\\infty}\\sum_{l=N_k+1}^{N_{k+1}} 2^{2l} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\frac{1}{2^{\\frac{k}{8}} }2^{l\/2}\\right)\\leq \\sum_{k=1}^{+\\infty}\\frac{1}{2^k} <\\infty.$$\n Define $\\delta_n= \\sqrt{2}\\epsilon_l$ for $2^{l-1} < n \\leq 2^{l}$. \n Thus, we exhibited a sequence of nonnegative numbers such that $\\delta_n \\downarrow 0$,\n \n $\\delta_n^2 \\sqrt{n} \\rightarrow +\\infty$ and $$\\sum_{n=1}^\\infty 2^{2n} \\mathbb{P} \\left( \\frac{Y}{\\sigma^*} >\\delta_{2^n} 2^{(n-1)\/2}\\right) <\\infty.$$\nLet us consider $X=Z_n\/\\sigma^*$.\nDefine for any $i\\geq 1, j\\geq 1$, $\\check X_{ij}= X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n} $.\nWe have for any $k$ large enough, \n \\begin{eqnarray*}\n\\mathbb{P}\\left(X\\neq \\check X \\; i.o \\right) &\\leq & \\sum_{l=k}^\\infty \\mathbb{P} \\left( \\bigcup_{2^{l-1}< n \\leq 2^{l}} \\bigcup_{1\\leq i,j\\leq n} \\left\\{\\vert X_{ij} \\vert > \\sqrt{n}\\delta_n\\right\\} \\right)\\\\\n &\\leq & \\sum_{l=k}^\\infty \\mathbb{P} \\left( \\bigcup_{2^{l-1}< n \\leq 2^{l}} \\bigcup_{1\\leq i,j\\leq n} \\left\\{\\vert X_{ij} \\vert > 2^{\\frac{l-1}{2}}\\delta_{2^l}\\right\\} \\right)\\\\\n &\\leq & \\sum_{l=k}^\\infty \\sum_{1\\leq i,j\\leq 2^{l}} \\mathbb{P} \\left( \\vert X_{ij} \\vert > 2^{\\frac{l-1}{2}}\\delta_{2^l} \\right)\\\\\n &\\leq & K\\sum_{l=k}^\\infty 2^{2l} \\mathbb{P} \\left( Y > 2^{\\frac{l-1}{2}} \\delta_{2^l } \\sigma^*\\right) \\rightarrow_{k \\rightarrow +\\infty} 0.\n\\end{eqnarray*}\n\n\n\n\\noindent Define for $i\\neq j$, $1\\leq i,j\\leq n,$ $ \\hat X_{ij}= X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n} $ and for any $i$, $\\hat X_{ii}=0$. Then, we have $\\left\\| \\frac{\\check X}{\\sqrt{n}}- \\frac{\\hat X}{\\sqrt{n}}\\right\\| \\leq \\delta_n\\rightarrow_{n\\rightarrow +\\infty} 0.$\n\\noindent Finally define for $i\\neq j$, $1\\leq i,j\\leq n$, $\\tilde X_{ij}= X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n} -\\mathbb{E}\\left( X_{ij}{\\1}_{\\vert X_{ij} \\vert \\leq \\sqrt{n} \\delta_n}\\right)$ and for any $i$, $\\tilde X_{ii}=0$.\nWe have for large $n$ (denoting by $\\Vert\\cdot\\Vert_2$ the Hilbert-Schmidt norm)\\begin{eqnarray*} \n\\left\\| \\frac{\\hat X}{\\sqrt{n}}- \\frac{\\tilde X}{\\sqrt{n}}\\right\\|&\\leq &\\left\\| \\frac{\\hat X}{\\sqrt{n}}- \\frac{\\tilde X}{\\sqrt{n}}\\right\\|_{2}\n\\\\&=&\\left( \\frac{1}{n}\\sum_{i,j=1;i\\neq j }^n \\left|\\mathbb{E}\\left( X_{ij}{\\1}_{\\vert X_{ij} \\vert > \\sqrt{n} \\delta_n} \\right)\\right|^2\\right)^{1\/2}\\\\\n&\\leq&\\left( \\frac{1}{n^3 \\delta_n^4}\\sum_{i,j=1}^n \\mathbb{E}\\left( \\left| X_{ij}^2\\right| \\right)\\mathbb{E}\\left( \\left| X_{ij}\\right|^4{\\1}_{\\vert X_{ij} \\vert > \\sqrt{n} \\delta_n} \\right)\\right)^{1\/2}\\\\\n&\\leq &\\left( \\frac{K}{n \\delta_n^4}\\mathbb{E}\\left( \\left| \\frac{Y}{\\sigma^*} \\right|^4{\\1}_{\\vert \\frac{Y}{\\sigma^*} \\vert > \\sqrt{n} \\delta_n} \\right)\\right)^{1\/2} \\rightarrow_{n\\rightarrow +\\infty} 0.\n\\end{eqnarray*}\nThus, we have that almost surely \\begin{equation}\\label{normetilde}\\left\\| \\frac{ X}{\\sqrt{n}}- \\frac{\\tilde X}{\\sqrt{n}}\\right\\|\\rightarrow_{n\\rightarrow +\\infty} 0.\\end{equation}\nNote that the entries of $\\tilde X$ satisfy \n\\begin{itemize}\n\\item $\\tilde X_{ii}=0$;\n\\item $\\tilde X_{ij}$, $i0 \\mbox{\\;and all \\;} i\\neq j, \\; l\\geq~3$.\n\\end{itemize}\nThen, sticking to the end of the proof of Theorem 5.1 pages 87-93 in \\cite{BaiSil06}, one can prove that for any even integer $k$, one has \n$$\\mathbb{E}\\left(\\Tr \\left(\\frac{\\tilde X}{\\sqrt{n}}\\right)^k \\right)\\leq n^2\\left[2+ (10(2\\delta_n)^{1\/3} k\/\\log n)^3\\right]^k.$$\nChebychev's inequality yields that for any $\\eta >2$, \n\\begin{equation}\\label{chebychev}\\mathbb{P}\\left(\\left\\|\\frac{\\tilde X}{\\sqrt{n}}\\right\\| >\\eta\\right)\\leq \\frac{1}{\\eta^k}\\mathbb{E}\\left(\\Tr \\left(\\frac{\\tilde X}{\\sqrt{n}}\\right)^k \\right)\n\\leq \nn^2\\left[\\frac{2}{\\eta}+ \\frac{(10 (2\\delta_n)^{1\/3} k\/\\log n)^3}{\\eta}\\right]^k.\\end{equation}\nSelecting the sequence of even integers $k_n=2 \\left\\lfloor \\frac{\\log n }{\\delta_n^{1\/6}}\\right\\rfloor$ with the properties $k_n\/\\log n \\rightarrow +\\infty$ and $k_n \\delta_n^{1\/3}\/\\log n \\rightarrow 0$, we obtain that the right hand side of \\eqref{chebychev} is summable.\nUsing Borel-Cantelli's Lemma, we easily deduce that almost surely $$\\limsup_{n\\rightarrow +\\infty} \\left\\|\\frac{\\tilde X}{\\sqrt{n}}\\right\\| \\leq 2$$ and then, using \\eqref{normetilde}, that almost surely $$\\limsup_{n\\rightarrow +\\infty} \\left\\|\\frac{ X}{\\sqrt{n}}\\right\\| \\leq 2.$$\n\\end{proof}\nAccording to Lemma \\ref{baiyin}, for any $v=1,\\ldots,r$, almost surely, \n$\\sup_n \\left\\| \\frac{X_n^{(v)}}{\\sqrt{n}} \\right\\| < +\\infty.$ Therefore by a simple approximation argument using polynomials with coefficients in $\\mathbb{Q} +i \\mathbb{Q}$, to establish Theorem \\ref{thprincipal}, it is sufficient to prove that for any polynomial, almost surely \\eqref{af} and \\eqref{saf} hold. Now, we are going to show that the proof of Theorem \\ref{thprincipal} can be reduced to the proof of \\eqref{saf} in the case where the $X_{ij}^{(v)}$'s satisfy $(H)$.\\\\\nLet $X = [X_{jk}]_{j,k=1}^n$ be a Hermitian $n\\times n$ matrix such that the random variables $X_{ii}$,\n$\\sqrt{2} \\Re (X_{ij})$, $\\sqrt{2} \\Im (X_{ij}), {i0$ and an integer number $n_0>0$ such that, for\nany $x>x_0$ and any integer number $n>n_0$, we have \\begin{equation}\\label{majquatre}\\frac{1}{n^2} \\sum_{i\\leq n, j\\leq n}\\mathbb{P}\\left( \\vert X_{ij}\\vert >x\\right) \\leq K\\mathbb{P}\\left(\\vert Z \\vert>x\\right).\\end{equation}\n\n\n\\noindent Define for any $C>0$, for any $1\\leq i , j \\leq n$,\n\\begin{eqnarray}Y_{ij}^C &=&\\Re X_{ij}\\1_{\\vert \\Re X_{ij} \\vert \\leq C} - \\mathbb{E}\\left( \\Re X_{ij}\\1_{\\vert \\Re X_{ij} \\vert \\leq C} \\right) \\nonumber \\\\&&+ \\sqrt{-1} \\left\\{\n\\Im X_{ij}\\1_{\\vert \\Im X_{ij} \\vert \\leq C} - \\mathbb{E}\\left( \\Im X_{ij}\\1_{\\vert \\Im X_{ij} \\vert \\leq C} \\right) \\right\\}.\\label{ycdef}\\end{eqnarray}\nWe have \n\\begin{eqnarray*}\\mathbb{E} \\left( \\vert X_{ij}-Y_{ij}^C\\vert^2 \\right)&=&\\mathbb{E} \\left( \\vert \\Re X_{ij}\\vert^2 \\1_{\\vert \\Re X_{ij} \\vert > C} \\right)\n+ \\mathbb{E} \\left( \\vert \\Im X_{ij}\\vert^2 \\1_{\\vert \\Im X_{ij} \\vert > C} \\right) \\\\&&\n-\\left\\{ \\mathbb{E} \\left( \\Re X_{ij} \\1_{\\vert \\Re X_{ij} \\vert > C} \\right)\\right\\}^2-\\left\\{ \\mathbb{E} \\left( \\Im X_{ij} \\1_{\\vert \\Im X_{ij} \\vert > C} \\right)\\right\\}^2\\\\& \\leq & \\frac{\\mathbb{E} \\left(\\vert \\Re X_{ij} \\vert^3\\right)+\\mathbb{E} \\left(\\vert \\Im X_{ij} \\vert^3\\right)}{C}\n\\end{eqnarray*}\nso that $$\\sup_{i\\geq 1,j\\geq 1}\\mathbb{E} \\left( \\vert X_{ij}-Y_{ij}^C\\vert^2 \\right) \\leq \\frac{2\\theta^*}{C}.$$\nAccording to Lemma \\ref{baiyin}, we have almost surely \\begin{equation}\\label{centragebis}\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{X}{\\sqrt{n}}-\\frac{Y^C}{\\sqrt{n}}\\right\\| \\leq 2\\frac{\\sqrt{2\\theta^*}}{\\sqrt{C}}.\\end{equation}\nNote that \n\\begin{eqnarray*}1 - 2 \\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right) &=& 1-2 \\mathbb{E} \\left\\{\\left(\\Re X_{ij} \\1_{\\vert \\Re X_{ij} \\vert \\leq C} - \\mathbb{E} \\left( \\Re X_{ij} \\1_{\\vert \\Re X_{ij} \\vert \\leq C}\\right)\\right)^2\\right\\}\n\\\\&=& 2\\left[ \\frac{1}{2} -\\mathbb{E} \\left( \\vert \\Re X_{ij}\\vert^2 \\1_{\\vert \\Re X_{ij} \\vert \\leq C} \\right)\\right] \n+2\\left\\{ \\mathbb{E} \\left( \\Re X_{ij} \\1_{\\vert \\Re X_{ij} \\vert \\leq C} \\right)\\right\\}^2\\\\&=& \n2 \\mathbb{E} \\left( \\vert \\Re X_{ij}\\vert^2 \\1_{\\vert\\Re X_{ij} \\vert > C} \\right)+ 2\\left\\{\\mathbb{E} \\left( \\Re X_{ij} \\1_{\\vert \\Re X_{ij} \\vert > C}\\right) \\right\\}^2. \\end{eqnarray*}\nso that $$\\sup_{i\\geq 1, j\\geq 1}\\vert 1 - 2\\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right) \\vert \\leq \\frac{4\\theta^*}{C}.$$\nSimilarly $$\\sup_{i\\geq 1,j\\geq 1}\\vert 1 - 2\\mathbb{E} \\left( \\vert \\Im Y_{ij}^C\\vert^2 \\right) \\vert \\leq \\frac{4\\theta^*}{C}.$$\nLet us assume that $C>8\\theta^*.$ Then, we have \n$$\\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right)> \\frac{1}{4} \\; \\mbox{and}\\; \\mathbb{E} \\left( \\vert \\Im Y_{ij}^C\\vert^2 \\right)> \\frac{1}{4}.$$\nNow define \\begin{equation}\\label{defxc}{X}_{ij}^C =\\frac{\\Re Y_{ij}^C}{\\sqrt{2\\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right)}} +\\sqrt{-1} \\frac{\\Im Y_{ij}^C}{\\sqrt{2\\mathbb{E} \\left( \\vert \\Im Y_{ij}^C\\vert^2 \\right)}}.\\end{equation}\nNote that \\begin{eqnarray*} {X}_{ij}^C-Y_{ij}^C&=& \\Re X_{ij}^C \\left( 1-\\sqrt{2} \\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right)^{1\/2}\\right)\n+\\sqrt{-1} \\Im X_{ij}^C \\left( 1-\\sqrt{2} \\mathbb{E} \\left( \\vert \\Im Y_{ij}^C\\vert^2 \\right)^{1\/2}\\right)\n\\\\&=&\\Re X_{ij}^C\\frac{1 - 2\\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right) }{1 + \\sqrt{2}\\mathbb{E} \\left( \\vert \\Re Y_{ij}^C\\vert^2 \\right)^{1\/2}}+\n\\sqrt{-1} \\Im X_{ij}^C\\frac{1 - 2\\mathbb{E} \\left( \\vert \\Im Y_{ij}^C\\vert^2 \\right) }{1 + \\sqrt{2}\\mathbb{E} \\left( \\vert \\Im Y_{ij}^C\\vert^2 \\right)^{1\/2}}.\\end{eqnarray*} \n\\noindent Thus, according to Lemma \\ref{baiyin}, we have almost surely \\begin{equation}\\label{normal}\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{X^C}{\\sqrt{n}}-\\frac{Y^C}{\\sqrt{n}}\\right\\| \\leq \\frac{{8\\theta^*}}{{C}}.\\end{equation}\nThus, by \\eqref{centragebis} and \\eqref{normal}, we obtain that almost surely \\begin{equation}\\label{xc}\\limsup_{n\\rightarrow+\\infty} \\left\\| \\frac{X}{\\sqrt{n}}-\\frac{X^C}{\\sqrt{n}}\\right\\| \\leq 2\\frac{\\sqrt{2\\theta^*}}{\\sqrt{C}}+ \\frac{{8\\theta^*}}{{C}}.\\end{equation}\nLet $[{\\cal G}_{ij}]_{i\\geq 1, j\\geq 1}$ be an infinite array which is independent of the $X_{ij}'s$ and such that $\\sqrt{2} \\Re {\\cal G}_{ij}$, $ \\sqrt{2} \\Im {\\cal G}_{ij}$, $i0$,\n\\begin{equation}\\label{xcdelta}X^{C,\\delta}= \\frac{ X^C +\\delta {\\cal G}}{\\sqrt{1+\\delta^2}}.\\end{equation}\nNote that the random variables $\\sqrt{2}\\Re (X^{(v)})^{C,\\delta}_{ij}$, $\\sqrt{2}\\Im (X^{(v)})^{C,\\delta}_{ij}$, $ i 8\\theta^*$, we have almost surely\n\\begin{equation}\\label{bybis} \\limsup_{n\\rightarrow+\\infty}\\left\\| \\frac{{X^C}}{\\sqrt{n}} \\right\\|\\leq 2.\\end{equation}\nNow, \\eqref{xc}, \\eqref{by} and \\eqref{bybis} yield that for any $C > 8 \\theta^*$, any $\\delta >0$, almost surely $ \\limsup_{n\\rightarrow +\\infty}\\left\\|\\frac{X -{X^{C,\\delta}}}{\\sqrt{n}} \\right\\| \\leq u_C +v_\\delta$ where $u_C$ and $v_\\delta$ are deterministic positive functions tending to zero when respectively $C$ goes to infinity and $\\delta$ goes to zero. \nHence, it is easy to see that for any $0<\\epsilon<1$, there exists $C_\\epsilon$ and $\\delta_\\epsilon$ such that almost surely for all large $n$, \n$$\\left\\|\\frac{X -{X^{C_\\epsilon,\\delta_\\epsilon}} }{\\sqrt{n}}\\right\\|\\leq \\epsilon.$$\nWe can deduce that for any polynomial $P$ in $r+t$ noncommutative variables, there exists some constant $L>0$ such that the following holds: for any $0<\\epsilon<1$, there exists $C_\\epsilon$ and $\\delta_\\epsilon$ such that almost surely for all large $n$, \\\\\n\n\\noindent $\\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}},\\ldots,\\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}\\right)\\right. $ \\begin{equation}\\label{approxnorme} \\left.- P\\left(\\frac{(X_n^{(1)})^{C_\\epsilon,\\delta_\\epsilon}}{\\sqrt{n}},\\ldots,\\frac{(X_n^{(r)})^{C_\\epsilon,\\delta_\\epsilon}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}\\right)\\right\\|\\leq L \\epsilon.\\end{equation} \nThen, it is clear that it is sufficient to establish Theorem \\ref{thprincipal} and Theorem \\ref{noeigenvalue} for the $(X_n^{(v)})^{C_\\epsilon,\\delta_\\epsilon}$'s. \\\\Moreover, we obviously have that for any $\\epsilon$ and for any $p\\in \\mathbb{N}$, \\begin{equation}\\label{moments}\\max_{v=1,\\ldots,r} \\sup_{i\\geq 1, j\\geq 1} \\mathbb{E}\\left(\\vert (X_{ij}^{(v)})^{C_\\epsilon,\\delta_\\epsilon}\\vert^p\\right) <+\\infty.\\end{equation}\n Then, \\eqref{af} is a consequence of Theorem 5.4.5 in \\cite{AGZ}.\\\\\n\n\n Moreover, note that, by definition, the distributions of the random variables\n$\\sqrt{2}\\Re (X^{(v)})_{ij}^{C_\\epsilon,\\delta_\\epsilon}, \\sqrt{2}\\Im (X^{(v)})_{ij}^{C_\\epsilon,\\delta_\\epsilon}, i0$ if the matrix $M$ is positive definite and $M\\ge0$ if it is nonnegative definite.\nIn general $M>P$ means that $M-P$ is positive definite.\n \\end{itemize}\n\\noindent We now define the random variables of interest. \nLet $({\\cal A}, \\tau)$ be a ${\\cal C}^*$-probability space with unit $1_{\\cal A}$,\n equipped with a faithful tracial state and \n $x=(x_1,\\ldots,x_r)$ be a semi-circular system in $({\\cal A}, \\tau)$.\n Let $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint random variables which is free from $x$ in $({\\cal A},\\tau$) and such that the distribution of $a_n$ in $({\\cal A},\\tau)$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$.\n\\begin{itemize}\n\n \\item[-] \n We define the random variable $S_n$ with values in $M_m(\\C) \\otimes M_n(\\C)$ by:\n \\begin{equation} \\label{defSn}\n S_n = \\gamma \\otimes I_n + \\sum_{v=1}^r \\alpha_v \\otimes \\frac{X^{(v)}_n}{\\sqrt{n}}+ \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}\n \\end{equation}\nand $s_n\\in M_m(\\C) \\otimes {\\cal A}$ by \n$$s_n=\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r\\alpha_v \\otimes x_v+ \\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}.$$\n \\item[-] For any matrix $\\lambda$ in $ M_m(\\C) $ such that $ \\Im(\\lambda)$ is positive definite,\n we define the $M_m(\\C)\\otimes M_n(\\C)$-valued respectively $M_m(\\C) \\otimes {\\cal A}$-valued random variables:\n \\begin{equation}\\label{rn}\n\tR_n(\\lambda)=(\\lambda \\otimes I_n - S_n)^{-1},\n\t\\end{equation}\n\t$$r_{n}(\\lambda)= \\left(\\lambda \\otimes 1_{\\cal A} -s_n \\right)^{-1},$$\n\tand \n\tthe $M_m(\\C)$-valued random variables:\n \\begin{equation} \\label{defSt}\n H_n(\\lambda) = ({\\rm id}_m \\otimes \\tr_n) [ (\\lambda \\otimes I_n - S_n)^{-1}],\n \\end{equation}\n \\begin{equation} \\label{espSt}\nG_n(\\lambda) =\\mathbb{ E}[ H_n(\\lambda) ]\n\\end{equation}\nand\n\\begin{equation}\\label{defGntilde}\\tilde G_{n}(\\lambda)={\\rm id}_m \\otimes \\tau \\left(r_n (\\lambda ) \\right).\\end{equation}\nSince $\\sum_{v=1}^r\\alpha_v \\otimes x_v$ is an $ M_m(\\mathbb C)$-valued semicircular of variance $\\eta\\colon b\\mapsto\\sum_{v=1}^r\\alpha_vb\\alpha_v$ which is free over $ M_m(\\mathbb C)$ from $\\gamma\\otimes 1_{\\cal A}+\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$, we know from \\cite{ABFN} (see proof of Theorem 8.3) that $\\tilde G_{n}$ satisfies (see Section \\ref{free} Theorem \\ref{resusub} for $p=1$) \\begin{equation}\\label{subor}\\tilde G_{n}(\\lambda)= G_{\\sum_{u=1}^t\\beta_s \\otimes a_n^{(u)}}(\\omega_n(\\lambda))\n\\end{equation}\nwhere \\begin{equation}\\label{omegan}\n\\omega_n(\\lambda)=\\lambda-\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde G_n(\\lambda)\\alpha_v\n\\end{equation} and $$G_{\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}}(\\lambda)={\\rm id}_m \\otimes \\tau \\left(\\lambda\\otimes 1_{\\cal A} -\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)} \\right)^{-1}.$$\n\\noindent For $z\\in \\C \\setminus \\R$, we also define\n \\begin{equation}\\label{defpetitg}g_n(z) = \\tr_m(G_n(zI_m))\\end{equation}and \\begin{equation}\\label{defpetitgtilde}\\tilde g_n(z) = \\tr_m(\\tilde G_n(z I_m)).\\end{equation}\n\n\n\n\n\n\n\n\n\n\n \n\n \n \\end{itemize}\n\n In the sequel, we will say that a random term in some $M_p(\\C)$, depending on $n$, $\\lambda \\in M_m(\\mathbb{C}) $ such that $\\Im \\lambda$ is positive definite, the $\\{\\alpha_v\\}_{v=1,\\ldots,r}$, $\\{\\beta_u\\}_{u=1,\\ldots,t}$ and $\\gamma$, is $O\\left(\\frac{1}{n^k}\\right)$ if its operator norm is smaller than\n $\\frac{ Q\\left(\\Vert (\\Im \\lambda)^{-1} \\Vert\\right)}{n^k}$ for some deterministic polynomial $Q$\n whose coefficients are nonnegative real numbers and can depend on $m$, $\\{\\alpha_v\\}_{v=1,\\ldots,r}$, $\\{\\beta_u\\}_{u=1,\\ldots,t}$, $\\gamma$.\\\\\n For a family of random terms $I_{pq}$, $(p,q) \\in \\{1,\\ldots,n\\}^2$, we will set $I_{pq}=O_{p,q}^{(u)} \\left(\\frac{1}{n^k}\\right)$ if for each $(p,q)$, $I_{pq}=O \\left(\\frac{1}{n^k}\\right)$ and moreover one can find a bound of \n\nthe norm of each $I_{p,q}$ as above involving a common polynomial $Q$.\\\\\n\nThroughout the paper, $K$, $C$ denote some positive constants and $Q$ denotes some deterministic polynomial in one variable\n whose coefficients are nonnegative real numbers; they can depend on $m$, $\\{\\alpha_v\\}_{v=1,\\ldots,r}$, $\\{\\beta_u\\}_{u=1,\\ldots,t}$ and $\\gamma$ and they may vary from line to line.\n\n\n\n\n\n\n\n\n\n\\section{Operator-valued subordination}\\label{free}\n\n\nIn this section we introduce one of the main tools used in describing joint distributions of\nrandom variables that do not necessarily commute. It was a crucial insight of Voiculescu\n\\cite{V2000, FreeMarkov, V1} that J. L. Taylor's theory of free noncommutative functions \\cite{taylor} \nprovides the appropriate analogue of the classical Stieltjes transform for encoding\noperator-valued distributions \\cite{V1995}, and hence joint distributions of $q$-tuples of\nnoncommuting random variables. We shall only present below the case of relevance to our\npresent work, and refer the reader to \\cite{V1995,V2000,V1} for the general case and for the\nproofs of the main results.\n\nGiven a tracial $\\mathcal C^*$-probability space $(\\mathcal A,\\tau)$ and a trace and order preserving\nunital $\\mathcal C^*$ inclusion $M_m(\\mathbb C)\\subseteq\\mathcal A$ (i.e. such that $I_m\\in \nM_m(\\mathbb C)$ identifies with the unit of $\\mathcal A$ and ${\\rm tr}_m(b)=\\tau(b)$ for all $b\\in \nM_m(\\mathbb C)$), there exists a conditional expectation $E\\colon\\mathcal A\\to M_m(\\mathbb C)$, i.e. \na linear map sending the unit to itself and such that $E(b_1yb_2)=b_1E(y)b_2$ for all $b_1,b_2\\in M_m(\\mathbb C),y\n\\in\\mathcal A$ - see \\cite[Section II.6.10.13]{Bruce}. We will only be concerned with the trivial case\nof the canonical inclusion $M_m(\\mathbb C)\\subseteq M_m(\\mathbb C)\\otimes\\mathcal A$ \ngiven by $b\\mapsto b\\otimes 1_{\\cal A}$, when the conditional expectation is the partial trace:\n$E(b\\otimes y)=({\\rm id}_m\\otimes\\tau)(b\\otimes y)=\\tau(y)b$.\nThe $M_m(\\mathbb C)$-valued distribution of an element $y\\in\\mathcal A$ with respect to $E$ is by \ndefinition the family of multilinear maps \n$\\mu_y=\n\\{\\Psi_q\\colon\\underbrace{M_m(\\mathbb C)\\times\\cdots\\times M_m(\\mathbb C)}_{q-1\\text{ times}}\n\\to M_m(\\mathbb C)\\colon \\Psi_q(b_1,\\dots,b_{q-1})=E[yb_1y\\cdots b_{q-1}y],q\\in\\mathbb N\\}$. \nBy convention, $\\Psi_0=1\\in M_m(\\mathbb C)$, $\\Psi_1=E[y]$.\n\nFor a given $y=y^*\\in\\mathcal A$ with distribution $\\mu_y$, define its noncommutative \nStieltjes transform to be the {\\em countable family} of maps\n$G_{\\mu_y,p}(b)=(E\\otimes{\\rm id}_{p})\\left[(b-y\\otimes I_p)^{-1}\\right], p\\in \\mathbb{N}\\setminus\\{0\\}.$\nThus, $G_{\\mu_y,1}(b)=E\\left[(b-y)^{-1}\\right],b\\in M_m(\\mathbb C)$,\n$G_{\\mu_y,2}(b)=(E\\otimes{\\rm id}_{2})\\left[\\left(\\begin{bmatrix}\nb_{11} & b_{12}\\\\\nb_{21} & b_{22}\n\\end{bmatrix}-\\begin{bmatrix}\ny & 0\\\\\n0 & y\n\\end{bmatrix}\\right)^{-1}\\right],$ $b_{11}, b_{12},\nb_{21}, b_{22}\\in M_m(\\mathbb C)$ etc. These maps are clearly analytic on the open sets $\\{b\\in \nM_m(\\mathbb C)\\otimes M_p(\\mathbb C)\\colon b-y\\otimes I_p\\text{ invertible in }\\mathcal A\\otimes M_p(\\mathbb C)\\}$. Two \nsuch sets will be important in this paper: the noncommutative upper half-plane $H^+_p(M_m(\\mathbb \nC))=\\{b\\in M_m(\\mathbb C)\\otimes M_p(\\mathbb C)\\colon \\Im b:=(b-b^*)\/2i>0\\}$, $p\\in \\mathbb{N}\\setminus\\{0\\}$, \nand the ``ball around infinity'' $\\{b\\in M_m(\\mathbb C)\\otimes M_p(\\mathbb C) \\colon b\\text{ invertible, \n}\\|b^{-1}\\|<\\|y\\|^{-1}\\}$ (actually only for $p=1$). The maps $b\\mapsto G_{\\mu_y,p}(b^{-1})$ have thus an analytic extension \naround zero, which maps zero to zero and has the identity as first (Frechet) derivative. While \n$G_{\\mu_y,p}$ does not map these ``balls around infinity'' into themselves, it does map \n$H^+_p(M_m(\\mathbb C))$ into $-H^+_p(M_m(\\mathbb C))$, and moreover $G_{\\mu_y,p}(b)^{-1}$ maps \n$H^+_p(M_m(\\mathbb C))$ into itself (see \\cite[Section 3.6]{V2000}). (In addition, the family of maps \n$\\{G_{\\mu_y,p}\\}_{p\\in \\mathbb{N}\\setminus\\{0\\}}$ satisfy certain compatibility conditions that make them into free \nnoncommutative maps - see \\cite{KVV}. It is known \\cite{W} that there is a bijection between such families \nof maps $G$ that send for any $p \\in \\mathbb{N}\\setminus\\{0\\}$, $H^+_p(M_m(\\mathbb C))$ into $-H^+_p(M_m(\\mathbb C))$ and have the \nabove-described behavior on ``balls around infinity'' and $M_m(\\mathbb C)$-valued distributions of \nself-adjoint elements; however, since we will not make use of this correspondence, we chose to only \nmention it in order to illustrate the parallel to the case of classical Stieltjes transforms, and direct the \ninterested reader to \\cite{W} for details.)\n\n\n\nAs in scalar-valued free probability, one defines \\cite{V1995} {\\em freeness with amalgamation}\nover $M_m(\\mathbb C)$ via an algebraic relation similar to \\eqref{freeness}, but involving $E$\ninstead of $\\tau$ and noncommutative polynomials with coefficients in $M_m(\\mathbb C)$.\nSince it is not important for us here, we refer the interested reader to \\cite{V1995} for \nmore details. The essential result of Voiculescu that we will need in this paper is the following\nanalytic subordination result:\n\\begin{theoreme}\\label{resusub}\nWith the above notations, assume that $y_1=y_1^*,y_2=y_2^*\\in\\mathcal A$ are free with amalgamation\nover $M_m(\\mathbb C)$. For any $p\\in\\mathbb N$ there exist analytic maps $\\omega_{1,p},\n\\omega_{2,p}\\colon H^+_p(M_m(\\mathbb C))\\to H^+_p(M_m(\\mathbb C))$ such that:\n\\begin{enumerate}\n\\item For all $b\\in H^+_p(M_m(\\mathbb C)),$ $\\Im \\omega_{j,n}(b)\\ge\\Im b$, $j=1,2$;\n\\item For all $b\\in H^+_p(M_m(\\mathbb C))$\n$$G_{\\mu_{y_1+y_2},p}(b)=G_{\\mu_{y_1},p}(\\omega_{1,p}(b))=G_{\\mu_{y_2},p}(\\omega_{2,p}(b))\n=\\left[\\omega_{1,p}(b)+\\omega_{2,p}(b)-b\\right]^{-1}$$ \n\\item $\\omega_{1,p},\n\\omega_{2,p}$ are noncommutative maps in the sense of $\\cite{taylor}$ $($see $\\cite{KVV})$.\n\\end{enumerate}\n\\end{theoreme}\nThe result as phrased here is a combination of parts of \\cite[Theorem 3.8]{V2000} and\n\\cite[Theorem 2.7]{BMS}. We shall use this theorem in the particular case when $y_2=s$ is a centred \n{\\em operator-valued semicircular} random variable (and actually only for $p=1$). As for the scalar-valued semicircular\ncentred random variables, it is uniquely determined by its variance $\\eta\\colon b\\mapsto E(sbs)$,\nwhich is a completely positive self-map of $M_m(\\mathbb C)$. A characterization in terms of moments\nand cumulants via $\\eta$ is provided by Speicher in \\cite{SMem}. Given the context of our paper,\nwe find it more useful to provide a characterization in terms of the noncommutative \nStieltjes transform \\cite{HRS}: the functions $G_{\\mu_s,p}$ are the unique solutions mapping\n$H^+_p(M_m(\\mathbb C))$ into $-H^+_p(M_m(\\mathbb C))$ of the functional equations\n$$\nG_p(b)^{-1}=b-(\\eta\\otimes{\\rm id}_p)(G_p(b)),\\quad b\\in H^+_p(M_m(\\mathbb C)),\np\\in\\mathbb N.\n$$\nStarting from this equation, it can be shown (see \\cite{ABFN}) that the subordination function associated \nto a semicircular operator-valued random variable is particularly nice: if $y_1$ and $s$ are free with\namalgamation over $M_m(\\mathbb C)$, then \n\\begin{equation}\n\\omega_{1,p}(b)=b-(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1},p}(\\omega_{1,p}(b)))=b-\n(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1+s},p}(b)),\n\\end{equation}\nfor $b\\in H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N,$ or, equivalently,\n\\begin{equation}\nG_{\\mu_{y_1},p}(b-(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1+s},p}(\\omega_{1,p}(b)))=G_{\\mu_{y_1+s},p}(b),\n\\quad b\\in H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N.\n\\end{equation}\nThis indicates that the $\\omega_{1,p}$'s are injective maps on $H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N.\n$ Their left inverses are defined as\n\\begin{equation}\\label{tauto}\n{\\Lambda}_{1,p}(w)=w+(\\eta\\otimes{\\rm id}_p)(G_{\\mu_{y_1},p}(w)),\n\\quad w\\in H^+_p(M_m(\\mathbb C)),p\\in\\mathbb N.\n\\end{equation}\n\nLet us explain how all this relates to the joint distributions of free\nrandom variables. It turns out (see, for example, \\cite{NSS}, but it can be easily verified directly)\nthat if $\\{x_1=x_1^*,\\dots,x_r=x_r^*\\}, \\{y_1=y_1^*,\\dots,y_t=y_t^*\\}\\subset\\mathcal A$ are free over \n$\\mathbb C$ and for $v=1,\\ldots,r$, $\\alpha_v=\\alpha_v^*$, for $u=1,\\ldots,t$, $\\beta_u=\\beta_u^*\\in M_m(\\mathbb C)$,\nthen $\\{\\alpha_1\\otimes x_1,\\dots,\\alpha_r\\otimes x_r\\}$ and $\\{\\beta_1\\otimes y_1,\\dots,\\beta_t\\otimes y_t\\}$ are free \nwith amalgamation over $M_m(\\mathbb C)$. Thus, they can be treated with the tools described above.\nMoreover, if $x_1,\\dots,x_r$ are free $\\mathbb C$-valued semicircular centred random variables \nof variance one and $\\alpha_1,\\dots,\\alpha_r$ are self-adjoint $m\\times m$ complex matrices, then\n$\\alpha_1\\otimes x_1+\\cdots+\\alpha_r\\otimes x_r$ is a centred $M_m(\\mathbb C)$-valued semicircular of \nvariance $b\\mapsto\\sum_{j=1}^r \\alpha_jb\\alpha_j$. These simple facts together with a linearization trick\n(see Section \\ref{linearisation} and Step 1 of Section \\ref{strategie}) will allow us in principle to treat, from the point of view of\nthe Stieltjes transform, an $r$-tuple of Wigner matrices and deterministic matrices as we would treat a \nsingle Wigner matrix together with a single deterministic matrix. \\\\\n~~\n\n\\noindent Let us conclude this section with the following invertibility property of matricial subordination maps related to semi-circular system that will fundamental in our approach.\n\n\\begin{lemme}\\label{inversion}\nUsing the notations of Section \\ref{Notations}, define for any $\\rho$ in $M_m(\\mathbb{C})$ such that $\\Im \\rho>0$,\n\\begin{equation}\\Lambda_n(\\rho)= \\gamma +\\rho + \\sum_{v=1}^r \\alpha_v G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\rho) \\alpha_v.\\end{equation}\nWith $\\omega_n$ defined in \\eqref{omegan},\nwhen $\\Im \\rho>0$ and $\\Im \\Lambda_n(\\rho)>0$ we have \n$$\\omega_n(\\Lambda_n(\\rho))=\\rho.$$\n\\end{lemme}\n\\begin{proof}\nThe equality $\\Lambda_n(\\omega_n(\\rho))=\\rho$ holds tautologically for all $\\rho$ with $\\Im \\rho>0$ (see \\eqref{tauto}).\nLet us first show that the equality $\\omega_n(\\Lambda_n(\\rho))=\\rho$ holds when $\\rho$ has \na small enough inverse. \nThe map $\\Lambda_n$ has a power series expansion\n$$\n\\Lambda_n(\\rho)=\\rho+\\gamma+\\sum_{v=1}^r\\alpha_v\\left(\n\\sum_{k=0}^\\infty({\\rm id}_m\\otimes\\tau)\n\\left(\\rho^{-1}\\left[\\sum_{u=1}^t(\\beta_u\\otimes a_n^{(u)})\\rho^{-1}\\right]^k\\right)\\right)\\alpha_v,\n$$\nconvergent when $\\|\\rho^{-1}\\|<\\left\\|\\sum_{u=1}^t\\beta_u\\otimes a_n^{(u)}\\right\\|^{-1}$. For simplicity\nwe let $h(\\lambda)=\\sum_{v=1}^r\\alpha_v\\left(\n\\sum_{k=0}^\\infty({\\rm id}_m\\otimes\\tau)\n\\left(\\lambda\\left[\\sum_{u=1}^t(\\beta_u\\otimes a_n^{(u)})\\lambda\\right]^k\\right)\\right)\\alpha_v,$ norm\nconvergent on a ball of radius $\\left\\|\\sum_{u=1}^t\\beta_u\\otimes a_n^{(u)}\\right\\|^{-1}$ and fixing zero. \nPerforming the change of variable $\\lambda=\\rho^{-1}$, we obtain $\\Lambda_n(\\rho)=\n\\Lambda_n(\\lambda^{-1})=\\lambda^{-1}+\\gamma+h(\\lambda)$. Then\n$(\\Lambda_n(\\lambda^{-1}))^{-1}=(\\lambda^{-1}+\\gamma+h(\\lambda))^{-1}=\n\\lambda(1+(\\gamma+h(\\lambda))\\lambda)^{-1}$, which is analytic on the set of all $\\lambda\\in\n M_m(\\mathbb C)$ such that $\\|\\lambda\\|<\\left\\|\\sum_{u=1}^t\\beta_u\\otimes a_n^{(u)}\\right\\|^{-1}$\nand $\\|\\gamma+h(\\lambda)\\|<\\|\\lambda\\|^{-1}.$ \n\nDefine ${\\check\\Lambda_n}(\\rho)=(\\Lambda_n(\\rho^{-1}))^{-1}$ and ${\\check\\omega_n}\n(\\rho)=(\\omega_n(\\rho^{-1}))^{-1}$. We have established above that ${\\check\\Lambda_n}$ is analytic\non a neighbourhood of zero, and a direct computation shows that ${\\check\\Lambda_n}(0)=0,\n{\\check\\Lambda_n}'(0)={\\rm id}$. The inverse function theorem for analytic maps allows us to\nconclude that there exists a neighbourhood of zero on which ${\\check\\Lambda_n}$ has a unique inverse\nwhich fixes zero and whose derivative at zero is equal to the identity. \nThe map ${\\check\\omega_n}$ is shown precisely the same way to satisfy the same properties as\n${\\check\\Lambda_n}$. In particular, for $\\|\\rho\\|$ small enough, \n${\\check\\Lambda_n}({\\check\\omega_n}(\\rho))=(\\Lambda_n({\\check\\omega_n}(\\rho)^{-1}))^{-1}=\n(\\Lambda_n(((\\omega_n(\\rho^{-1}))^{-1})^{-1}))^{-1}=(\\Lambda_n(\\omega_n(\\rho^{-1}))^{-1}=\n(\\rho^{-1})^{-1}=\\rho$ for any $\\rho$ with strictly positive imaginary part. Since zero is in the\nclosure of $\\{\\rho\\in M_m(\\mathbb C)\\colon\\Im \\rho>0\\}$, it follows that ${\\check\\omega_n}$\nand ${\\check\\Lambda_n}$ are compositional inverses to each other on a small enough neighbourhood of\nzero. We conclude that for all $\\rho$ such that the lower bound of the spectrum of $\\Im \\rho$ is \nsufficiently large, $\\omega_n(\\Lambda_n(\\rho))=\\rho$.\n\n\nLet now $\\rho$ be fixed in $ M_m(\\mathbb{C})$ such that $\\Im \\rho>0$ and $\\Im \\Lambda_n(\\rho)>0$.\nLet $\\phi$ be a positive linear functional on $ M_m(\\mathbb{C})$ such that\n$\\phi(1)=1$ (i.e. a state). Define $\\varphi_\\rho(\\cdot )=\\phi(\\cdot )\/\\phi(\\Im \\rho)$. It is linear and \npositive (well defined because $\\Im \\rho\\ge\\frac{1}{\\|(\\Im \\rho)^{-1}\\|}1$, so that $\\phi(\\Im \\rho)\\ge\n\\frac{1}{\\| (\\Im \\rho)^{-1}\\|}>0$). Define \n$$\nf_\\rho(z)=\\varphi_\\rho\\left(\\Lambda_n(\\Re \\rho+z\\Im \\rho)\\right),\\quad z\\in\\mathbb C^+.\n$$\nNote that $$f_\\rho(z)=z+ \\varphi_\\rho(\\gamma+\\Re \\rho))+F(z)$$\nwhere $$F(z)=\\frac{\\phi\\left[ \\sum_{v=1}^r \\alpha_v {\\rm id}_m\\otimes \\tau \\left\\{\\left( (\\Re \\rho +z \\Im \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}\\alpha_v\\right] }{\\phi(\\Im \\rho)}.$$\n$F(z)$ is analytic on $\\mathbb{C}\\setminus \\mathbb{R}$ and satisfies $\\overline{F(z)}=F(\\bar{z}).$\nLet $z\\in \\mathbb{C}^+$.\nWe have \\\\\n\n\n\\noindent $\\Im F(z)$ $$= \\frac{\\phi\\left[ \\sum_{v=1}^r \\alpha_v{\\rm id}_m\\otimes \\tau \\left\\{ \\Im \\left\\{\\left( (\\Re \\rho +z \\Im \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\} \\right\\}\\alpha_v\\right] }{\\phi(\\Im \\rho)}$$\nwhere \\\\\n\n\\noindent $\\Im \\left\\{\\left( (\\Re \\rho +z \\Im \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}$ \\begin{eqnarray*}&=&\n-\\Im z \\left(( \\Re \\rho +z \\Im \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1} (\\Im \\rho \\otimes 1_{\\cal A})\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\left( (\\Re \\rho +\\bar{z} \\Im \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}<0.\n\\end{eqnarray*}\nIt follows by the complete positivity of the trace $\\tau$ that $${\\rm id}_m\\otimes \\tau\\left\\{ \\Im \\left\\{\\left(( \\Re \\rho +z \\Im \\rho)\\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}\\right\\}<0.$$\nNow, according to Remark \\ref{remarqueinversible}, we can assume the $\\alpha_v$'s invertible so that \n$\\sum_{v=1}^r \\alpha_v{\\rm id}_m\\otimes \\tau \\left\\{\\Im \\left\\{\\left( (\\Re \\rho +z \\Im \\rho) \\otimes 1_{\\cal A} -\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)} \\right)^{-1}\\right\\}\\right\\}\\alpha_v <0$ and then $\\Im F(z) <0$.\nThus for any $z\\in \\mathbb{C}\\setminus \\mathbb{R}$, we have $\\Im z \\Im F(z)<0.$\nFinally \n$$\\lim_{y \\rightarrow +\\infty} iy F(iy)= \\varphi_\\rho \\left( \\sum_{v=1}^r\\alpha_v (\\Im \\rho )^{-1} \\alpha_v \\right):=c_\\rho >0. $$\nThus, by Akhiezer-Krein's Theorem (\\cite{AK} page 93), there exists a probability measure $\\mu$ on $\\mathbb {R}$ such that\n$$\nF(z)= c_\\rho \\int_\\mathbb R\\frac{d\\mu(s)}{z-s}\n,\\quad z\\in\\mathbb C^+.\n$$\nThen\n$$\nf_{\\rho}(z)=z+\\varphi_\\rho(\\gamma+\\Re \\rho)+c_\\rho \\int_\\mathbb R\\frac{d\\mu(s)}{z-s}\n,\\quad z\\in\\mathbb C^+.\n$$\nThus $\\Im f_{\\rho}(u+iv)=v\\left(1-c_\\rho\\int_\\mathbb R\\frac{d\\mu(s)}{(u-s)^2+v^2}\\right)$. We observe \nthat what's under parenthesis is strictly increasing in $v$. Since by hypothesis, we have $\\Im \\Lambda_n(\\rho)>0$ and thus $\\Im f_{\\rho}(i)=\\Im \\varphi_\\rho\\left(\\Lambda_n(\\Re \\rho+i\\Im \\rho)\\right)\n>0$, we obtain \nimmediately that $\\Im f_{\\rho}(iv)>0$ for all $v\\ge1$. This means that $\\Im \n\\phi(\\Lambda_n(\\Re \\rho+iv\\Im \\rho))>0$ for all $v\\in[1,+\\infty)$ and all states $\\phi$, so that \\begin{equation}\\label{ray}\\Im \\Lambda_n (\\Re \\rho+iv\\Im \\rho)\n>0, {\\rm~ for~ all~} v\\ge1.\\end{equation}\nNow it is clear that $$\\Omega=\\{z \\in \\C^+, \\Im \\Lambda_n(\\Re \\rho + z\\Im \\rho) >0\\}$$\nis an open set which contains $d=\\{iv, v\\geq 1\\}$.\nLet $\\Omega_d$ be the connected component of $\\Omega$ which contains $d$. Note that $\\Omega_d $ is an open set.\n\nAs we have shown at the beginning of our proof, for given $\\rho,\\Im \\rho>0$, there exists an $M>0$ (possibly depending on $\\rho$) such that $\\omega_n(\\Lambda_n(\\Re \\rho + iv\\Im \\rho))=\\Re \\rho + iv\\Im \\rho$ for all $v>M.$\nBy the identity principle for analytic functions, we immediately obtain that \n$ \\omega_n(\\Lambda_n(\\Re \\rho +z \\Im \\rho))=\\Re \\rho +z \\Im \\rho$ for all $z\\in \\Omega_d$ and in particular for $z=i$. The \nproof of Lemma \\ref{inversion} is complete.\n\\end{proof}\n\n\n\\section{Proof of Lemma \\ref{inclu2}}\\label{lemmefonda} \\subsection{ Sharp estimates of Stieltjes transforms} The proof of \\eqref{spectre3} requires the sharp estimate \\eqref{estimdiffeqno} we are going to prove here.\n\n\n\\noindent According to Section \\ref{troncation}, from now on, we assume that the $X_{ij}^{(v)}$'s satisfy (H).\nNote that this assumption implies that for any $v\\in \\{1,\\ldots,r\\}$, \n$$\\forall i\\geq 1, \\forall j \\geq 1, \\;\\kappa_1^{i,j,v}=0, \\;\\kappa_2^{i,j,v}=1,$$\n $$\\forall i\\geq 1, \\forall j \\geq 1,\\;, i\\neq j, \\; \\tilde \\kappa_1^{i,j,v}=0,\\; \\tilde \\kappa_2^{i,j,v}=1$$ and \n for any $p\\in \\mathbb{N}\\setminus\\{0\\}$, \\begin{equation}\\label{cumulants}\\max_{v=1,\\ldots,r} \\sup_{i\\geq 1, j \\geq 1} \\vert \\kappa_p^{i,j,v}\\vert<+\\infty, \\; \\max_{v=1,\\ldots,r} \\sup_{i\\geq 1, j\\geq 1} \\vert \\tilde \\kappa_p^{i,j,v}\\vert<+\\infty, \\end{equation}\nwhere for $i\\neq j$, $(\\kappa_p^{i,j,v})_{p\\geq1}$ and $(\\tilde \\kappa_p^{i,j,v})_{p\\geq 1}$ denote the classical cumulants of $\\sqrt{2}\\Re X_{ij}^{(v)}$ and $\\sqrt{2}\\Im X_{ij}^{(v)}$ respectively and $(\\kappa_p^{i,i,v})_{p\\geq 1}$ denotes the classical cumulants of $ X_{ii}^{(v)}$ (we set $(\\tilde \\kappa_p^{i,i,v})_{p\\geq 1}\\equiv 0$).\\\\\n\n\n\n\\noindent Now, we present our main technical tool (see \\cite{KKP}):\n \\begin{lemme} \\label{lem1}\nLet $\\xi$ be a real-valued random variable such that $\\mathbb{E}(\\vert \\xi\n\\vert^{p+2})<\\infty$. Let $\\phi$ be a function from $\\R$ to $\\C$\nsuch that the first $p+1$ derivatives are continuous and bounded. Then,\n\\begin{equation}\\label{IPP}\\mathbb E(\\xi \\phi(\\xi)) = \\sum_{a=0}^p\n\\frac{\\kappa_{a+1}}{a!}\\mathbb{E}(\\phi^{(a)}(\\xi)) + \\epsilon\\end{equation}\nwhere $\\kappa_{a}$ are the classical cumulants of $\\xi$, $\\epsilon \\leq C\n\\sup_t \\vert \\phi^{(p+1)}(t)\\vert \\mathbb{E}(\\vert \\xi \\vert^{p+2})$, $C$\ndepends on $p$ only.\n\\end{lemme}\nIn the following, we shall apply this identity with a function\n$\\phi(\\xi)$ given by the entries of the resolvent of $S_n$. It\nfollows from Lemma \\ref{lem2} and (\\ref{resolvente}) below\nthat the conditions of Lemma \\ref{lem1} (bounded derivatives) are\nfulfilled.\nWe first need the following preliminary lemma.\n \n\\begin{lemme}\\label{inverseY}\nFor any $\\lambda \\in M_m(\\mathbb{C}) $ such that $\\Im \\lambda$ is positive definite, $(\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}$ and $(\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde G_n(\\lambda)\\alpha_v)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}$ are invertible.\nSet \\begin{equation}\\label{y}Y_n(\\lambda)=\\left((\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}\\end{equation}\nand \\begin{equation}\\label{ytilde}\\tilde Y_n(\\lambda)=\\left((\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde G_n(\\lambda)\\alpha_v)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}.\\end{equation}\nWe have \\begin{equation} \\label{Y}\\left\\| Y_n(\\lambda) \\right\\| \\leq \\Vert ( \\Im \\lambda)^{-1} \\Vert\\end{equation}\nand \\begin{equation} \\label{Y2}\\left\\| \\tilde Y_n(\\lambda) \\right\\| \\leq \\Vert ( \\Im \\lambda)^{-1} \\Vert.\\end{equation}\n\n\\end{lemme}\n\\begin{proof}\nWe only present the proof for $Y_n(\\lambda)$ since the proof is similar for $\\tilde Y_n(\\lambda)$.\nNote that \n\\begin{eqnarray*}\n\\Im \\left[ \\left( \\lambda\\otimes I_n -S_n\\right)^{-1}\\right]\n&=& \\frac{1}{2i}\\left[\\left( \\lambda\\otimes I_n -S_n\\right)^{-1}-\\left( \\lambda^*\\otimes I_n -S_n\\right)^{-1}\\right]\\\\\n&=& -\\left( \\lambda\\otimes I_n -S_n\\right)^{-1}\\left( \\Im \\lambda \\otimes I_n\\right) \\left( \\lambda^*\\otimes I_n -S_n\\right)^{-1}.\n\\end{eqnarray*}\nThis yields that $-\\Im R_n(\\lambda) $ is positive definite. Since the map ${\\rm id }_m\\otimes \\tr_n$ is positive we can deduce that $-\\Im H_n(\\lambda)$ is positive and then that $-\\Im G_n(\\lambda)$ is positive. It readily follows that\n\\begin{equation}\\label{image} \\Im \\left[\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v \\right] \\geq \\Im \\lambda\\end{equation} and then\n$$\\Im \\left[\\left(\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v\\right)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}\\right] \\geq \\Im \\lambda\\otimes I_n.$$ Hence Lemma \\ref{inverseY} follows by lemma 3.1 in \\cite{HT}.\n\\end{proof}\n\\begin{theoreme}\\label{resolvante}\nFor any $\\lambda \\in M_m(\\mathbb{C}) $ such that $\\Im \\lambda$ is positive definite, we have \n\\begin{equation}\\label{mast}\\mathbb{E} \\left(R_n(\\lambda)\\right)=Y_n(\\lambda)+Y_n(\\lambda)\\Xi(\\lambda)\\end{equation}\nwhere $Y_n(\\lambda)$ is defined in Lemma \\ref{inverseY}\n\\noindent and $\\Xi(\\lambda)=\\sum_{l,j}\\Xi_{lj}(\\lambda)\\otimes E_{lj}$ satisfies that for all $l,j\\in \\{1, \\ldots,n\\}$, $$ \\Xi_{lj}(\\lambda)=\\Psi_{lj}(\\lambda) +O_{lj}^{(u)}( \\frac{1}{n^2})$$\nwhere\n \\begin{eqnarray}\\Psi_{lj}(\\lambda)& =&\\sum_{v=1}^r\\bigg\\{\\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v [R_n(\\lambda ) ]_{lj} \\right]\\nonumber\n\\\\&&+ \\frac{1}{2\\sqrt{2}n\\sqrt{n}} \\sum_{i=1}^n \\left\\{1-\\delta_{il}\\left(1-\\frac{1}{\\sqrt{2}}\\right)\\right\\}M^{(3)}(v,i,l,j)\\nonumber\\\\&&+ \\frac{1}{4n^2} \\sum_{i=1}^n \\left(1-\\frac{1}{2}\\delta_{il}\\right) M^{(4)}(v,i,l,j)\\nonumber\\\\&&\\left.\n+ \\frac{1}{4\\sqrt{2}n^2\\sqrt{n}}\\sum_{i=1}^n \\left[1-\\delta_{il}\\left(1-\\frac{1}{2\\sqrt{2}}\\right)\\right]M^{(5)}(v,i,l,j)\\right\\},\\label{psi}\\end{eqnarray}\nwith \\\\\n\n\n\\noindent $M^{(3)}(v,i,l,j)$\n\\begin{eqnarray}=& \\mathbb{E} \\{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{lj}\\nonumber \\\\\n &+(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ij}\\label{except}\n \\\\\n&+(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{lj} \\nonumber \\\\\n&+(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij}\\}, \\nonumber \\end{eqnarray}\n\n\n\n\\noindent $M^{(4)}(v,i,l,j)$\n\\begin{eqnarray}=&(\\kappa_4^{i,l,v}+\\tilde \\kappa_4^{i,l,v}) \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v \n(R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij}\\label{premierk4}\n\\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{lj}\\label{deuxcas4} \\\\&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v(R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{lj}\\label{troiscas4}\\\\& + \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v(R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{ij}\\} \\label{quatrecas4}\\\\\n&\\hspace*{-0.4cm}+(\\kappa_4^{i,l,v}~\\hspace*{-0.4cm}-\\tilde \\kappa_4^{i,l,v}) \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n(R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{lj} \\label{tilde1}\n\\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{ij} \\label{tilde2} \\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v(R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{lj} \\label{tilde3}\\\\& + \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v(R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ij}\\}, \\label{tilde4}\n\\end{eqnarray}\n\n\\noindent $M^{(5)}(v,i,l,j)$\n\\begin{eqnarray*}=& \\mathbb{E} \\{(\\kappa_5^{i,l,v}+\\tilde\\kappa_5^{i,l,v}\\sqrt{-1}) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\& \\times \\left[\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ij}\\right.\\\\\n &+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{lj} \\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{lj} \\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij} \\\\&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{lj} \\\\\n&+ \\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij} \\\\\n&+ \\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ij} \\\\\n&+\\left. \\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{lj} \\right]\\\\\n&+(\\kappa_5^{i,l,v}-\\tilde\\kappa_5^{i,l,v}\\sqrt{-1}) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\&\\left[\n\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{li}\\alpha_v \n (R_n(\\lambda))_{li}\\alpha_v (R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{lj}\\right.\\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij} \\\\\n&+\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda))_{ij} \\\\\n&+\\left. \\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{lj} \\right]\\\\\n&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ij} \\\\\n&+\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{lj} \\\\\n&+ \\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ii} \\alpha_v (R_n(\\lambda))_{li} \\alpha_v (R_n(\\lambda))_{lj} \\\\\n&+\\left. \\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{ij}\\right]\\}.\n\\end{eqnarray*}\n\\end{theoreme}\n \n\n \\begin{proof}\n We shall apply formula \\eqref{IPP} to the ${ M}_m(\\C)$-valued function $\\phi(\\xi) = (R_n(\\lambda))_{ij}$ for $1 \\leq i,j \\leq n$ and $\\xi$ is one of the variable $\\frac{X^{(v)}_{kk}}{\\sqrt{n}}$,\n$\\sqrt{2} \\frac{Re(X^{(v)}_{kl})}{\\sqrt{n}}$, $\\sqrt{2} \\frac{Im(X^{(v)}_{kl})}{\\sqrt{n}}$ for $1 \\leq k0$ such that for every $k,l,i,j \\in\\{1,\\ldots,n\\}$ and every $v\\in \\{1,\\ldots,r\\}$, \\begin{equation}\\label{majuniv}\\left\\| O_{k,l,i,j}\\left(\\frac{1}{n^3}\\right)\\right\\|\\ \\leq \\frac{C\\Vert \\alpha_v \\Vert^5 \\Vert ( \\Im \\lambda)^{-1} \\Vert^6}{n^3},\\end{equation}\nwith the analogous equations with $f_{kl}$ and $e_{kk}$ replacing the $\\kappa_i^{k,l,v}$'s by the $\\tilde \\kappa_i^{k,l,v}$'s and $\\kappa_i^{k,k,v}$'s respectively. \\\\\n\n\\noindent\nNoticing that for $ k0$ such that for every $l,j \\in\\{1,\\ldots,n\\}$ and any $v \\in \\{1,\\ldots,r\\}$, \\begin{equation}\\label{majuniv2}\\left\\| O_{l,j,v}\\left(\\frac{1}{n^2}\\right)\\right\\|\\ \\leq \\frac{C\\Vert \\alpha_v \\Vert^5 \\Vert ( \\Im \\lambda)^{-1} \\Vert^6}{n^2}.\\end{equation}\n\n\\noindent Now,\n\\begin{eqnarray*}\n\\sum_{v=1}^r(\\alpha_v \\otimes X_n^{(v)})R_n (\\lambda)\n &=& (S_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}-\\gamma \\otimes I_n) (\\lambda \\otimes I_n - S_n)^{-1} \\\\\n &=& -I_m\\otimes I_n +\\left[ (\\lambda - \\gamma) \\otimes I_n -\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right] R_n(\\lambda)\n \\end{eqnarray*}\n implying\n\\begin{eqnarray}\\sum_{v=1}^r \\mathbb{E}[(\\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}) R_n(\\lambda ) ]_{lj} &=& -\\delta_{jl} I_m +(\\lambda-\\gamma) \\mathbb{E} (R_n(\\lambda))_{lj}\\nonumber\n\\\\&&- \\left[\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\mathbb{E}( R_n(\\lambda))\\right]_{lj}. \\label{identiteresol}\\end{eqnarray}\nOn the other hand, we have \n\\begin{eqnarray}\\mathbb{E}\\left[ \\alpha_v H_n(\\lambda) \\alpha_v [R_n(\\lambda ) ]_{lj} \\right]&= & \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v [R_n(\\lambda ) ]_{lj} \\right]\\nonumber\\\\ &&\n+\n\\alpha_v G_n(\\lambda) \\alpha_v \\mathbb{E}\\left[ [R_n(\\lambda ) ]_{lj} \\right]\n.\\label{centrage}\\end{eqnarray}\nHence \\eqref{4}, \\eqref{identiteresol} and \\eqref{centrage} yield \\\\\n\n $ -\\delta_{jl}I_m +(\\lambda-\\gamma) \\mathbb{E} (R_n(\\lambda))_{lj}\n- \\left[\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\mathbb{E}( R_n(\\lambda))\\right]_{lj}$ \\begin{equation}\\label{me} = \\sum_{v=1}^r\\alpha_v G_n(\\lambda) \\alpha_v \\mathbb{E}\\left[ [R_n(\\lambda ) ]_{lj} \\right] +\n\\Xi_{lj}(\\lambda) \\end{equation}\nwhere $$\\Xi_{lj}(\\lambda)=\\Psi_{lj}(\\lambda)\n+O^{(u)}_{l,\nj}(1\/n^2)$$ and $\\Psi_{lj}$ is defined in Theorem \\ref{resolvante}.\nThus, we have \n$$ \\left[\\left(\\lambda-\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda) \\alpha_v\\right)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right] \\mathbb{E} (R_n(\\lambda))\n = I_n\\otimes I_m +\n\\Xi(\\lambda).$$\n\\eqref{mast} readily follows. \\end{proof}\n\n\n\\begin{proposition}\\label{55}\nFor any $p,q\\in \\{1,\\ldots,n\\}^2$, for any $mn\\times mn $ deterministic matrix $F_n(\\lambda)$ such that \n$ F_n(\\lambda)=O(1)$, setting $\\Psi(\\lambda)=\\sum_{l,j}\\Psi_{lj}(\\lambda)\\otimes E_{lj}$ where $\\Psi_{lj}$ is defined by \\eqref{psi}, we have\\\\\n\n$\\left\\{Y_n(\\lambda)\\Psi(\\lambda)F_n(\\lambda)\\right\\}_{pq}$\n\\begin{eqnarray} &=&\\frac{1}{2\\sqrt{2}n\\sqrt{n}} \\sum_{v=1}^r\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n)_{pl}\\nonumber \\\\&& ~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda)F_n(\\lambda))_{iq}\\} \\nonumber\\\\&&+O^{(u)}_{p,q}(\\frac{1}{{n}}). \\label{YPSIF}\\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nLet us fix $v \\in \\{1,\\ldots,r\\}$. Using \\eqref{cumulants}, \\eqref{Y} and \\eqref{norme}, one can easily deduce from \\eqref{Oderacine} (respectively from \\eqref{Oden}) that all the terms in $\\left\\{Y_n(\\lambda)\\Psi(\\lambda)F_n(\\lambda)\\right\\}_{pq}$ corresponding to the $M^{(3)}(v,i,l,j)$'s in \\eqref{psi} excluding $$(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})\\alpha_v (R_n(\\lambda))_{ii}\\alpha_v (R_n(\\lambda))_{ll} \\alpha_v (R_n(\\lambda))_{ij} $$ (respectively all the terms corresponding to the $M^{(4)}(v,i,l,j)$'s) are equal to $O(1\/n)$.\n\n\n\n\n\n\n\n\nLet $C$ be some constant such that $\\sup_{i,l,v} \\{\\vert \\kappa_5^{i,l,v}\\vert +\\vert \\tilde \\kappa_5^{i,l,v} \\vert\\}\\leq C$.\nFor the terms in $\\left\\{Y_n(\\lambda)\\Psi(\\lambda)F_n(\\lambda)\\right\\}_{pq}$ corresponding to $M^{(5)}(v,i,l,j)$'s, note that using \\eqref{norme} they can be all obviously bounded by \n\\\\\n\n\\noindent $\n\\frac{C\\Vert F_n(\\lambda) \\Vert \\Vert \\alpha_v\\Vert^5 \\Vert (Im(\\lambda))^{-1}\\Vert^5}{n\\sqrt{n}}\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl}\\right\\|$\n\\begin{eqnarray*}&\\leq &\n\\frac{C\\Vert F_n (\\lambda)\\Vert \\Vert \\alpha_v\\Vert^5 \\Vert (Im(\\lambda))^{-1}\\Vert^5}{n}\\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl}\\right\\|^2\\right\\}^{\\frac{1}{2}}\\\\&\\leq & \\frac{\\sqrt{m}C\\Vert F_n(\\lambda) \\Vert \\Vert \\alpha_v\\Vert^5 \\Vert (Im(\\lambda))^{-1}\\Vert^6}{n}\\\\&=&O(1\/n)\\end{eqnarray*}\nwhere we used \\eqref{l} and \\eqref{Y}.\\\\\n\n\\noindent Finally define\n$$\\hat {\\cal R}= \\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v\\right] \\otimes I_n $$\nso that there exists some constant $C>0$ such that \n\\\\\n\n\\noindent $\\sum_{j,l=1}^n (Y_n(\\lambda))_{pl} \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v [R_n(\\lambda ) ]_{lj}(F_n(\\lambda)_{jq} \\right]\n$\\begin{eqnarray*}&=& \\mathbb{E}\\left[ [Y_n (\\lambda)\\hat {\\cal R} R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]\\\\\n&\\leq&\\Vert F_n(\\lambda) \\Vert \\Vert (Im(\\lambda))^{-1}\\Vert^2 \\mathbb{E}\\left[ \\Vert \\hat {\\cal R} \\Vert \\right]\\\\\n&\n\\leq&\\frac{C \\sqrt{m} \\Vert F_n (\\lambda)\\Vert \\Vert \\alpha_v\\Vert^2}{n}\n \\Vert (Im(\\lambda))^{-1}\\Vert^4\\\\&=&O(1\/n).\\end{eqnarray*}\n where we used \\eqref{Y}, \\eqref{norme} and \\eqref{varhn} in the last lines.\\\\\n\\noindent It is moreover clear that one can find a common polynomial to bound the involved $nO_{p,q}(1\/n)$. \\eqref{YPSIF} follows.\n\\end{proof}\n\n\\begin{corollaire} \\label{estimenunsurn}\nFor any $mn\\times mn$ deterministic matrix $F_n(\\lambda)$ such that $F_n(\\lambda)=O(1)$, we have \\\\\n\n\\noindent \n$\\mathbb{E}\\left[ [ R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]$ \\begin{eqnarray*}&=& \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} \n\\\\&&+\\ \\sum_{v=1}^r \\sum_{i,l=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}} (Y_n(\\lambda))_{pl} \\alpha_v(Y_n(\\lambda))_{ii}\\alpha_v (Y_n(\\lambda))_{ll}\\alpha_v\\\\~&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\mathbb{E}\\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right]\\\\&& +O^{(u)}_{p,q}(1\/n).\\end{eqnarray*}\n\\end{corollaire}\n\\begin{proof} \nNoticing that\n\n\\noindent $\\left\\|\n\\sum_{l,j=1}^n \\left(Y_n(\\lambda)\\right)_{pl} O^{(u)}_{l,j}\\left(1\/n^2\\right) \\left(F_n(\\lambda)\\right)_{jq} \\right\\|$\n \\begin{eqnarray*}&\\leq & \n\n O\\left(\\frac{1}{n}\\right)\\left\\{ \\sum_{l=1}^n \\Vert \\left(Y_n(\\lambda)\\right)_{pl} \\Vert^2 \\right\\}^{1\/2} \\left\\{ \\sum_{j=1}^n \\Vert \\left(F_n(\\lambda)\\right)_{jq} \\Vert^2 \\right\\}^{1\/2}\\\\\n\n &=&O_{p,q}^{(u)}(1\/n) ,\n\\end{eqnarray*}\n(using Lemma \\ref{majcarre} and \\eqref{Y} in the last line)\nit readily follows from Theorem \\ref{resolvante} and Proposition \\ref{55} that \\\\\n\n\n\n\\noindent $\\mathbb{E}\\left[ [ R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]$ \\begin{eqnarray*}&\\hspace*{-0.5cm}=& \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} \n\\\\&&+ \\sum_{v=1}^r \\sum_{i,l=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}} (Y_n(\\lambda))_{pl} \\alpha_v\\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda)F_n(\\lambda))_{iq} \\right]\\\\&& +O^{(u)}_{p,q}(1\/n).\\end{eqnarray*}\n\nTo simplify the writing let us set $U_i=\\alpha_v (R_n(\\lambda))_{ii}$, $V_l=\\alpha_v(R_n(\\lambda))_{ll}$ and $W_i= \\alpha_v \n (R_n(\\lambda)F_n(\\lambda))_{iq}$.\nWe have \\\\\n\n$\\mathbb{E}\\left[U_i V_l W_i\\right]$\n\\begin{eqnarray}&=&\\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i)) (V_l-\\mathbb{E}(V_l))W_i\\right] \\nonumber\\\\&&+ \\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i))\\mathbb{E}(V_l) (W_i-\\mathbb{E}(W_i))\\right] \\nonumber\n\\\\&&+ \\mathbb{E}(U_i) \\mathbb{E}\\left[ (V_l-\\mathbb{E}(V_l))(W_i-\\mathbb{E}(W_i))\\right]+\\mathbb{E}\\left[U_i\\right]\n\\mathbb{E}\\left[V_l\\right]\\mathbb{E}\\left[W_i\\right]. \\label{decomposition}\n\\end{eqnarray}\n\n\\noindent Now,\\\\\n\n\\noindent $\\left\\|\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda))_{pl} \\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i)) (V_l-\\mathbb{E}(V_l))W_i\\right]\\right\\|$ \n\n\\begin{eqnarray*}&\\leq& C\\Vert \\alpha_v \\Vert \\Vert ( \\Im \\lambda)^{-1} \\Vert \\Vert F_n(\\lambda)\\Vert\n\\\\&&~~~~~~\\times \\sum_{i,l=1}^n \\left\\| \n(Y_n(\\lambda))_{pl} \\right\\|\\left\\{\n\\mathbb{E} \\left( \\left\\| U_{i} -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\n\\left\\{\\mathbb{E} \\left( \\left\\| V_l -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\\\\n&\\leq &\\sqrt{n} C\\Vert \\alpha_v \\Vert \\Vert {( \\Im \\lambda)}^{-1} \\Vert \\Vert F_n(\\lambda)\\Vert \\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|^2\\right\\}^{1\/2}\n\\\\&&~~~~~~~~~~~~~~~~~\\times \n\\left\\{\\sum_{l=1}^n \\mathbb{E} \\left( \\left\\|V_l -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\left\\{\\sum_{i=1}^n \\mathbb{E} \\left( \\left\\|U_i -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}.\\end{eqnarray*}\n\n\\noindent Moreover,\n$$\\left\\|\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda))_{pl} \\mathbb{E}\\left[ (U_i-\\mathbb{E}(U_i))\\mathbb{E}(V_l) (W_i-\\mathbb{E}(W_i))\\right]\\right\\|$$\n\\begin{eqnarray*}&\\leq& C\\Vert \\alpha_v \\Vert \\Vert ( \\Im \\lambda)^{-1} \\Vert \\sum_{i,l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|\\\\&&~~~~~~~~~~~~~~~~~\\times \\left\\{\n\\mathbb{E} \\left( \\left\\| U_{i} -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\n\\left\\{\\mathbb{E} \\left( \\left\\| W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\\\\n&\\leq &\\sqrt{n} C\\Vert \\alpha_v \\Vert \\Vert {( \\Im \\lambda)}^{-1} \\Vert \n \\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|^2\\right\\}^{1\/2}\n\\\\&& \\times \\left\\{\\sum_{l=1}^n \\mathbb{E} \\left( \\left\\|W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2} \\left\\{\\sum_{i=1}^n \\mathbb{E} \\left( \\left\\|U_i -\\mathbb{E}\\left( U_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}.\\end{eqnarray*}\nFinally\n$$\\left\\|\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda))_{pl} \\mathbb{E}\\left[ U_i\\right]\\mathbb{E}\\left[ (V_l-\\mathbb{E}(V_l)) (W_i-\\mathbb{E}(W_i))\\right]\\right\\|$$\n\\begin{eqnarray*}&\\leq& C\\Vert \\alpha_v \\Vert \\Vert ( \\Im \\lambda)^{-1} \\Vert \\sum_{i,l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|\n\\\\&& \\times \\left\\{\n\\mathbb{E} \\left( \\left\\| V_{l} -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\n\\left\\{\\mathbb{E} \\left( \\left\\| W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\\\\n&\\leq &\\sqrt{n} C\\Vert \\alpha_v \\Vert \\Vert {( \\Im \\lambda)}^{-1} \\Vert \n \\left\\{\\sum_{l=1}^n \\left\\| (Y_n(\\lambda))_{pl} \\right\\|^2\\right\\}^{1\/2}\n\\\\&& \\times \\left\\{\\sum_{l=1}^n \\mathbb{E} \\left( \\left\\|V_l -\\mathbb{E}\\left( V_l\\right)\\right\\|^2\\right)\\right\\}^{1\/2}\\left\\{\\sum_{i=1}^n \\mathbb{E} \\left( \\left\\|W_i -\\mathbb{E}\\left( W_i\\right)\\right\\|^2\\right)\\right\\}^{1\/2}.\\end{eqnarray*}\n\n\n\n\n \\noindent Using Lemma \\ref{var}, (\\ref{l}) and \\eqref{Y}, we readily deduce that \\\\\n\n\n\\noindent $\\mathbb{E}\\left[ [ R_n(\\lambda ) F_n(\\lambda)]_{pq} \\right]$ \\begin{eqnarray}&=& \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} \\nonumber\n\\\\&&+\\frac{1}{2\\sqrt{2}n\\sqrt{n}}\\sum_{v=1}^r \\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})(Y_n(\\lambda)_{pl} \\alpha_v\\\\&&~~~~~~~~~~~~~~~~~~~~~~~\\times\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right]\n\\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ll}\\right]\\alpha_v \\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\nonumber \\\\&& +O^{(u)}_{p,q}(1\/{n}). \\label{eq}\\end{eqnarray}\n\\noindent Now, define \n$${\\cal R}= \\sum_{v=1}^r\\sum_{i,l=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right]\\alpha_v\\mathbb{E}\n\\left[(R_n(\\lambda))_{ll}\\right]\\alpha_v\\otimes E_{li},$$\nIt is easy to see that $\\Vert {\\cal R}\\Vert\\leq C \\Vert (\\Im \\lambda)^{-1}\\Vert^2 n.$\nWe have \\\\\n\n\\noindent $ \\sum_{v=1}^r \\sum_{i,l=1}^n(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) (Y_n(\\lambda))_{pl} \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right] $ $$ ~~~~~~~~~~\\times \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ll}\\right]\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right]= \\left[ Y_n(\\lambda) {\\cal R} \\mathbb{E} (R_n(\\lambda) F_n(\\lambda))\\right]_{pq}.$$\nSo that if we define $$T=\\sum_{p,q=1}^n T_{pq}\\otimes E_{pq}$$ where \n$$T_{pq}\n= \\frac{1}{2\\sqrt{2}n\\sqrt{n}}\\sum_{v=1}^r \\sum_{l,i=1}^n (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) (Y_n(\\lambda))_{pl} \\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ii}\\right]$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times\\alpha_v\\mathbb{E}\\left[(R_n(\\lambda))_{ll}\\right]\n\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right],$$\nwe have \\begin{equation}\\label{Q}\\Vert T\\Vert=\\left\\| \\frac{1}{2\\sqrt{2}n\\sqrt{n}} Y_n(\\lambda) {\\cal R} \\mathbb{E} (R_n(\\lambda) F_n(\\lambda)) \\right\\|=O(1\/\\sqrt{n}).\\end{equation}\nHence, in particular we have \\begin{equation}\\label{estimenracinen} \\mathbb{E}\\left[ [ R_n(\\lambda ) F_n]_{pq} \\right]= \\left(Y_n(\\lambda)F_n(\\lambda)\\right)_{pq} +O^{(u)}_{p,q}(1\/\\sqrt{n}). \\end{equation}\n\n \n\\noindent Now, \nwe have \n$$\\left\\| \\sum_{i,l=1}^n \\frac{ (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}} (Y_n(\\lambda))_{pl} \\alpha_vO^{(u)}_{i,i}(\\frac{1}{\\sqrt{n}})\\alpha_v(Y_n(\\lambda))_{ll}\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\right\\|$$\n\\begin{eqnarray}& \\leq& \\frac{C \\Vert(\\Im \\lambda)^{-1} \\Vert }{n} \\left( \\sum_{l=1}^n \\Vert (Y_n(\\lambda))_{pl} \\Vert^2 \\right)^{1\/2} \\left\\{\\sum_{i=1}^n \n\\left\\| \\mathbb{E} \\left[(R_n(\\lambda)F_n(\\lambda))_{iq}\\right] \\right\\|^2 \\right\\}^{1\/2}\\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~\\times \\left\\{\\sum_{i=1}^n \\Vert O^{(u)}_{i,i}(1\/{\\sqrt{n}})\\Vert^2\\right\\}^{1\/2} \\nonumber \\\\ & =& O^{(u)}_{p,q}(1\/n) \\label{eq2} \n\\end{eqnarray}\nwhere we used (\\ref{l}) twice and \\eqref{Y} and \\eqref{norme} in the last line.\nSimilarly $$\n\\left\\| \\sum_{i,l=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}} (Y_n(\\lambda))_{pl} \\alpha_v(Y_n(\\lambda))_{ii}\\alpha_vO_{l,l}(\\frac{1}{\\sqrt{n}})\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\right\\|$$ \\begin{equation}\\label{eq3}=O^{(u)}_{p,q}(\\frac{1}{{n} }), \\end{equation} \n$$\n\\left\\| \\sum_{i,l=1}^n \\frac{ (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}n\\sqrt{n}}(Y_n(\\lambda))_{pl} \\alpha_vO_{i,i}(\\frac{1}{\\sqrt{n}})\\alpha_vO_{l,l}(1\/\\sqrt{n})\\alpha_v\\mathbb{E} \\left[ (R_n(\\lambda)F_n(\\lambda))_{iq} \\right] \\right\\|$$ \\begin{equation}\\label{eq4}=O^{(u)}_{p,q}(\\frac{1}{{n \\sqrt{n}} }) \\end{equation} \n\n\n(\\ref{eq}), (\\ref{estimenracinen}) and (\\ref{eq2}), (\\ref{eq3}), \\eqref{eq4} readily yields Corollary \\ref{estimenunsurn}.\n\\end{proof}\n\\begin{corollaire} \\label{ME} With the notations of Section \\ref{Notations},\n \n \\begin{eqnarray*}G_n(\\lambda) &=&\\mathbb{E} \\left({\\rm id}_m\\otimes tr_n R_n(\\lambda)\\right)\\\\\n&=&G_{\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}}\\left(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v \\right)\n\\\\&&+L_n(\\lambda)\n+\\epsilon_n(\\lambda)\n \\end{eqnarray*}\n where \n$$L_n(\\lambda)=\\frac{1}{n}\\sum_{p=1}^n \\left[Y_n(\\lambda)\n\\Psi(\\lambda) \\right]_{pp},$$\n(with $\\Psi(\\lambda)$ defined in Theorem \\ref{resolvante} and $Y_n(\\lambda)$ defined in Lemma \\ref{inverseY})\n and $$ \\epsilon_n(\\lambda) =O \\left( \\frac{1}{n\\sqrt{n}}\\right).$$\n Moreover \\begin{equation} \\label{L} L_n(\\lambda) =O \\left( \\frac{1}{\\sqrt{n}}\\right).\\end{equation}\n\n\n \\end{corollaire}\n\\begin{proof}\nFirst note that, since the distribution of $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ in $({\\cal A},\\tau)$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$, we have \\begin{eqnarray*}{\\rm id}_m\\otimes tr_n Y_n(\\lambda)&= &{\\rm id}_m\\otimes \\tau \\left((\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v)\\otimes 1_{\\cal A}-\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)} \\right)^{-1}\n\\\\&=& G_{\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}}\\left(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v \\right).\\end{eqnarray*}\nThen, the corollary readily follows from Theorem \\ref{resolvante} and Proposition \\ref{55} by noting that \n\\begin{itemize}\n\\item \n\n\\noindent \n$\\frac{1}{n}\\Vert \\sum_{p,l=1}^n (Y_n(\\lambda))_{pl}O_{lp}^{(u)}(1\/n^2)\\Vert$ \\begin{eqnarray*} & \\leq &\\frac{1}{n}\\left(\\sum_{p,l=1}^n \\Vert (Y_n(\\lambda))_{pl}\\Vert ^2 \\right)^{1\/2}\\left( \\sum_{p,l=1}^n \\Vert O^{(u)}_{lp}(1\/n^2)\\Vert ^2 \\right)^{1\/2}\\\\&=&O(\\frac{1}{n\\sqrt{n}})\\end{eqnarray*}\nwhere we used Lemma \\ref{majcarre} and \\eqref{Y} in the last line.\n\\item $$ \\Vert \\sum_{i,l,p=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{n^2\\sqrt{n}}(Y_n(\\lambda))_{pl} \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{ii}\\alpha_v(R_n(\\lambda))_{ll}\\alpha_v \n (R_n(\\lambda))_{ip} \\Vert $$\n$$\\leq \\frac{C\\Vert \\alpha_v \\Vert^3 \\Vert (\\Im \\lambda) ^{-1}\\Vert^2}{n^2} \\sum_{l,p=1}^n \\Vert (Y_n(\\lambda))_{pl} \\Vert \\left(\\sum_{i=1}^n \\Vert(R_n(\\lambda))_{ip} \\Vert^2\\right)^{1\/2}$$\n$$ \\leq \\frac{ C\\sqrt{m}\\Vert \\alpha_v \\Vert^3 \\Vert (\\Im \\lambda )^{-1}\\Vert^3}{n} \\left( \\sum_{p,l=1}^n \\Vert (Y_n(\\lambda))_{pl}\\Vert ^2 \\right)^{1\/2} =O(1\/\\sqrt{n})$$\nwhere we used Lemma \\ref{majcarre}, \\eqref{norme} and \\eqref{Y} in the two last lines.\n\\end{itemize}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\\begin{theoreme}\\label{difftilde}\nLet $\\lambda$ be in $M_m(\\mathbb{C})$ such that \n $\\Im \\lambda>0$, and $\\tilde G_n(\\lambda)$ as defined in \\eqref{defGntilde}. We have \n\\begin{equation} \\label{prediff}\nG_n(\\lambda)-\\tilde G_n(\\lambda)+{E_n(\\lambda)}= O(\\frac{1}{n\\sqrt{n}}),\n\\end{equation}\nwhere $E_n(\\lambda)$ is given by\\\\\n\n\\noindent $E_n(\\lambda) =$\n\\begin{equation}\n \\sum_{v=1}^r \\tilde G_n'(\\lambda) \\cdot \\alpha_v L_n(\\lambda) \\alpha_v -\\frac{1}{2} \\tilde G_n''(\\lambda) \\cdot\\left( \\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v, \\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v\\right) -L_n(\\lambda) \n\\end{equation}\nwith $L_n(\\lambda)$ defined in Corollary \\ref{ME}.\n\\end{theoreme}\n\\begin{proof}\nLet $\\lambda$ be in $M_m(\\mathbb{C})$ such that \n $\\Im \\lambda>0$. Note that according to \\eqref{image}, we have $\\Im \\left( \\lambda-\\gamma - \\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v\\right)>0.$\nDefine (using the notations of Section \\ref{Notations})\n\\begin{equation}\\label{lambdan}\\Lambda_n(\\lambda)= \\gamma +\\lambda + \\sum_{v=1}^r \\alpha_v G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\lambda) \\alpha_v\\end{equation}\nand $\\lambda'=\\Lambda_n(\\lambda-\\gamma - \\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v).$\nUsing Corollary \\ref{ME}, we have \n\\begin{eqnarray}\\lambda'-\\lambda &= & - \\sum_{v=1}^r \\alpha_v G_n(\\lambda) \\alpha_v + \\sum_{v=1}^r \\alpha_v G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda ) \\alpha_v) \\alpha_v \\nonumber \\\\&=& -\\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v +O(\\frac{1}{n\\sqrt{n}})\\label{lambdaprimemoinslambdaavant}\\\\&=& O(1\/\\sqrt{n}). \\label{lambdaprimemoinslambda}\n\\end{eqnarray}\n\n\n\n\\noindent Thus there exists a polynomial $Q$ with nonnegative coefficients \nsuch that $$\\left\\|\\lambda'-\\lambda\\right\\|\\leq \\frac{Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}}.$$\n-On the one hand, if $$\\frac{Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}}\\geq \\frac{1}{2\\Vert (\\Im \\lambda )^{-1}\\Vert},$$ \nor equivalently \n\\begin{equation} \\label{1=O(1\/n)}\n1\\leq \\frac{2\\Vert (\\Im \\lambda )^{-1}\\Vert Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}},\n\\end{equation}\nto prove \\eqref{prediff}\nit is enough to prove that \n\\begin{equation} \\label{O(1)}\nG_n(\\lambda)-\\tilde G_n(\\lambda)+E_n(\\lambda) = O(1).\n\\end{equation}\nIndeed, if we assume that \\eqref{1=O(1\/n)} and \\eqref{O(1)} hold, \nthen there exists a polynomial $\\tilde Q$ with nonnegative coefficients \nsuch that \n\\begin{eqnarray*}\n\\left\\|G_n(\\lambda)-\\tilde G_n(\\lambda)+E_n(\\lambda) \\right\\|&\\leq &\\tilde Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)\\\\\n&\\leq &\\tilde Q(\\Vert ( \\Im \\lambda )^{-1}\\Vert)\\frac{2\\Vert ( \\Im \\lambda )^{-1}\\Vert Q(\\Vert( \\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}}\\\\\n&\\leq &\\tilde Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)(\\frac{2\\Vert (\\Im \\lambda) ^{-1}\\Vert Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}})^4.\n\\end{eqnarray*}\nHence, $$G_n(\\lambda)-\\tilde G_n(\\lambda)+E_n(\\lambda) = O(\\frac{1}{n^2})$$\nso that \\eqref{prediff} holds.\nTo prove \\eqref{O(1)}, one can notice that, using \\eqref{norme} and \\eqref{normeG},\n both $G_n(\\lambda)$ and $\\tilde G_n(\\lambda)$ \nare bounded by $\\Vert (\\Im \\lambda) ^{-1}\\Vert$, and that \\\\\n\n\\noindent \n$\\left\\|E_n(\\lambda)\\right\\|$ $$\\leq \\left\\{r \\max_{v=1}^r\\Vert \\alpha_v\\Vert^2 \\Vert (\\Im \\lambda )^{-1}\\Vert^2 +1\\right\\} \\left\\|L_n(\\lambda)\\right\\|+r^2 \\max_{v=1}^r\\Vert \\alpha_v\\Vert^4 \\Vert (\\Im \\lambda )^{-1}\\Vert^3 \\left\\|L_n(\\lambda)\\right\\|^2 ,$$\nwhere $L_n(\\lambda)=O(1\/\\sqrt{n})$ according to \\eqref{L}.\\\\\n~~\n\n\\noindent -On the other hand, if $$\\frac{Q(\\Vert (\\Im \\lambda )^{-1}\\Vert)}{\\sqrt{n}}\\leq \\frac{1}{2\\Vert (\\Im \\lambda )^{-1}\\Vert},$$ \none has : \n\\begin{equation}\\label{lambdaprime}\\left\\|\\Im \\lambda'-\\Im \\lambda\\right\\|\\leq \\left\\|\\lambda'-\\lambda \\right\\|\\leq \\frac{1}{2\\Vert ( \\Im \\lambda) ^{-1}\\Vert}\\end{equation}\nDenoting for any Hermitian matrix $H$ by $l_1(H)$ the smallest eigenvalue of $H$, we readily deduce from \\eqref{lambdaprime} that \n $l_1(\\Im \\lambda')\\geq \\frac{l_1(\\Im \\lambda)}{2}$ and \ntherefore \\begin{equation}\\label{lambdaprimepositif}\\Im \\lambda' >0. \\end{equation}\nThen, it makes sense to consider $\\tilde G_{n}(\\lambda')$ which satisfies according to \\eqref{subor}\n\\begin{eqnarray}\\tilde G_{n}(\\lambda')&= &G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}(\\lambda'-\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde G_n(\\lambda')\\alpha_v) \\nonumber\\\\&=&\nG_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}\\left(\\omega_n(\\lambda')\\right) \\nonumber\\\\& =&G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}\\left(\\omega_n(\\Lambda_n(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v ))\\right). \\label{subordin}\\end{eqnarray}\n\n\n\n\n\n\n\n\n\nApplying Lemma \\ref{inversion} to $\\rho=\\lambda -\\gamma -\\sum_{v=1}^r\\alpha_v G_n(\\lambda)\\alpha_v $ ( using \\eqref{image} and \\eqref{lambdaprimepositif}) we obtain that \n since $\\Im \\lambda'=\\Im \\Lambda_n(\\rho)>0$ we have $\\omega_n(\\Lambda_n(\\rho))=\\rho$ and according to \\eqref{subordin}\n$$ \\tilde G_{n}\\left(\\lambda'\\right)\n=G_{\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}}\\left(\\lambda -\\gamma -\\sum_{v=1}^r \\alpha_v G_n(\\lambda)\\alpha_v \\right). $$\nHence, by Corollary \\ref{ME}, we have\n\\begin{equation} \\label{termone}\nG_n(\\lambda)-\\tilde G_n(\\lambda')-{L_n(\\lambda)}=O(\\frac{1}{n\\sqrt{n}}).\n\\end{equation}\nNow, we have \\\\\n\n\\noindent $\\tilde G_n(\\lambda')-\\tilde G_n(\\lambda)$\n\\begin{eqnarray*}\n&=& {\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda')\\left[ (\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\}\\\\\n&=& {\\rm id}_m\\otimes \\tau \\left\\{ r_n(\\lambda)\\left[(\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\} \\\\&&+ \n{\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda)\\left[ (\\lambda-\\lambda') \\otimes1_{\\cal A}\\right] r_n(\\lambda) \\left[ (\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda) \\right\\}\n\\\\&&+ \n{\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda')\\left[ (\\lambda-\\lambda') \\otimes1_{\\cal A}\\right] r_n(\\lambda)\\left[ (\\lambda-\\lambda') \\otimes1_{\\cal A}\\right] r_n(\\lambda) \\left[(\\lambda-\\lambda') \\otimes 1_{\\cal A}\\right] r_n(\\lambda) \\right\\}\\\\\n&=&\\sum_{v=1}^r {\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda)\\left[\\alpha_v L_n(\\lambda)\\alpha_v \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\} \\\\&&+\\sum_{v,v'=1}^r {\\rm id}_m \\otimes \\tau \\left\\{ r_n(\\lambda)\\left[\\alpha_v L_n(\\lambda)\\alpha_v \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\left[\\alpha_{v'} L_n(\\lambda)\\alpha_{v'} \\otimes 1_{\\cal A}\\right] r_n(\\lambda)\\right\\}\n\\\\&& \n+O(\\frac{1}{n\\sqrt{n}})\n\\end{eqnarray*}\nwhere we used \\eqref{lambdaprimemoinslambdaavant}, \\eqref{lambdaprimemoinslambda} and \\eqref{normeG} in the last line.\nHence we have \n$$\n\\tilde G_n(\\lambda')-\\tilde G_n(\\lambda)+\\sum_{v=1}^r\\tilde G_n'(\\lambda) \\cdot \\alpha_v L_n(\\lambda) \\alpha_v\n-\\frac{1}{2}\\tilde G_n''(\\lambda) \\cdot\\left(\\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v, \\sum_{v=1}^r \\alpha_v L_n(\\lambda) \\alpha_v\\right)$$\n\\begin{equation} \\label{termtwo}~~~~~~~~~~~~~~~~~~~~~~~~~~=O(\\frac{1}{n\\sqrt{n}}).\n\\end{equation}\n(\\ref{prediff}) follows from \\eqref{termone} and \\eqref{termtwo} since \\\\\n\n\\noindent \n$\\left\\|G_n(\\lambda)-\\tilde G_n(\\lambda)+{E_n(\\lambda)}\\right\\|\\leq \n\\left\\|G_n(\\lambda)-\\tilde G_n(\\lambda')-{L_n(\\lambda)}\\right\\|$\n$$+\\left\\|\\tilde G_n(\\lambda')-\\tilde G_n(\\lambda)+\\sum_{p=1}^r\\tilde G_n'(\\lambda)\\cdot \\alpha_v{L_n(z)}\\alpha_v -\\frac{1}{2}\\tilde G_n''(\\lambda) \\cdot\\left(\\sum_{p=1}^r \\alpha_v L_n(\\lambda) \\alpha_v, \\sum_{p=1}^r \\alpha_v L_n(\\lambda) \\alpha_v\\right)\\right\\|.\n$$\n\n\\end{proof}\n\\begin{remarque}\n\\eqref{L} and \\eqref{normeG} readily yield that $E_n(\\lambda)=O(\\frac{1}{\\sqrt{n}})$. Thus, we can deduce from (\\ref{prediff}) that \\begin{equation}\\label{difgngntilde}G_n( \\lambda)-\\tilde G_n(\\lambda) =O(\\frac{1}{\\sqrt{n}}).\\end{equation}\n\\end{remarque}\n\n\n\n\\begin{proposition} \\label{estimdiff}\n For $z \\in \\C \\setminus \\R$, let $g_n(z)$ and $\\tilde g_n(z)$ as defined in \\eqref{defpetitg} and \\eqref{defpetitgtilde} respectively. We have \n\\begin{equation} \\label{estimdiffeqn}\ng_n(z)-\\tilde g_n(z)+{\\tilde{E}_n(z)} = O(\\frac{1}{n\\sqrt{n}}),\n\\end{equation}\nwhere $\\tilde{E}_n(z)$ is given by\n\\begin{equation}\\label{defentilde}\n\\tilde{E}_n(z) = \\sum_{v=1}^r tr_m\\left( \\tilde G_n'(zI_m)\\cdot \\alpha_v \\tilde{L}_n(z)\\alpha_v\\right) - \\tr_m \\tilde L_n(z)\n\\end{equation}\n{with }\\\\\n\n$\\displaystyle{\\tilde{L}_n(z)=\\sum_{v=1}^r}$ \\begin{eqnarray*} && \\bigg\\{\n\\sum_{i,l,p=1}^n \\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v})}{4n^3} \\tilde Y_n(zI_m)_{pl} \\alpha_v (\\tilde Y_n(zI_m))_{ii} \\alpha_v (\\tilde Y_n(zI_m))_{ll} \\alpha_v (\\tilde Y_n(zI_m))_{ii} \\alpha_v \\left(\\tilde Y_n(zI_m)\\right)_{lp}\\\\\n&&+ \\sum_{i,l,p=1}^n \\frac{(\\kappa_3^{i,l,v} +\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} } \\tilde Y_n(zI_m)_{pl} \\alpha_v (\\tilde Y_n(zI_m))_{ii} \\alpha_v (\\tilde Y_n(zI_m))_{li} \\alpha_v \\left( \\tilde Y_n(zI_m)\\right)_{lp} \\\\\n&&+\\sum_{i,l,p=1}^n \\frac{ (\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} } \\tilde Y_n(zI_m)_{pl} \\alpha_v (\\tilde Y_n(zI_m))_{ii} \\alpha_v (\\tilde Y_n(zI_m))_{ll} \\alpha_v \\left(\\tilde Y_n(zI_m)\\right)_{ip}\\\\\n&&+\\sum_{i,l,p=1}^n \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } \\tilde Y_n(zI_m)_{pl} \\alpha_v (\\tilde Y_n(zI_m))_{il} \\alpha_v (\\tilde Y_n(zI_m))_{ii} \\alpha_v \\left (\\tilde Y_n(zI_m)\\right)_{lp}\\bigg\\}\n\\end{eqnarray*}\nwhere $\\tilde Y_n$ and $\\tilde G_n$ were defined in \\eqref{ytilde} and \\eqref{defGntilde} respectively so that \n \\begin{eqnarray*}\\tilde Y_n(zI_m)&= &\\left((zI_m -\\gamma -\\sum_{v=1}^r\\alpha_v \\tilde G_n(zI_m)\\alpha_v)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}\\\\&=&\\left(\\omega_n(zI_m)\\otimes I_n - \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)} \\right)^{-1}\\end{eqnarray*}\nand \\begin{eqnarray*}\\tilde G_n(zI_m)&=&{\\rm id}_m\\otimes \\tau\\left( (zI_m-\\gamma) \\otimes 1_{\\cal A} - \\sum_{v=1}^r \\alpha_v \\otimes x_v-\\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)}\\right)^{-1}\\\\&=&{\\rm id}_m \\otimes \\tau\\left( zI_m\\otimes 1_{\\cal A} - s_n\\right)^{-1} .\\end{eqnarray*}\n\\end{proposition}\n\\begin{proof}\nLet $z\\in \\mathbb{C}\\setminus \\mathbb{R}$ such that $\\Im z >0$.\nTheorem \\ref{difftilde} yields $$ g_n(z)- \\tilde g_n(z)+\\sum_{v=1}^r tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v L_n(zI_m) \\alpha_v -tr_mL_n(zI_m)\n$$ \\begin{equation}\\label{doubleetoile}-\\frac{1}{2} \\tr_m \\tilde G_n''(zI_m) \\cdot\\left( \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v, \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v\\right)=O(\\frac{1}{n\\sqrt{n}}).\\end{equation}\nFirst note that, by Riesz-Fr\\'echet's Theorem (and using \\eqref{normeG} and \\eqref{L}), there exists $B_n^{(1)}(z)$ and $B_n^{(2)}(z)$ in $M_m(\\C)$ such that $$\\Vert B_n^{(1)}(z) \\Vert_2 \\leq \\vert \\Im z \\vert^{-2} \\left( \\sum_{v=1}^r \\Vert \\alpha_v\\Vert^2 \\right)=O(1), $$ \\begin{equation}\\label{B2}\\Vert B_n^{(2)}(z) \\Vert_2=O(1\/\\sqrt{n}), \\end{equation} and \n$$\\sum_{v=1}^r \\tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v L_n(zI_m) \\alpha_v = \\Tr_m \\left[ B_n^{(1)}(z) L_n(zI_m) \\right],$$\n\\begin{equation}\\label{etoilehat}\\tr_m \\tilde G_n''(zI_m) \\cdot\\left( \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v, \\sum_{v=1}^r \\alpha_v L_n(zI_m) \\alpha_v\\right)= \\Tr_m \\left[ B_n^{(2)}(z) L_n(zI_m) \\right].\\end{equation}\n\n\\noindent Recall that for $\\lambda \\in M_m(\\C)$ such that $\\Im \\lambda>0$, $$L_n(\\lambda)=\\frac{1}{n}\\sum_{p=1}^n \\left[Y_n(\\lambda)\n\\Psi(\\lambda) \\right]_{pp},$$ where $\\Psi$ is defined in \\eqref{psi}.\nFirst, note that according to \\eqref{YPSIF}, we have (setting $c_{l,i,v}= \\frac{(\\kappa_3^{i,l,v} -\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2}} $)\\\\\n~~\n\n\\noindent $\\Tr_m \\left[ B_n^{(2)}(z) L_n(zI_m) \\right]$\n\\begin{eqnarray*} \n\\hspace*{-1.6cm}=&\\hspace*{-0.3cm}\n\\displaystyle{ \\sum_{v=1}^r \\sum_{i,l=1}^n} \\frac{c_{l,i,v}}{n^2\\sqrt{n}} \\mathbb{E} \\Tr_m \\left\\{\\alpha_v(R_n(zI_m))_{ii}\\alpha_v(R_n(zI_m))_{ll}\\alpha_v \n \\left(R_n(zI_m) (B_n^{(2)}(z)\\otimes I_n)Y_n(zI_m)\\right)_{il}\\right\\}\n\\\\&+O(\\frac{1}{n\\sqrt{n}}).\n\\end{eqnarray*}\nMoreover, we have \n $$ \\sum_{v=1}^r\\sum_{i,l=1}^n \\frac{ c_{l,i,v}}{n^2\\sqrt{n}} \\mathbb{E} \\Tr_m \\left\\{\\alpha_v(R_n(zI_m))_{ii}\\alpha_v(R_n(zI_m))_{ll}\\alpha_v \n \\left(R_n(zI_m) (B_n^{(2)}(z)\\otimes I_n)Y_n(zI_m)\\right)_{il}\\right\\}$$ \n$$= O(\\frac{1}{n\\sqrt{n}}),$$\nwhere we used \\eqref{Odenracinepas}, \\eqref{norme}, \\eqref{Y} and \\eqref{B2}. \nHence \\begin{equation}\\label{etoiletilde}\\Tr_m \\left[ B_n^{(2)}(z) L_n(zI_m) \\right]= O(\\frac{1}{n\\sqrt{n}}).\\end{equation}\n\\eqref{doubleetoile}, \\eqref{etoilehat} and \\eqref{etoiletilde} yield \\begin{equation}\\label{doubleetoilehat}g_n(z)- \\tilde g_n(z)+\\sum_{v=1}^r tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v L_n(zI_m) \\alpha_v -tr_mL_n(zI_m)=O(\\frac{1}{n\\sqrt{n}}).\\end{equation}\n Thus, in the following we will consider $\\frac{1}{n}\\sum_{p=1}^n \\tr_m B_n(\\lambda) \\left[Y_n(\\lambda)\nT(\\lambda) \\right]_{pp}$ for any $\\lambda\\in M_m\\left( \\mathbb{C}\\right)$ such that $\\Im \\lambda >0$, for each term $T(\\lambda)$ involving in \\eqref{psi} and any $m\\times m$ matrix $B_n(\\lambda)=O(1)$ (in the interests of simplifying notations, we deal with any $\\lambda\\in M_m\\left( \\mathbb{C}\\right)$ such that $\\Im \\lambda >0$ instead of $zI_m$).\nWe set ${\\cal B}(\\lambda)= B_n(\\lambda) \\otimes I_{n}$. \n\n\n\\noindent First, for any fixed $v \\in \\{1,\\ldots,r\\}$,\\\\\n \n$\\left|\\frac{1}{n}\\sum_{p,l=1}^n \\tr_m B_n(\\lambda)(Y_n(\\lambda))_{pl} \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v [R_n(\\lambda ) ]_{lp} \\right]\\right|$\n\\begin{eqnarray*}\n&=& \\left|\\tr_m \\mathbb{E}\\left[ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v id\\otimes tr_n [R_n(\\lambda ) {\\cal B}(\\lambda)Y_n(\\lambda)] \\right]\\right|\\\\\n&=& \\left|\\tr_m \\mathbb{E}\\left\\{ \\alpha_v [H_n(\\lambda) -\\mathbb{E}(H_n(\\lambda))] \\alpha_v\\right. \\right.\\\\&&\\left.\\left.~~~~~~\\times \\left[id\\otimes tr_n [R_n(\\lambda ) {\\cal B}(\\lambda)Y_n(\\lambda)]\n-\\mathbb{E}(id\\otimes tr_n [R_n(\\lambda ) {\\cal B}(\\lambda)Y_n(\\lambda)]) \\right]\\right\\}\\right|\\\\&\n\\leq&\\frac{\\Vert B_n(\\lambda) \\Vert \\Vert \\alpha_v\\Vert^2 C m}{n^2}\n \\Vert (Im(\\lambda))^{-1}\\Vert^5\\\\&=& O(1\/n^2).\\end{eqnarray*}\nwhere we used Cauchy Schwarz's inequality, \\eqref{norme}, \\eqref{Y} and Lemma \\ref{var} in the last line.\\\\\nWe also have\n$$\\left|\\sum_{i,p,l=1}^n \\frac{ (\\kappa_3^{i,l,v} +\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{n^2\\sqrt{n}} \\tr_m B_n(\\lambda)(Y_n(\\lambda))_{pl} \\mathbb{E} \\{\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v \n (R_n(\\lambda))_{ip}\\}\\right|$$\n\\begin{eqnarray*}&=&\\frac{1}{n^2\\sqrt{n}} \\left| \\sum_{i,l=1}^n (\\kappa_3^{i,l,v} +\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) \\mathbb{E}\\tr_m \\{\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v \n (R_n(\\lambda){\\cal B}(\\lambda) Y_n(\\lambda))_{il}\\}\\right|\n\\\\&=&O(\\frac{1}{n\\sqrt{n}})\n\\end{eqnarray*}\nwhere we used \\eqref{Odenpas}, \\eqref{norme} and \\eqref{Y}.\n\n\n\\noindent Now, let us investigate the terms corresponding to the the $M^{(4)}(v,i,l,j)$'s in \\eqref{psi}. We have \n$$ \\frac{1}{4n^3} \\sum_{i,p,l=1}^n (\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v})\\mathbb{E} \\tr_m B_n(\\lambda)(Y_n(\\lambda))_{pl} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{il} \\alpha_v (R_n(\\lambda))_{il}\\alpha_v (R_n(\\lambda))_{ip}$$\n\\begin{eqnarray*}\n&=& \\frac{1}{4n^3} \\sum_{i,l=1}^n (\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v}) \\mathbb{E}\\tr_m \\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il}\\alpha_v(R_n(\\lambda))_{il} \\alpha_v \\left(R_n(\\lambda){\\cal B}(\\lambda)Y_n(\\lambda)\\right)_{il}\n\\\\&=&O(1\/n^2) \\end{eqnarray*}\nwhere we used \\eqref{Odenpas}, \\eqref{norme} and \\eqref{Y}.\nSimilarly the terms corresponding to \\eqref{deuxcas4}, \\eqref{quatrecas4}, \\eqref{tilde1}, \\eqref{tilde2}, \\eqref{tilde3} and \\eqref{tilde4}\nare $O(1\/n^2).$\\\\\n\n\n\n\\noindent Since moreover each term in \\eqref{psi} corresponding to the $M^{(3)}(v,i,i,j)$, $M^{(4)}(v,i,i,j)$ and $M^{(5)}(v,i,l,j)$ leads obviously to a term which is a $O(\\frac{1}{n\\sqrt{n}})$, it readily follows from \\eqref{doubleetoilehat} that $$ g_n(z)- \\tilde g_n(z)+ \\sum_{v=1}^r tr_m\\tilde G_n'(zI_m) \\cdot \\alpha_v \\hat L_n(zI_m) \\alpha_v -tr_m \\hat L_n(zI_m)=O(\\frac{1}{n\\sqrt{n}}),$$\nwhere \\\\\n\n\\noindent $\\hat L_n (zI_m)=\\sum_{v=1}^r \\sum_{i,p,l=1}^n $ $$\\bigg\\{ \\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v})}{4n^3} ( Y_n(zI_m)) _{pl} \\alpha_v \\mathbb{E} \\left\\{(R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll} \\alpha_v(R_n(zI_m))_{ii}\\alpha_v (R_n(zI_m))_{lp}\\right\\}$$\n$$+\\frac{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } ( Y_n(zI_m)) _{pl} \\alpha_v \\mathbb{E} \\left\\{(R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{li} \\alpha_v (R_n(zI_m))_{lp}\\right\\}$$\n$$+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } ( Y_n(zI_m)) _{pl} \\alpha_v \\mathbb{E} \\left\\{ (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll} \\alpha_v (R_n(zI_m))_{ip}\\right\\}$$\n$$+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } ( Y_n(zI_m)) _{pl} \\alpha_v \\mathbb{E} \\left\\{(R_n(zI_m))_{il} \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{lp}\\right\\}\\bigg\\}$$\nFor any $m\\times m$ deterministic matrix $B_n(z)$, \\\\\n~~\n\n\\noindent $\\tr_m B_n (z)\\hat L_n (zI_m)=\\sum_{v=1}^r\\sum_{i,l=1}^n$ $$\\bigg\\{\\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v}) }{4n^3} \\mathbb{E} \\left\\{\\tr_m \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll}\\alpha_v (R_n(zI_m))_{ii}\\alpha_v (R_n(zI_m){\\cal B}(z)Y_n(zI_m))_{ll}\\right\\}$$\n$$+ \\frac{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} } \\mathbb{E} \\tr_m \\left\\{ \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{li} \\alpha_v \\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right\\}$$\n$$+ \\frac{ (\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1}) }{2\\sqrt{2} n^2 \\sqrt{n} } \\mathbb{E} \\tr_m\\left\\{ \\alpha_v (R_n(zI_m))_{ii} \\alpha_v (R_n(zI_m))_{ll} \\alpha_v \\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{il}\\right\\}$$\n$$+\\frac{ (\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } \\mathbb{E} \\tr_m\\left\\{ \\alpha_v (R_n(zI_m))_{il} \\alpha_v (R_n(zI_m))_{ii} \\alpha_v\\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right\\}\\bigg\\}$$\n\n\\noindent Hence (using a decomposition similar to \\eqref{decomposition}) Lemma \\ref{var} readily yields that \\\\\n\n\n\\hspace*{-0.8cm} $\\tr_m B_n \\hat L_n (zI_m) + O(\\frac{1}{n\\sqrt{n}})= \\sum_{v=1}^r \\sum_{i,l=1}^n \\tr_m$ $$\\hspace*{-0.8cm}\\bigg\\{\\frac{(\\kappa_4^{i,l,v} +\\tilde \\kappa_4^{i,l,v}) }{4n^3} \\alpha_v \\mathbb{E}\\left[ (R_n(zI_m))_{ii} \\right] \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{ll}\\right] \\alpha_v\\mathbb{E}\\left[ (R_n(zI_m))_{ii} \\right]\\alpha_v [ \\mathbb{E}\\left[\\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right]$$\n$$\\hspace*{-0.8cm}+ \\frac{(\\kappa_3^{i,l,v}+\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{ii}\\right] \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{li}\\right] \\alpha_v \\mathbb{E}\\left[\\left(R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right] $$\n$$\\hspace*{-0.8cm}+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{ii}\\right] \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{ll}\\right] \\alpha_v \\mathbb{E}\\left[ \\left(R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{il}\\right]$$\n$$\\hspace*{-0.8cm}+ \\frac{(\\kappa_3^{i,l,v}-\\tilde \\kappa_3^{i,l,v}\\sqrt{-1})}{2\\sqrt{2} n^2 \\sqrt{n} } \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{il} \\right] \\alpha_v \\mathbb{E}\\left[(R_n(zI_m))_{ii}\\right] \\alpha_v \\mathbb{E}\\left[\\left (R_n(zI_m){\\cal B}(z)Y_n(zI_m)\\right)_{ll}\\right]\\bigg\\}.$$\nNote that, with $Y_n$ and $\\tilde Y_n$ defined in \\eqref{y} and \\eqref{ytilde}, we have $$Y_n(zI_m)-\\tilde Y_n(zI_m) =\\sum_{v=1}^r Y_n(zI_m) \\left[\\alpha_v (G_n(zI_m)-\\tilde G_n(zI_m)) \\alpha_v \\otimes I_n\\right] \\tilde Y_n(zI_m)$$ so that, using (\\ref{difgngntilde}), \\eqref{Y} and \\eqref{Y2}, we can deduce that \\begin{equation}\\label{YmoinsYtilde} \\Vert Y_n(zI_m) -\\tilde Y_n(zI_m)\\Vert =O(1\/\\sqrt{n}). \\end{equation}\nNow, (\\ref{estimenracinen}) and (\\ref{YmoinsYtilde}) obviously yield that, up to a $O(\\frac{1}{n\\sqrt{n}})$ correction term, one can replace any $R_n(zI_m)$ and $Y_n(zI_m)$ by $\\tilde Y_n(zI_m)$ in any term in the sum corresponding to the fourth cumulants. Now, using (\\ref{lp}), (\\ref{estimenracinen}) and (\\ref{YmoinsYtilde}) also yield that, up to a $O(\\frac{1}{n\\sqrt{n}})$ correction term, for any $p=1,\\ldots,n$, one can replace $(R_n(zI_m))$ and $(Y_n(zI_m))$ by $\\tilde Y_n(zI_m)$ in any diagonal term $(R_n(zI_m))_{pp}$ or $(R_n(zI_m){\\cal B}(z)Y_n(zI_m))_{pp} $ in the sums corresponding to the third cumulants.\\\\\nFinally, \nassume that for $i=1,2,$ $Q^{(i)}= \\tilde Y_n(zI_m) $ or $\\tilde Y_n(zI_m){\\cal B}(z)\\tilde Y_n(zI_m)$. Let us consider any $ Q^{(3)}= \\sum_{i,l=1}^nQ^{(3)}_{il}\\otimes E_{il}$.\nIt is clear that if there exists some polynomial $Q$ with nonnegative coefficients such that for any $i,l \\in \\{1,\\ldots,n\\}^2$, $\\Vert Q^{(3)}_{il}\\Vert \\leq \\frac{Q(\\vert \\Im z \\vert^{-1})}{n}$, then \n$$\\frac{1}{n^2 \\sqrt{n}} \\sum_{i,l=1}^n\\Vert Q^{(1)}_{ii}\\Vert \\Vert Q^{(2)}_{ll}\\Vert \\Vert Q^{(3)}_{il}\\Vert =O(1\/n\\sqrt{n}).$$\nNow, if $\\Vert Q^{(3)}\\Vert =O(1\/\\sqrt{n})$,\nwe have $$\\frac{1}{n^2 \\sqrt{n}} \\sum_{i,l=1}^n\\Vert Q^{(1)}_{ii}\\Vert \\Vert Q^{(2)}_{ll}\\Vert \\Vert Q^{(3)}_{il}\\Vert $$ $$\\leq \\frac{1}{n \\sqrt{n}} \\vert (\\Im z)^{-1}\\vert^q\\left(\\sum_{i,l=1}^n\\Vert Q^{(3)}_{il}\\Vert^2 \\right)^{1\/2} \\leq \\frac{\\sqrt{m}}{n } \\vert (\\Im z)^{-1}\\vert^q\\Vert Q^{(3)}\\Vert =O(1\/n\\sqrt{n}) $$ for some $q\\in \\mathbb{N}\\setminus{\\{0\\}},$ where we used \\eqref{lp}.\n\\\\\nIt is then clear that using Corollary \\ref{estimenunsurn}, \\eqref{Q} and (\\ref{YmoinsYtilde}), \nup to a $0(\\frac{1}{n\\sqrt{n}})$ correction term, for any $(i,l)\\in \\{1,\\ldots,n\\}^2$, one can replace $R_n(zI_m)$ and $Y_n(zI_m)$ by $\\tilde Y_n(zI_m)$ in any non-diagonal term $(R_n(zI_m))_{il}$, $(R_n(zI_m))_{li}$ or $(R_n(zI_m){\\cal B}Y_n)_{il}$ in the sums corresponding to the third cumulants.\nHence \\eqref{estimdiffeqn} is proved for any $z\\in \\mathbb{C}$ such that $\\Im z >0$. \\\\Set $\\alpha=(\\alpha_1,\\ldots \\alpha_r)$, $\\beta=(\\beta_1,\\ldots, \\beta_t)$. Let us denote for a while $g_n=g_n^{\\alpha,\\beta,\\gamma}$, $\\tilde g_n=\\tilde g_n^{\\alpha,\\beta,\\gamma}$ and $\\tilde E_n=\\tilde E_n^{\\alpha,\\beta,\\gamma}$. Note that we have similarly for any $z\\in \\mathbb{C}$ such that $\\Im z >0$, \n\\begin{equation} \\label{estimdiffeqnmoins}\ng_n^{-\\alpha,-\\beta,-\\gamma}(z)-\\tilde g^{-\\alpha,-\\beta,-\\gamma}_n(z)+{\\tilde{E}^{-\\alpha,-\\beta,-\\gamma}_n(z)} = O(\\frac{1}{n\\sqrt{n}}).\n\\end{equation} Thus, since $g_n^{-\\alpha,-\\beta,-\\gamma}(z)=-g_n^{\\alpha,\\beta,\\gamma}(-z)$, $\\tilde g_n^{-\\alpha,-\\beta,-\\gamma}(z)=-\\tilde g_n^{\\alpha,\\beta,\\gamma}(-z)$ and $\\tilde E_n^{-\\alpha,-\\beta,-\\gamma}(z)=-\\tilde E_n^{\\alpha,\\beta,\\gamma}(-z)$, it readily follows that \n\\eqref{estimdiffeqn} is also valid for any $z \\in \\mathbb{C}$ such that $\\Im z <0$.\n\\end{proof}\n\n\n\n\\subsection{ From Stieltjes transform estimates to spectra.}\nWe start with the following key lemma.\n\\begin{lemme}\\label{LSt}\nFor any fixed large n, $ \\tilde E_n$ defined in Proposition \\ref{estimdiff} is the Stieltjes transform of a compactly supported distribution $\\nabla_n$ on $\\mathbb{R}$ whose support is\nincluded in the spectrum of $s_n=\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$ and such that $\\nabla_n(1)=0$ .\n\\end{lemme}\n\n\nThe proof relies on the following characterization already used in \\cite{Schultz05}. \n\n\\begin{theoreme}\\label{TS}\\cite{Tillmann53}\n\\begin{itemize}\n\\item Let $\\Lambda $ be a distribution on $\\R$ with compact support. \nDefine the Stieltjes transform of $\\Lambda $, \n$ l:\\C\\setminus \\R \\rightarrow \\C$ by \n$$l(z)=\\Lambda \\left( \\frac{1}{z-x}\\right) .$$\n\\noindent Then $l$ is analytic on $\\C\\setminus \\R$\nand has an analytic continuation to $\\C\\setminus {\\rm supp}(\\Lambda )$. \nMoreover\n\\begin{itemize}\n\\item[($c_1$)] $l(z)\\rightarrow 0$ as $|z|\\rightarrow \\infty ,$\n\\item[($c_2$)] there exists a constant $C > 0$, \nan integer $q\\in \\N$ and a compact set $K\\subset \\R$ containing ${\\rm supp}(\\Lambda )$, \nsuch that for any $z\\in \\C\\setminus \\R$, \n$$|l(z)|\\leq C\\max \\{ {\\rm dist}(z,K)^{-q}, 1\\} ,$$\n\\item[($c_3$)] for any $\\phi \\in \\cal C^\\infty (\\R, \\R)$ with compact support\n$$\\Lambda (\\phi )=\\frac{i}{2\\pi }\\lim _{y\\rightarrow 0^+} \\int _\\R\\phi (x)[l(x+iy)-l(x-iy)]dx.$$\n\\end{itemize}\n\\item Conversely, if $K$ is a compact subset of $\\R$ \nand if $l:\\C \\setminus K\\rightarrow \\C$ is an analytic function\nsatisfying ($c_1$) and ($c_2$) above, \nthen $l$ is the Stieltjes transform of a compactly supported distribution $\\Lambda $ on $\\R$. \nMoreover, ${\\rm supp}(\\Lambda )$ is exactly the set of singular points of $l$ in $K$. \n\\end{itemize}\n\\end{theoreme}\n\\begin{lemme} The singular points of $\\tilde{E}_N$ defined in \\eqref{defentilde} are included in the \nspectrum of $s_n=\\gamma \\otimes 1_{\\cal A}+\\sum_{v=1}^r \\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$. \n\\end{lemme}\n\\begin{proof}\nLet us start by noting that, as it follows from the definition \\eqref{defentilde} of $\\tilde{E}_N$,\nit is enough to show that $\\omega_n(zI_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\n\\in GL_{nm}(\\mathbb C)$ (the group of invertible $nm\\times nm$ complex matrices) for any $z$\nin the domain of definition of $\\mathbb C\\ni z\\mapsto\\omega_n(zI_m)\\in M_{m}(\\mathbb C).$\n\nAssume towards contradiction that $x_0\\in\\mathbb C$ is in the domain of $\\omega_n(\\cdot I_m)$, and \nyet $\\omega_n(x_0I_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}$ is not invertible. First,\nobserve that a point $x_0$ with this property must be isolated and real. Indeed, otherwise the zeros of\nthe analytic map $\\mathbb C\\ni z\\mapsto\\det\\left(\\omega_n(zI_m)\\otimes I_n-\n\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\\right)\\in\\mathbb C$ would have $x_0$ as a cluster point\nin the interior of its domain (which coincides with the domain of $\\omega_n(\\cdot I_m)$), and thus\nit would be identically equal to zero. However, \n$\\omega_n(zI_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}$ is invertible when $\\Im z\\neq0$,\nproviding us with a contradiction. Consider now such an isolated $x_0$. Recall \n\\begin{eqnarray*}\n\\tilde{g}_n(z) & = & (\\tr_m\\otimes\\tau)\\left(\\left(\n\\omega_n(zI_m)\\otimes1_\\mathcal A-\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}\\right)^{-1}\\right)\\\\\n& = & (\\tr_m\\otimes\\tr_n)\\left(\\left(\n\\omega_n(zI_m)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\\right)^{-1}\\right),\n\\end{eqnarray*}\nfrom before (the second equality is justified by the hypothesis that the distribution of $(A_n^{(1)},\n\\dots,A_n^{(t)})$ with respect to $\\tr_n$ coincides with the distribution of $(a_n^{(1)},\\dots,a_n^{(t)})$\nwith respect to $\\tau$). We have seen that $\\tilde{g}_n$ is defined exactly on the complement of the \nspectrum of $s_n$. Thus, it is enough to show that, given an analytic function \n$f\\colon\\mathbb C^+\\to H^{-}(M_p(\\mathbb C))=\\{b\\in M_p(\\mathbb{C}), \\Im b <0\\}$, and $x_0\\in\\mathbb R$ with the property that there\nexists some $\\epsilon>0$ such that $f$ extends analytically through $(x_0-\\epsilon,x_0)\\cup(x_0,\nx_0+\\epsilon)$ with self-adjoint values, then either both or none of $f$ and $\\tr_p\\circ f$ extend \nanalytically to $x_0$. We shall then apply this to $p=mn$ and $f(z)=\\left(\\omega_n(zI_m)\\otimes I_n-\n\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)}\\right)^{-1}$ to conclude.\n \nIt is clear that if $f$ extends analytically through $x_0$, then so does $\\tr_p\\circ f$. Assume \ntowards contradiction that $\\tr_p\\circ f$ extends analytically through $x_0$, but $f$ does not.\nConsider an arbitrary system $\\{e_1,\\dots,e_p\\}$ of minimal mutually orthogonal projections\nin $M_p(\\mathbb C)$. It is clear that $z\\mapsto e_jf(z)e_j\\in e_jM_p(\\mathbb C)e_j\\simeq\\mathbb C$ is \nanalytic wherever $f$ is. Moreover, $\\Im e_jf(z)e_j\\leq0$ whenever $z\\in\\mathbb C^+$, and \n$e_jf(z)e_j\\in\\mathbb R$ whenever $f(z)$ is self-adjoint. If $z\\mapsto e_jf(z)e_j$ extends analytically\nthrough $x_0$ for any $e_j\\in\\{e_1,\\dots,e_p\\}$ and all systems $\\{e_1,\\dots,e_p\\}$, then \n$z\\mapsto\\varphi(f(z))$ extends analytically through $x_0$ for all linear functionals $\\varphi\\colon\nM_p(\\mathbb C)\\to\\mathbb C$. This implies that $f$ itself is analytic around $x_0$. On the other hand,\nif there exists a system $\\{e_1,\\dots,e_p\\}$ of minimal mutually orthogonal projections\nin $M_p(\\mathbb C)$ which contains an $e_j$ such that $z\\mapsto e_jf(z)e_j$ does not \nextend analytically through $x_0$, then, as $z\\mapsto e_jf(z)e_j$ does extend analytically \nwith real values through $(x_0-\\epsilon,x_0)\\cup(x_0,x_0+\\epsilon)$ and maps $\\mathbb C^+$\ninto $\\mathbb C^-\\cup\\mathbb R$, it follows that $x_0$ is a simple pole of \n$z\\mapsto e_jf(z)e_j$ and $\\lim_{y\\to0}iye_jf(x_0+iy)e_j\\in(0,+\\infty)$ by the Julia-Carath\\'eodory\nTheorem applied to $1\/e_jf(z)e_j$ (see (2) Theorem 2.1 in \\cite{Serban}). But then\n$\\tr_p(f(z))=\\sum_{k=1}^pe_kf(z)e_k$, so that \n$$\n\\lim_{y\\to0}iy\\tr_p(f(x_0+iy))=\\sum_{k=1}^p\\lim_{y\\to0}iye_kf(x_0+iy)e_k>0,\n$$\nwhich contradicts the assumption that $\\tr_p\\circ f$ extends analytically through $x_0$.\n\n\\end{proof}\n\n\nNow, we are going to show that for any fixed large $n$, \n$\\tilde{E}_n$ satisfies ($c_1$) and ($c_2$) of Theorem \\ref{TS}. \n\n\n\n\\noindent First note that there exists a polynomial Q in two variables with positive coefficients such that \n\\begin{equation}\\label{normedetildeEn} \\vert \\tilde {E}_n(z)\\vert \\leq \\Vert \\tilde Y(zI_m))\\Vert^4 Q(\\Vert \\tilde Y(zI_m))\\Vert, \\Vert r_n(zI_m)\\Vert ).\n\\end{equation}\n\n\\noindent Let $C > 0$ be such that, for all $n$, $\\mbox{sp}(\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)})\\subset [-C;C]$ and\n$\\mbox{sp}(\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r \\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)})\\subset [-C;C]$.\n\n\\noindent Let $d > C + \\sqrt{r}\\max_{v=1}^r\\Vert \\alpha_v \\Vert $. \nFor any $z\\in \\C$ such that $|z| > \\Vert \\gamma \\Vert +d $, \n\\begin{eqnarray*}\\Vert \\gamma +\\sum_{v=1}^r \\alpha_v \\tilde G_n(zI_m) \\alpha_v \\Vert &\\leq& \\Vert \\gamma \\Vert + \\frac{r \\max_{v=1}^r\\Vert \\alpha_v \\Vert ^2}{\\vert z\\vert -C}\\\\&\\leq &\\Vert \\gamma \\Vert + \\frac{r \\max_{v=1}^r\\Vert \\alpha_v \\Vert ^2}{d-C} \\\\&\n<& \\Vert \\gamma \\Vert + \\frac{(d -C)^2}{d -C} \\\\&=& \\Vert \\gamma \\Vert + d-C\\end{eqnarray*}\n Thus, $$\\Vert \\gamma+\\sum_{v=1}^r\\alpha_v\\tilde G_n(zI_m) \\alpha_v +\\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)}\\Vert \\leq\\Vert \\gamma \\Vert +d $$\nso that we get that for any $z\\in \\C$ such that $|z| > \\Vert \\gamma\\Vert +d $, \n\\begin{eqnarray*}\\Vert \\tilde Y_n(zI_m)\\Vert &=&\n\\Vert ((zI_m- \\gamma -\\sum_{v=1}^r \\alpha_v \\tilde G_n(zI_m) \\alpha_v)\\otimes I_n-\\sum_{u=1}^t\\beta_u \\otimes A_n^{(u)})^{-1}\\Vert\\\\\n& \\leq &\\frac{1}{\\vert z\\vert -\\Vert \\gamma \\Vert -d }.\n\\end{eqnarray*}\nWe get readily from \\eqref{normedetildeEn} that, for $|z| > \\Vert \\gamma\\Vert +d $, \n\\begin{equation}\\label{bound} \\vert \\tilde{E}_n(z)\\vert \\leq \\frac{1} {(\\vert z\\vert -\\Vert \\gamma \\Vert -d )^4}Q\\left(\\frac{1} {(\\vert z\\vert -\\Vert \\gamma \\Vert -d )},\\frac{1}{(|z|-C)} \\right) .\\end{equation}\nThen, it is clear than $\\vert\\tilde{E}_n(z)\\vert \\rightarrow 0$ \nwhen $|z|\\rightarrow +\\infty $ and ($c_1$) is satisfied.\\\\\n\\noindent \nNow we are going to prove ($c_2$) using the approach of \\cite{Schultz05}(Lemma 5.5). \nDenote by $\\mathcal {E}_n$ the convex envelope of the spectrum of $s_n=\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$\nand define $$K_n:=\\left\\{ x\\in \\R; {\\rm dist}(x, \\mathcal {E}_n)\\leq 1\\right\\} $$\n\\noindent \nand $$D_n=\\left\\{ z\\in \\C; 0 < {\\rm dist}(z, K_n)\\leq 1\\right\\} .$$\n\\begin{itemize}\n\\item Let $z\\in D_n\\cap (\\C\\setminus \\R)$ with $\\Re (z)\\in K_n$. \nWe have ${\\rm dist}(z, K_n)=|\\Im z|\\leq 1$. \nWe have from \\eqref{normedetildeEn}, (\\ref{normeG}) and \\eqref{Y2} that \n$$\\vert \\tilde{E}_n(z)\\vert \\leq {\\vert \\Im z \\vert^{-4}} Q\\left(\\vert \\Im z \\vert^{-1},\\vert \\Im z \\vert^{-1}\\right).$$\nNoticing that $1\\leq {\\vert \\Im z\\vert^{-1}}$, \nwe easily deduce that there exists some constant $C_0$ and some number $q_0 \\in \\mathbb{N}\\setminus \\{0\\}$ such that \nfor any $z\\in D_n\\cap \\C\\setminus \\R$ with $\\Re (z)\\in K_n$, \n\\begin{eqnarray*}\n\\vert \\tilde{E}_n(z)\\vert &\\leq &C_0|\\Im z|^{-q_0}\\\\\n&\\leq &C_0{\\rm dist}(z, K_n)^{-q_{0}}\\\\\n&\\leq &C_0\\max ({\\rm dist}(z, K_n)^{-q_{0}}; 1)\n\\end{eqnarray*}\n\\item Let $z\\in D_n\\cap (\\C\\setminus \\R)$ with $\\Re (z)\\notin K_n$. \nThen ${\\rm dist}(z,{\\rm sp}(s_n))\\geq 1$. \nSince $\\tilde{E}_n$ is bounded on compact subsets of $\\C\\setminus {\\rm sp}(\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)})$,\nwe easily deduce that there exists some constant $C_1(n)$ \nsuch that for any $z\\in D_n$ with $\\Re (z)\\notin K_n$, \n$$\\vert \\tilde{E}_n(z)\\vert \\leq C_1(n)\\leq C_1(n)\\max ({\\rm dist}(z, K_n)^{-q_0}; 1).$$\n\\item Since $\\vert \\tilde{E}_n(z)\\vert \\rightarrow 0$ when $|z|\\rightarrow +\\infty $, \n$\\tilde{E}_n$ is bounded on $\\C\\setminus \\overline{D_n}$. \nThus, there exists some constant $C_2(n)$ such that for any $z\\in \\C\\setminus \\overline{D_n}$, \n$$\\vert \\tilde{E}_n(z)\\vert \\leq C_2(n)=C_2(n)\\max ({\\rm dist}(z, K_n)^{-q_0}; 1).$$\n\\end{itemize}\nHence ($c_2$) is satisfied with $C(n)=\\max (C_0, C_1(n), C_2(n))$ and $l=q_0$. Thus, Theorem \\ref{TS} implies that for any fixed large n, $ \\tilde E_n$ defined in Proposition \\ref{estimdiff} is the Stieltjes transform of a compactly supported distribution $\\nabla_n$ on $\\mathbb{R}$ whose support is\nincluded in the spectrum of $s_n=\\gamma \\otimes 1_{\\cal A} +\\sum_{v=1}^r\\alpha_v \\otimes x_v +\\sum_{u=1}^t\\beta_u \\otimes a_n^{(u)}$. Following the proof of Lemma 5.6 in \\cite{Schultz05} and using (\\ref{bound}), \none can show that $\\nabla _n(1)=0$. \nThe proof of Lemma \\ref{LSt} is complete. $\\Box$\\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent Now, \\eqref{spectre3} can be deduced from \\eqref{estimdiffeqno} by an approach inspired by \\cite{HT} and \\cite{Schultz05} as follows. \\\\\nUsing the inverse Stieltjes tranform, we get respectively that, \nfor any $\\varphi _n $ in ${\\cal C}^\\infty (\\R, \\R)$ with compact support, \n$$\\mathbb{E} [\\tr_m \\otimes \\tr _n(\\varphi _n(S_n))]-\\tr_m\\otimes \\tau(\\varphi_n(s_n))\n+{\\nabla _{n}(\\varphi _n)}$$\n$$=\\frac{1}{\\pi }\\lim _{y\\rightarrow 0^+}\\Im \\int _\\R\\varphi _n(x)\\epsilon_n(x+iy)dx,$$\nwhere $\\epsilon_n(z)=\\tilde g_n(z)-g_n(z)-\\tilde{E}_n(z)$ satisfies, \naccording to Proposition \\ref{estimdiff}, for any $z\\in \\C\\setminus \\R$, \n\\begin{equation*}\\label{estimgdif}\n\\vert \\epsilon_n(z)\\vert \\leq \\frac{1}{n\\sqrt{n}}P(\\vert \\Im z \\vert ^{-1}) .\n\\end{equation*} \nWe refer the reader to the Appendix of \\cite{CD07} \nwhere it is proved using the ideas of \\cite{HT} that if $h$ is an analytic function on $\\C\\setminus \\R$ which satisfies\n\\begin{equation*}\\label{nestimgdif}\n\\vert h(z)\\vert \\leq P(\\vert \\Im z\\vert ^{-1})\n\\end{equation*} \n\\noindent for some polynomial $P$ with nonnegative coefficients and degree $k$, then\nthere exists a polynomial $Q$ such that\n$$\\limsup _{y\\rightarrow 0^+}\\vert \\int _\\R\\varphi _n(x)h(x+iy)dx\\vert $$\n$$\\leq \\int _\\R\\int _0^{+\\infty }\\vert (1+D)^{k+1}\\varphi _n(x)\\vert Q(t)\\exp(-t)dtdx$$\nwhere $D$ stands for the derivative operator.\nHence, if there exists $K > 0$ such that, for all large $n$, \nthe support of $\\varphi _n$ is included in $[-K, K]$ and \n$\\sup _n\\sup _{x \\in [-K, K]}\\vert D^p\\varphi _n(x)\\vert =C_p < \\infty$ for any $p\\leq k+1$, \ndealing with $h(z) =n\\sqrt{n}\\epsilon_n(z)$, we deduce that there exists $C>0$ such that for all large $n$,\n\\begin{equation*} \\label{majlimsup1} \n\\limsup _{y\\rightarrow 0^+}\\vert \\int _\\R \\varphi _n(x)\\epsilon_n(x+iy)dx\\vert \\leq \\frac{C}{n\\sqrt{n}}\n\\end{equation*} \nand then \n\\begin{equation}\\label{StS} \n\\mathbb{E} [\\tr_m \\otimes \\tr _n(\\varphi _n(S_n))]-\\tr_m\\otimes \\tau(\\varphi_n(s_n))\n+{\\nabla _{n}(\\varphi _n)}=O(\\frac{1}{n\\sqrt{n}}). \n\\end{equation}\nLet $\\rho \\geq 0$ be in ${\\cal C}^\\infty (\\R, \\R)$ \nsuch that its support is included in $[-1;1]$ and $\\int \\rho (x)dx=1$. \nLet $0 < \\epsilon < 1$. \nDefine for any $x\\in \\mathbb{R}$, $$\\rho _{\\frac{\\epsilon }{2}}(x)=\\frac{2}{\\epsilon }\\rho(\\frac{2x}{\\epsilon }).$$\nSet $$K_n(\\epsilon )=\\{ x, {\\rm dist}(x, {\\rm sp}(s_n))\\leq \\epsilon \\}$$ \nand define for any $x\\in \\mathbb{R}$, $$f_n(\\epsilon )(x)=\\int _\\mathbb{R} \\1 _{K_n(\\epsilon )}(y)\\rho _{\\frac{\\epsilon }{2}}(x-y)dy.$$\nThe function $f_{n}(\\epsilon )$ is in ${\\cal C}^\\infty (\\mathbb{R}, \\mathbb{R})$, \n$f_{n}(\\epsilon )\\equiv 1$ on $K_n(\\frac{\\epsilon }{2})$; \nits support is included in $K_n(2\\epsilon )$. \nSince there exists $K$ such that, for all large $n$, the spectrum of $s_n$ \nis included in $[-K;K]$, for all large $n$ the support of $f_n(\\epsilon )$ is included in $[-K-2;K+2]$ \nand for any $p > 0$, \n$$\\sup _{x\\in [-K-2;K+2]}\\vert D^pf_n(\\epsilon )(x)\\vert \\leq \n\\sup _{x\\in [-K-2;K+2]}\\int _{-K-1}^{K+1} \\vert D^p \\rho _{\\frac{\\epsilon }{2}}(x-y)\\vert dy \\leq C_p(\\epsilon ).$$\nThus, according to \\eqref{StS}, \n\\begin{equation} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n(f_n(\\epsilon )(S_n))]-\\tr_m \\otimes \\tau f_n(\\epsilon )(s_n)\n+{\\nabla _n(f_n(\\epsilon ))}=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}})\n\\end{equation}\nand \n\\begin{equation}\\label{prime} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n((f_n'(\\epsilon ))^2(S_n))]-\\tr_m \\otimes \\tau (f_n'(\\epsilon )(s_n))^2\n+{\\nabla _n((f_n'(\\epsilon ))^2)}=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}}).\n\\end{equation}\nAccording to Lemma \\ref{LSt}, we have $\\nabla _n(1)=0$. \nThen, the function $\\psi _n(\\epsilon )\\equiv 1-f_n(\\epsilon )$ also satisfies\n\\begin{equation} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))]-\\tr_m \\otimes \\tau \\left(\\psi _n(\\epsilon )(s_n)\\right)\n+\\nabla _n(\\psi _n(\\epsilon ))=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}}). \n\\end{equation}\nMoreover, since $\\psi _n'(\\epsilon )=-f_n'(\\epsilon )$, \nit comes readily from \\eqref{prime} that \n$$\\mathbb{E} [\\tr _n((\\psi _n'(\\epsilon ))^2(S_n))]-\\tr_m \\otimes \\tau (\\psi _n'(\\epsilon )(s_n))^2\n+{\\nabla_n((\\psi _n'(\\epsilon ))^2)}=O_{\\epsilon }(\\frac{1}{n\\sqrt{n}}).$$\nNow, since $\\psi _n(\\epsilon )\\equiv 0$ on the spectrum of $s_n$, \nwe deduce that \n\\begin{equation}\\label{psi2} \n\\mathbb{E} [\\tr_m\\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))]=O_{\\epsilon }\\left(\\frac{1}{n\\sqrt{n}}\\right)\n\\end{equation}\nand \n\\begin{equation} \\label{psiprime} \\mathbb{E} [\\tr_m\\otimes \\tr _n((\\psi _n'(\\epsilon ))^2(S_n))]=O_{\\epsilon }\\left(\\frac{1}{n\\sqrt{n}}\\right).\n\\end{equation}\nBy Lemma \\ref{variance} (sticking to the proof of Proposition 4.7 in \\cite{HT} with $\\varphi=f_n(\\epsilon)$), \nwe have\n$$\\mathbf{V}{\\left[ \\tr_m \\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))\\right] }\n\\leq \\frac{C }{n^2}\\mathbb{E} \\left[\\tr_m \\otimes \\tr _n\\{ (\\psi _n'(\\epsilon )(S_n))^2\\} \\right] .$$\nHence, using \\eqref{psiprime}, one can deduce that \n\\begin{equation} \\label{variancepsi}\n\\mathbf{V}{\\left[ \\tr_m \\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))\\right] }=O_{\\epsilon }\\left(\\frac{1}{n^3\\sqrt{n}}\\right).\n\\end{equation}\nFix $0<\\delta<\\frac{1}{4}$.\nSet $$Z_{n, \\epsilon }:=\\tr_m \\otimes \\tr _n(\\psi _n(\\epsilon )(S_n))$$ \nand $$\\Omega _{n, \\epsilon }=\\{ \\left| Z_{n, \\epsilon }-\\mathbb{E}\\left(Z_{n, \\epsilon }\\right)\\right| > n^{-(1+\\delta)}\\}.$$\nHence, using \\eqref{variancepsi}, we have $$\\mathbb{P}(\\Omega _{n, \\epsilon })\\leq n^{2+2\\delta}\\mathbf{V}\\{ Z_{n, \\epsilon }\\}\n=O_{\\epsilon }\\left(\\frac{1}{n^{1+\\frac{1}{2}-2\\delta}}\\right).$$\nBy Borel-Cantelli lemma, we deduce that, almost surely for all large $n$, \\begin{equation}\\label{concentrezn} \\left|Z_{n, \\epsilon }-\n\\mathbb{E}\\left(Z_{n, \\epsilon }\\right)\\right| \\leq n^{-(1+\\delta)}.\\end{equation}\nFrom \\eqref{psi2} and \\eqref{concentrezn}, we deduce that there exists some constant $C_\\epsilon$ such that, almost surely for all large $n$,\n$$\\left| Z_{n, \\epsilon }\\right|\\leq n^{-1}\\left(n^{-\\delta}+C_\\epsilon n^{-1\/2}\\right).$$\nSince $\\psi_{n}( \\epsilon )\\geq \\1 _{\\mathbb{R}\\setminus K_n({2\\epsilon })}$, \nit readily follows that, almost surely for all large $n$, \nthe number of eigenvalues of $S_n$ which are in $\\R\\setminus K_n({2\\epsilon })$ \nis lower than $m\\left(n^{-\\delta}+C_\\epsilon n^{-1\/2}\\right)$ and thus obviously, almost surely for all large $n$, the number of eigenvalues of $S_n$ which are in $\\R\\setminus K_n({2\\epsilon })$ \n has \nto be equal to zero. Thus we have the following\n\\begin{theoreme} Let $\\epsilon>0$.\nAlmost surely for all large $n$, the spectrum of $S_n$ is included in $K_n(\\epsilon )=\\{ x, {\\rm dist}(x, {\\rm sp}(s_n))\\leq \\epsilon \\}$. \n\\end{theoreme}\nSince the above theorem holds for any $m\\times m$ Hermitian matrices $\\gamma$, $\\{\\alpha_v\\}_{v=1,\\ldots,r}$, $\\{\\beta_u\\}_{u=1,\\ldots,t}$, the proof of Lemma \\ref{inclu2} is complete.\n\n\\section{Proof of Theorem \\ref{noeigenvalue}}\\label{sectionnoeigenvalue}\n\\subsection{Linearization}\\label{linearisation}\n\nLinearization procedures are by no means unique, and no agreed upon definition of what a linearization\nis exists in the literature. We use the procedure introduced in \\cite[Proposition 3]{A}, which has several \nadvantages, to be described below.\n\n\n\n\nIt is shown in \\cite{A} that, given a polynomial $P\\in\\mathbb C\\langle X_1,\\dots,X_k\\rangle$,\nthere exist $m\\in\\mathbb N$ and matrices $\\zeta_1,\\dots,\\zeta_k,\\gamma\\in M_m(\\mathbb C)$\nsuch that $(z-P(X_1,\\dots,X_k))^{-1}=\\left[\\left(z\\hat{E}_{11}\\otimes1-\\gamma\\otimes 1-\\sum_{j=1}^k\n\\zeta_j\\otimes X_j\\right)^{-1}\\right]_{11}$. Moreover, if $P=P^*$, then $\\gamma$ and $\\zeta_1,\\dots,\n\\zeta_k$ can be chosen to be self-adjoint. We denote $L_P=\\gamma\\otimes1+\\sum_{j=1}^k\\zeta_j\n\\otimes X_j\\in M_m(\\mathbb C\\langle X_1,\\dots,X_k\\rangle)$ and call it a {\\em linearization} of $P$. \nThe size $m$ and the matrix coefficients $\\gamma,\\zeta_1,\\dots,\\zeta_k$ aren't unique. Following \\cite{BMS} (see also \\cite{Mai}), we provide \na very brief outline of a recursive construction for a linearization $L_P$ such that $$L_P := \\begin{pmatrix} 0 & u\\\\v & Q \\end{pmatrix} \\in M_m(\\mathbb{C}) \\otimes \\mathbb{C} \\langle X_1,\\ldots, X_k \\rangle$$\nwhere\n\\begin{enumerate}\n\\item $ m \\in \\mathbb{N}$,\n\\item $ Q \\in M_{m-1}(\\mathbb{C})\\otimes \\mathbb{C} \\langle X_1,\\ldots, X_k \\rangle$ is invertible,\n\\item \n u is a row vector and v is a column vector, both of size $m-1$ with\nentries in $\\mathbb{C} \\langle X_1,\\ldots, X_k \\rangle$,\n\\item the polynomial entries in $Q, u$ and $v$ all have degree $\\leq 1$,\\\\\n\\item\n$${P=-uQ^{-1}v} ,$$\n\\item and moreover, if $P$ is self-adjoint, $L_P$ is self-adjoint.\n\\end{enumerate}\nThus, to linearize a monomial $P=X_{i_1}X_{i_2}X_{i_3}\\cdots X_{i_{k-1}}X_{i_l}$, write\n$$\nL_P=-\\begin{bmatrix}\n0 & 0 & \\cdots & 0 & 0 & X_{i_1}\\\\\n0 & 0 & \\cdots & 0 & X_{i_2} & -1\\\\\n0 & 0 & \\cdots & X_{i_3} & -1 & 0\\\\\n\\vdots&\\vdots& \\cdots&\\vdots&\\vdots&\\vdots\\\\\n0 & X_{i_{l-1}}&\\cdots&0&0&0\\\\\nX_{i_l}&-1&\\cdots&0&0&0\n\\end{bmatrix},\n$$\nwith the obvious adaptations if $l=1,2$. The $(l-1)\\times(l-1)$ lower right corner of the \nabove matrix is invertible in the algebra $M_{l-1}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle)$ and \nits inverse has as entries polynomials of degree up to $l-1$ (again with the obvious modifications when \n$l\\leq 2$). The constant term in the inverse's formula is simply the matrix having $-1$ on its second \ndiagonal, and its spectrum included in $\\{-1,1\\}$.\n If the matrices $\\begin{bmatrix}\n0 & u_j\\\\\nv_j & Q_j\\end{bmatrix},$ \nwith $u_j\\in M_{1\\times n_j}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle),v_j\\in M_{n_j\\times1}\n(\\mathbb C\\langle X_1,\\dots,X_k\\rangle),Q_j\\in M_{n_j}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle)$\nare linearizations for $P_j,$ $j=1,2$, then \n$$\nL_{P_1+P_2}=\\begin{bmatrix}\n0 & u_1 & u_2\\\\\nv_1 & Q_1 & 0\\\\\nv_2 & 0 &Q_2\n\\end{bmatrix}\\in M_{n_1+n_2+1}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle).\n$$\nIn particular, we have again that $\\begin{bmatrix} Q_1 & 0\\\\ 0 & Q_2\\end{bmatrix}^{-1}\\in\nM_{n_1+n_2}(\\mathbb C\\langle X_1,\\dots,X_k\\rangle),$ with entries of degrees no more $\\max\\{n_1,n_2\\}$. \nAgain, its constant term has all its eigenvalues of absolute value equal to one.\nThe above construction does not necessarily provide a self-adjoint $L_P$, even if\n$P=P^*$. However, any self-adjoint polynomial $P$ is written as a sum $P=P_0+P_0^*$ for some \nother polynomial $P_0$ (self-adjoint or not) of the same degree. Let $L_{P_0}=\\begin{bmatrix}\n0 & u_0\\\\\nv_0 & Q_0\n\\end{bmatrix}.$ To insure that the linearization we obtain is self-adjoint, we write\n$$\nL_P=L_{P_0+P_0^*}=\\begin{bmatrix}\n0 & u_0 & v_0^*\\\\\nu_0^* & 0 & Q_0^* \\\\\nv_0 & Q_0 & 0\n\\end{bmatrix},\n$$\nwhich satisfies $L_P=L_P^*$ and linearizes $P$. Moreover, since $\\begin{bmatrix} 0 & Q_0^* \\\\\nQ_0 & 0\n\\end{bmatrix}^{-1}=\\begin{bmatrix} 0 & Q_0^{-1} \\\\\n(Q_0^*)^{-1} & 0\n\\end{bmatrix}$, the matrix $\\begin{bmatrix} 0 & Q_0^* \\\\\nQ_0 & 0\n\\end{bmatrix}^{-1}$ has entries which are polynomials in $X_1,\\dots,X_k$, all of them of degree \nmajorized by the degree of $P$, and its constant term is a complex matrix having \nspectrum included in the unit circle of the complex plane. These remarks, which the reader can find in\n\\cite{Mai}, will be most useful in our analysis below.\n\n\n\nIt follows from the above that if $X_1,\\dots,X_k$ are elements in some complex algebra $\\mathcal R$ \nwith unit $1$, then $z1-P(X_1,\\dots,X_k)$ is invertible in $\\mathcal R$ if and only if $z\\hat{E}_{11}-L_{P}\n(X_1,\\dots,X_k)$ is invertible in $M_m(\\mathcal R)$ ($m$ being the size of the matrix $L_P$). Moreover, \nthe construction above guarantees that the matrix $Q$ in the linearization $L_{P}(X_1,\\dots,X_k)\n=\\begin{bmatrix}\n0 & u\\\\\nv & Q\n\\end{bmatrix}$ is invertible independently of the elements $X_1,\\dots,X_k\\in\\mathcal R$.\nWe apply this to the case when $\\mathcal R\\subseteq\\mathcal B(\\mathcal H)$ is \na unital ${\\cal C}$${}^*$-algebra of bounded linear operators on the Hilbert space $\\mathcal H$, \nfor various separable Hilbert spaces $\\mathcal H$. We formalize this result in the following\n\n\n\\begin{lemme}\\label{inversible}\nLet $P=P^*\\in\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle$ and let\n$L_P \\in M_m(\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle)$\nbe a linearization of P with the properties outlined above. Let $y = (y_1,\\ldots, y_k)$ be a k-tuple of self-adjoint operators in a ${\\cal C}^*$-algebra ${\\cal A}$. Then, for any $z\\in \\mathbb{C}$, $z\\hat E_{11}\\otimes 1_{\\cal A}-L_P(y)$\nis invertible if and only if $z 1_{\\cal A}-P(y)$ is invertible.\n\\end{lemme}\n\nBeyond the property described above, we want also to compare the norms of the inverses of \n$z\\hat E_{11}\\otimes 1_{\\cal A}-L_P(y)$ and $z 1_{\\cal A}-P(y)$ when one (and hence the other) exists.\n\n\\begin{lemme}\\label{distanceauspectre} Let $P=P^*\\in\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle$ and let\n$L_P \\in M_m(\\mathbb{C}\\langle X_1,\\ldots,X_k\\rangle)$\nbe a linearization of P constructed as above. Let $y_n = (y_1^{(n)},\\ldots, y_k^{(n)})$ be a k-tuple of self-adjoint operators in a ${\\cal C}^*$-algebra ${\\cal A}$ such that $\\sup_n \\max_{i=1}^k \\Vert y_n^{(i)}\\Vert=C<+\\infty$. Let $z_0 \\in \\mathbb{C}$ be such that, for all large $n$, the distance from $z_0$\nto $sp(P(y_n))$ is greater than $\\delta$. Then, there exists a constant $\\epsilon > 0$, depending only on $ \\delta$, $L_P$ and $C$ such that the distance from $0$ to $sp(z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P(y_n))$\nis at least $\\epsilon$.\n\\end{lemme}\n\\begin{proof}\nIn this proof we only consider $P,u$ and $Q$ evaluated in $y_n$, so we will suppress $y_n$ from the \nnotation without any risk of confusion. Let $L_P=\\begin{bmatrix} 0 & u^*\\\\ u & Q\\end{bmatrix}$,\nso that $z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P=\\begin{bmatrix} z_0 &- u^*\\\\ -u & -Q\\end{bmatrix}$, as \nabove. We seek an $\\epsilon>0$ such that \n$z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P-z(I_m\\otimes 1_{\\cal A})=\\begin{bmatrix} z_0-z &- u^*\\\\ -u & -Q-\nz\\end{bmatrix}$ is invertible for all $|z|<\\epsilon$. We naturally require first that $-Q-z$ remains\ninvertible. As $Q=Q^*$, it follows by functional calculus that the spectrum of $-Q$ is at \ndistance equal to $\\|Q^{-1}\\|^{-1}$ from zero. As noted before, $Q^{-1}\\in M_{m-1}(\\mathbb C\n\\langle y_1^{(n)},\\dots,y_k^{(n)}\\rangle)$, with entries depending only on $P$, so that we can \nmajorize $\\|Q^{-1}\\|$ by a constant $\\kappa>0$ depending only on $C$ and $L_P$ (and independent of\nthe particular $y_n$). Thus, our first condition on $\\epsilon$ is $\\epsilon\\leq\\kappa^{-1}\/2$. \nNote that it follows that $\\|(Q+z)^{-1}\\|< 2 \\kappa.$\nNext,\nwe require that in addition $(z_0-z-u^*(-Q-z)^{-1}u)$ is invertible for all $|z|<\\epsilon$. By the openness\nof the resolvent set, we do know that such an $\\epsilon>0$ exists. More precisely, by a geometric series \nargument, if $a$ is invertible, then $b=((b-a)a^{-1}+1)a$ is invertible whenever $\\|b-a\\|<\\|a^{-1}\\|^{-1}\n$. We apply this to $a=z_0-P=z_0+u^*Q^{-1}u$ (so that $\\|a^{-1}\\|^{-1}>\\delta$) and \n$b=z_0-z-u^*(-Q-z)^{-1}u$.\nWe have \\\\\n\n\\noindent $\\|z_0+u^*Q^{-1}u-z_0+z+u^*(-Q-z)^{-1}u\\|$ \n\\begin{eqnarray*}\n& \\leq &|z|+ \\|u^*[(-Q-z)^{-1}+Q^{-1}]u\\|\\\\\n& \\leq &|z|+ |z|\\|u\\|^2\\|Q^{-1}\\|\\|(Q+z)^{-1}\\|\\\\\n&\\leq & |z|(1+ 2\\kappa^2 \\|u\\|^2).\n\\end{eqnarray*}\nSince the norm of $u$ is majorized in terms of $C$ and $L_P$ only by \nsome constant $\\ell>0$, we deduce that \n $\\|z_0+u^*Q^{-1}u-z_0+z+u^*(-Q-z)^{-1}u\\|\\leq |z|(1+ 2\\kappa^2 \\ell^2).$\nThus, we require $|z|(1+ 2\\kappa^2 \\ell^2)<\\delta$. \n This yields that , if $|z|<\\min\\{\\kappa^{-1}\/2,\\frac{\\delta}{(1+2\n\\kappa^2\\ell^2)}\\}$, then $z_0\\hat E_{11}\\otimes 1_{\\cal A}-L_P-z(I_m\\otimes 1_{\\cal A})$ is invertible. This\nconcludes the proof of our lemma.\n\n\\end{proof}\n\n\\subsection{From Lemma \\ref{inclu2} to Theorem \\ref{noeigenvalue}}\n\nLet $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint random variables which \nis free with the semicircular system $x=(x_1,\\ldots,x_r)$ in $({\\cal A},\\tau)$, such that the distribution of \n$a_n$ in $({\\cal A},\\tau)$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$.\nLet $P$ be a Hermitian polynomial in t+r noncommutative indeterminates.\nLet\n$L_P\\in M_m\\left(\\mathbb{C}\\langle X_1,\\ldots,X_{t+r}\\rangle\\right)$ be a linearization of $P$ as constructed in Section \\ref{linearisation}. Fix $\\delta>0$ and let $z\\in \\mathbb{R}$ be such that for all large $n$, the distance from $z$ to the spectrum of $\\left(P\\left(x_1,\\ldots,x_r, a_n^{(1)},\n\\ldots,a_n^{(t)}\\right)\\right)$ is greater than $\\delta$. According to Lemma \\ref{distanceauspectre} (and \n\\eqref{normeAn}), there exists a constant $\\epsilon > 0$, depending only on $\\delta,L_P$ and $\\sup_{n}\n\\max_{1\\le u\\le t}\\|A_n^{(u)}\\|$ such that the distance from 0 to $sp(z\\hat E_{11}\\otimes 1_{\\cal A}-\nL_P(x_1,\\ldots,x_r, a_n^{(1)},\\ldots,a_n^{(t)}))$ is as least $\\epsilon$. Now, according to Lemma \n\\ref{inclu2}, almost surely, for all large $n$, the distance from 0 to the spectrum of $(z\\hat E_{11}\\otimes I_n-L_P(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}))$ is as least \n$\\epsilon\/2$. Hence, \nfor any $z'\\in ]z-\\epsilon\/4;z+\\epsilon\/4[$, 0 is not in the spectrum of $(z' \\hat E_{11}\\otimes I_n-\nL_P(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)}))$.\n Finally, according to Lemma \\ref{inversible}, almost surely, for large $n$, $]z-\\epsilon\/4;z+\\epsilon\/4[$ lies outside the spectrum of $P(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}}, A_n^{(1)},\\ldots,A_n^{(t)})$.\nA compacity argument readily yields Theorem \\ref{noeigenvalue}.\n \n \n \n \n \n \n \n \n\\section{Proof of \\eqref{safbis}}\\label{strategie}\n\n Our approach is then very similar to that of \\cite{HT} and \\cite{Schultz05}. Therefore, we will recall the main steps.\\\\ First, the almost sure minoration\n$$ \\liminf_{n \\vers +\\infty} \\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}},A_n^{(1)},\\ldots,A_n^{(t)}\\right) \\right\\| \\geq \\left\\| P(x_1, \\ldots x_r,a_1,\\ldots,a_t)\\right\\| \\ $$\ncomes rather easily from \\eqref{af}; this can be proved by closely following the proof of Lemma 7.2 in \\cite{HT}. So, the main\ndifficulty is the proof of the almost sure reverse inequality:\n\\begin{equation}\\label{limsup}\n\\limsup_{n \\vers +\\infty} \\left\\| P\\left(\\frac{X_n^{(1)}}{\\sqrt{n}}, \\ldots, \\frac{X_n^{(r)}}{\\sqrt{n}},,A_n^{(1)},\\ldots,A_n^{(t)}\\right) \\right\\|\\leq \\left\\| P(x_1, \\ldots x_r,,a_1,\\ldots,a_t)\\right\\|_{\\cal A}. \\ \n\\end{equation}\nThe proof of\n(\\ref{limsup}) consists in two steps.\\\\\n\n\n \\noindent\n{\\bf Step 1: A linearization trick} (see Section 2 and the proof of Proposition 7.3 in \\cite{HT}) \\\\\nIn order to prove (\\ref{limsup}), it is sufficient to prove:\n\\begin{lemme} \\label{inclu} For all $m \\in \\N$, all self-adjoint matrices $\\gamma, \\alpha_1, \\ldots,\\alpha_r, \\beta_1, \\ldots, \\beta_t$ of size $m\\times m$ and\nall $\\epsilon >0$, almost surely for all large $n$,\n$$ \nsp(\\gamma \\otimes I_n + \\sum_{v=1}^r \\alpha_v \\otimes \\frac{X_n^{(v)}}{\\sqrt{n}}+ \\sum_{u=1}^t \\beta_u \\otimes A_n^{(u)})$$ \\begin{equation} \\label{spectre}\\subset\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_v + \\sum_{u=1}^t \\beta_u \\otimes a_u) + ]-\\epsilon, \\epsilon[.\n\\end{equation}\n\n\\end{lemme}\n\\noindent {\\bf Step 2: An intermediate inclusion}\nLet $a_n=(a_n^{(1)},\\ldots,a_n^{(t)})$ be a t-tuple of noncommutative self-adjoint random variables which is free with the semicircular system $x=(x_1,\\ldots,x_r)$ in $({\\cal A},\\tau$), such that the distribution of $a_n$ coincides with the distribution of $(A_n^{(1)},\\ldots, A_n^{(t)})$ in $({ M}_n(\\mathbb{C}),\\tr_n)$.\nMale \\cite{CamilleM} proved that if $(A_n^{(1)},\\ldots, A_n^{(t)})$ converges strongly to $(a_1,\\ldots, a_t)$, then, for any $\\epsilon >0$, for all large $n$,\\\\\n\n\\noindent \n$\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_v + \\sum_{u=1}^t \\beta_u \\otimes a_n^{(u)})$ \\begin{equation} \\label{spectre2} \\subset\nsp(\\gamma \\otimes 1_{\\cal A} + \\sum_{v=1}^r \\alpha_v \\otimes x_i + \\sum_{u=1}^t \\beta_u \\otimes a_u) + ]-\\epsilon, \\epsilon[.\n\\end{equation}\nTherefore, Lemma \\ref{inclu} can be deduced from Lemma \\ref{inclu2}.\n\n \n\n\\section{Appendix}\n\\subsection{Basic identities and inequalities}\n\\begin{lemme}\\label{majcarre}\nFor any matrix $M \\in M_m(\\mathbb{C})\\otimes M_n(\\mathbb{C})$ \nand for any fixed $k$, we have \\begin{equation}\\label{O} \\sum_{l =1}^n ||M_{lk} ||^2 \\leq m ||M ||^2\\end{equation}\n(or equivalently \\begin{equation}\\label{l} \\sum_{l =1}^n ||M_{kl} ||^2 \\leq m ||M ||^2.)\\end{equation}\nTherefore, we have \n \\begin{equation}\\label{lp}\n \\frac{1}{n} \\sum_{k,l =1}^n ||M_{kl} ||^2 \\leq m ||M ||^2.\n \\end{equation}\n\\end{lemme}\n\\begin{proof}\nNote that \\begin{eqnarray*} \\sum_{l =1}^n ||M_{lk} ||^2 &\\leq& \\sum_{l =1}^n ||M_{lk} ||_2^2\\\\&=& \\Tr M ( I_m\\otimes E_{kk}) M^*\\\\&=&\\Tr ( I_m\\otimes E_{kk}) M^*M ( I_m\\otimes E_{kk})\\\\&\\leq& \n \\Vert M \\Vert^2 \\Tr ( I_m\\otimes E_{kk})=m \\Vert M \\Vert^2 .\\end{eqnarray*}\nNow, since $$ \\sum_{l =1}^n ||M_{kl} ||^2=\\sum_{l =1}^n ||M_{kl}^* ||^2 = \\sum_{l =1}^n ||(M^*)_{lk} ||^2 \\; \\mbox{and} \\; \\Vert M^* \\Vert= \\Vert M \\Vert,$$\n\\eqref{O} and \\eqref{l} can be deduced from each other thanks to conjugate transposition.\nFinally \\eqref{l} readily yields \\eqref{lp}.\n\\end{proof}\n\\begin{lemme}\nLet $k\\geq 1$. Let $M^{(0)}, M^{(1)},\\ldots, M^{(k)}, M^{(k+1)}$, be $nm\\times nm$ matrices depending on $\\lambda \\in \\{\\rho\\in M_m(\\mathbb{C}), \\Im \\rho>0\\}$ such that \n$\\forall w=0, \\ldots, k+1, \\; \\left\\| M^{(w)} \\right\\| =O(1).$ Assume that for any $ (i,l)\\in \\{1,\\ldots,n\\}^2$,\n$z_{i,l}$ are complex numbers such that $\\sup_{i,l} \\vert z_{i,l} \\vert \\leq C$ for some constant $C$ \nand $\\{ i_w(i,l) \\}_{w=1,\\ldots,k+1}$ and \n $\\{j_w(i,l)\\}_{w=0,\\ldots,k}$ are equal to either $i$ or $l$.\\\\\n Then, \n \\begin{itemize} \\item for any $(p,q)\\in \\{1,\\ldots,n\\}^2,$ \\begin{equation}\\label{Oden}\\sum_{i,l=1}^n z_{i,l} M^{(0)}_{pj_0} M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k} M^{(k+1)}_{i_{k+1}q}=O_{p,q}^{(u)}(n),\\end{equation}\n\\item if there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$, then for any $(p,q)\\in \\{1,\\ldots,n\\}^2,$ \\begin{equation}\\label{Oderacine}\\sum_{i,l=1}^n z_{i,l} M^{(0)}_{pj_0} M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k} M^{(k+1)}_{i_{k+1}q} =O_{p,q}^{(u)}(\\sqrt{n}),\\end{equation}\n\\item if there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$, then\n \\begin{equation}\\label{Odenracinepas}\\sum_{i,l=1}^n z_{i,l} M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k} =O(n\\sqrt{n}),\\end{equation}\n \\item if there exists $(w_0, w_1)\\in \\{1,\\ldots,k\\}^2$, $w_0\\neq w_1$, such that \n $\\{(i_{w_{0}}, j_{w_0}), (i_{w_{1}}, j_{w_1})\\}$ is a subset of $\\{(i,l),(l,i)\\}^2$\n then \n \\begin{equation}\\label{Odenpas}\\sum_{i,l=1}^n z_{i,l} M^{(1)}_{i_1 j_1}\\cdots M^{(k)}_{i_k j_k} =O(n).\\end{equation}\n\\end{itemize}\n\\end{lemme}\n\\begin{proof}\nIf $(j_0, i_{k+1}) \\in \\{(i,l),(l,i)\\}$, noticing (using Lemma \\ref{majcarre}) that \\begin{eqnarray*}\\sum_{i,l=1}^n \\left\\|M^{(0)}_{pi} \\right\\| \\left\\| M^{(k+1)}_{lq}\\right\\|\n&\\leq& \\left( \\sum_{i=1}^n \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2} \\sqrt{n} \\left( \\sum_{l=1}^n \\left\\|M^{(k+1)}_{lq} \\right\\|^2 \\right)^{1\/2} \\sqrt{n} \\\\&=&O_{p,q}^{(u)}(n),\n\\end{eqnarray*}\n\\eqref{Oden} follows. \\\\\nNow, if $(j_0, i_{k+1}) \\in \\{(i,i),(l,l)\\}$, noticing (using Lemma \\ref{majcarre}) that \\begin{eqnarray*}\\sum_{i,l=1}^n \\left\\|M^{(0)}_{pi} \\right\\| \\left\\| M^{(k+1)}_{iq}\\right\\|\n&\\leq& n \\left( \\sum_{i=1}^n \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2} \\left( \\sum_{i=1}^n \\left\\|M^{(k+1)}_{iq} \\right\\|^2 \\right)^{1\/2} \n \\\\&=&O_{p,q}^{(u)}(n),\n\\end{eqnarray*}\n\\eqref{Oden} follows. The proof of \\eqref{Oden} is complete.\\\\\n\n\\noindent Now, assume that there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$. Let $\\tilde M^{(w_0)}$ be either $ M^{(w_0)}$ or $ (M^{(w_0)})^*$.\\\\\nIf $(j_0, i_{k+1}) \\in \\{(i,l),(l,i)\\}$, noticing (using Lemma \\ref{majcarre}) that\n\\\\\n\n$\\sum_{i,l=1}^n \\left\\|M^{(0)}_{pi} \\right\\|\\left\\| \\tilde M^{(w_0)}_{il}\\right\\| \\left\\| M^{(k+1)}_{lq}\\right\\|$ \\begin{eqnarray*}\n&\\leq& \\sum_{l=1}^n \\left\\|M^{(k+1)}_{lq} \\right\\| \\left( \\sum_{i=1}^n \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2} \\left( \\sum_{i=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2} \\\\&\\leq & \\left( \\sum_{i=1}^n \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2} \n\\left( \\sum_{i=1}^n \\left\\|M^{(k+1)}_{lq} \\right\\|^2 \\right)^{1\/2} \\left( \\sum_{i,l=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2}\n \\\\&=&O_{p,q}^{(u)}(\\sqrt{n}),\n\\end{eqnarray*}\n\\eqref{Oderacine} follows. \\\\\nNow, if $(j_0, i_{k+1}) \\in \\{(i,i),(l,l)\\}$, noticing (using Lemma \\ref{majcarre}) that \\\\\n\n\\noindent $\\sum_{i,l=1}^n \\left\\|M^{(0)}_{pi} \\right\\| \\left\\| \\tilde M^{(w_0)}_{il}\\right\\| \\left\\| M^{(k+1)}_{iq}\\right\\|$ \\begin{eqnarray*}\n&\\leq& n \\left\\| \\tilde M^{(w_0)}\\right\\| \\left( \\sum_{i=1}^n \\left\\|M^{(0)}_{pi} \\right\\|^2 \\right)^{1\/2} \\left( \\sum_{i=1}^n \\left\\|M^{(k+1)}_{iq} \\right\\|^2 \\right)^{1\/2} \n \\\\&=&O_{p,q}^{(u)}(n),\n\\end{eqnarray*}\n\\eqref{Oderacine} follows. The proof of \\eqref{Oderacine} is complete.\\\\\n\n\\noindent Assume that there exists $w_0\\in \\{1,\\ldots,k\\}$ such that $(i_{w_{0}}, j_{w_0})\\in \\{(i,l),(l,i)\\}$; then noticing (using Lemma \\ref{majcarre}) that \\begin{eqnarray*} \n\\sum_{i,l=1}^n \\left\\| \\tilde M^{(w_0)}_{il} \\right\\|& \\leq& n \\left( \\sum_{i,l=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2}\\\\&=& O(n\\sqrt{n}),\n\\end{eqnarray*}\n\\eqref{Odenracinepas} follows.\\\\\n\n\\noindent Now, assume that there exists $(w_0, w_1)\\in \\{1,\\ldots,k\\}^2$, $w_0\\neq w_1$, such that \n $\\{(i_{w_{0}}, j_{w_0}), (i_{w_{1}}, j_{w_1})\\}$ is a subset of $\\{(i,l),(l,i)\\}^2$. Let for $h=0,1$, $\\tilde M^{(w_h)}$ be either $ M^{(w_h)}$ or $ (M^{(w_h)})^*$; then noticing (using Lemma \\ref{majcarre}) that \\begin{eqnarray*} \n\\sum_{i,l=1}^n \\left\\| \\tilde M^{(w_0)}_{il} \\right\\| \\left\\| \\tilde M^{(w_1)}_{il} \\right\\|& \\leq& \\left( \\sum_{i,l=1}^n \\left\\|\\tilde M^{(w_0)}_{il} \\right\\|^2 \\right)^{1\/2} \\left( \\sum_{i,l=1}^n \\left\\| \\tilde M^{(w_1)}_{il} \\right\\|^2 \\right)^{1\/2}\\\\&=& O(n),\n\\end{eqnarray*}\n\\eqref{Odenpas} follows.\\\\\n\\end{proof}\n We end by recalling some properties of resolvents.\n First, one can easily see that for any $\\lambda$ and $\\lambda^{'}$ in $M_m(C)$ such that $\\Im(\\lambda)$ and $\\Im(\\lambda^{'})$ are positive\ndefinite,\n \\begin{equation}\\label{difStieljes}(\\lambda \\otimes 1_{{\\cal A}} - s)^{-1}-(\\lambda^{'} \\otimes 1_{{\\cal A}} -\n s)^{-1}=(\\lambda \\otimes 1_{{\\cal A}} - s)^{-1}(\\lambda^{'} -\n \\lambda)(\\lambda^{'} \\otimes 1_{{\\cal A}} - s)^{-1}.\n \\end{equation}\nFor a Hermitian matrix $M$, the derivative w.r.t $M$ of the resolvent $R(z) = (z-M)^{-1}$ satisfies:\n \\begin{equation} \\label{resolvente}\n R'_M(z) . H = R(z) H R(z) \\mbox{ for all Hermitian matrix $H$}. \\end{equation}\n\\begin{lemme} \\label{G}\nLet $\\lambda$ in $M_m(C)$ such that $\\Im(\\lambda)$ is positive\ndefinite and $h$ be a self-adjoint element in $M_m(\\C)\\otimes {\\cal A}$ where ${\\cal A}$ is a ${\\cal C}^*$-algebra endowed with some state $\\tau$. Then\n \\begin{equation}\\label{normeG}\n \\Vert (\\lambda \\otimes 1_{{\\cal A}} - h)^{-1}\\Vert \\leq ||\n \\Im(\\lambda)^{-1}\\Vert \\mbox{~~and~~~}\n || G(\\lambda ) || \\leq || \\Im(\\lambda)^{-1}\n ||,\\end{equation}\nwhere $G(\\lambda) = ({\\rm id}_m \\otimes \\tau) [ (\\lambda \\otimes 1_{{\\cal A}} - h)^{-1}].$\n\\end{lemme}\n\n\n\n \\begin{lemme} \\label{lem2}\n Let $\\lambda$ in $M_m(C)$ such that $\\Im(\\lambda)$ is positive definite, then for any $mn\\times mn$ Hermitian matrix $H$\n \\begin{equation}\\label{norme}\n || (\\lambda \\otimes I_n - H)^{-1} || \\leq || \\Im(\\lambda)^{-1}\n ||,\\end{equation}\n $$ \\forall 1 \\leq k,l \\leq n, || (\\lambda \\otimes I_n - H)^{-1}_{kl} || \\leq || \\Im(\\lambda)^{-1} ||,$$\n and for $p \\geq 2$,\n \\begin{equation}\\label{pplusgrand}\n \\frac{1}{n} \\sum_{k,l =1}^n ||(\\lambda \\otimes I_n - H)^{-1}_{kl} ||^p \\leq m ||\\Im(\\lambda)^{-1} ||^p.\n \\end{equation}\n\n\n\n \\end{lemme}\n \\subsection{Variance estimates}\nWe refer the reader to the book \\cite{Tou}. \nA probability measure $\\mu$ on $\\mathbb{R}$ is said to satisfy the Poincar\\' e inequality with constant $C_{PI}$ if\n for any\n${\\cal C}^1$ function $f: \\R\\rightarrow \\C$ such that $f$ and\n$f' $ are in $L^2(\\mu)$,\n$$\\mathbf{V}(f)\\leq C_{PI}\\int \\vert f' \\vert^2 d\\mu ,$$\n\\noindent with $\\mathbf{V}(f) = \\int \\vert\nf-\\int f d\\mu \\vert^2 d\\mu$. \\\\\n\n\n\\begin{remarque}\\label{multiple} If the law of a random variable $X$ satisfies the Poincar\\'e inequality with constant $C_{PI}$ then, for any fixed $\\alpha \\neq 0$, the law of $\\alpha X$ satisfies the Poincar\\'e inequality with constant $\\alpha^2 C_{PI}$.\\\\\nAssume that probability measures $\\mu_1,\\ldots,\\mu_M$ on $\\mathbb{R}$ satisfy the Poincar\\'e inequality with constant $C_{PI}(1),\\ldots,C_{PI}(M)$ respectively. Then the product measure $\\mu_1\\otimes \\cdots \\otimes \\mu_M$ on $\\mathbb{R}^M$ satisfies the Poincar\\'e inequality with constant $\\displaystyle{C_{PI}^*=\\max_{i\\in\\{1,\\ldots,M\\}}C_{PI}(i)}$ in the sense that for any differentiable function $f$ such that $f$ and its gradient ${\\rm grad} f$ are in $L^2(\\mu_1\\otimes \\cdots \\otimes \\mu_M)$,\n$$\\mathbf{V}(f)\\leq C_{PI}^* \\int \\Vert {\\rm grad} f \\Vert_2 ^2 d\\mu_1\\otimes \\cdots \\otimes \\mu_M$$\n\\noindent with $\\mathbf{V}(f) = \\int \\vert\nf-\\int f d\\mu_1\\otimes \\cdots \\otimes \\mu_M \\vert^2 d\\mu_1\\otimes \\cdots \\otimes \\mu_M$ (see Theorem 2.5 in \\cite{GuZe03}) .\n\\end{remarque}\n\n\n\n\n\n\n\n\n\\begin{lemme}\\label{zitt}[Theorem 1.2 in \\cite{BGMZ}]\nAssume that the distribution of a random variable $X$ is supported in $[-C;C]$ for some constant $C>1$. Let $g$ be an independent standard real Gaussian random variable. Then $X+\\delta g$ satisfies a Poincar\\'e inequality with constant \n$C_{PI}\\leq \\delta^2 \\exp \\left( 4C^2\/\\delta^2\\right)$.\n\\end{lemme}\n\nConsider the linear isomorphism $\\Psi_0$ between $M_n(\\C)_{sa}$ and $ \\mathbb{R}^{n^2}$ defined for any $[b_{kl}]_{k,l=1}^{n} \\in M_n(\\C)_{sa}$ by $$\\Psi_0([b_{kl}]_{k,l=1}^{n}) = ((b_{kk})_{1\\leq k\\leq n}, (\\sqrt{2} \\Re (b_{kl}))_{1\\leq k0$ such that, for any $n\\geq 1$ and for any $\\lambda$ in $M_m(\\mathbb{C})$ such that $\\Im \\lambda$ is positive definite,\nwe have, \n for any deterministic $nm\\times nm$ matrices $F_n^{(1)}$ and $F_n^{(2)}$ such that $\\Vert F_n^{(1)}\\Vert \\leq K$ and $ \\Vert F_n^{(2)}\\Vert \\leq K$, for any $(p,q)\\in \\{1,\\ldots,n\\}^2$,\\\\\n \n \\noindent \n$ \\mathbb{E}\\{\\Vert{\\rm id}_m\\otimes tr_n( F_n^{(1)}R_n(\\lambda)F_n^{(2)})-\\mathbb{E}({\\rm id}_m\\otimes tr_n(F_n^{(1)}R_n(\\lambda)F_n^{(2)}))\\Vert^2\\}$ \\begin{equation}\\label{varhn}\\leq\\frac{K^4C m^3}{n^2}\n \\Vert (\\Im (\\lambda))^{-1}\\Vert^4,\\end{equation}\n $ \\mathbb{E}\\{\\Vert (F_n^{(1)}R_n(\\lambda)F_n^{(2)})_{pq}-\\mathbb{E}((F_n^{(1)}R_n(\\lambda)F_n^{(2)})_{pq})\\Vert^2\\}$ $$~~~~~~~~~~\\leq\\frac{K^4C m^3}{n}\n \\Vert (\\Im(\\lambda))^{-1}\\Vert^4.$$\n\n \\end{lemme}\n\n{\\bf Acknowledgements.} The authors wish to thank an anonymous referee for pertinent comments\nwhich led to an improvement of this paper.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThere are several reasons to increase the still rather\nmeagre data on very high-$z$, powerful radio galaxies (e.g.\nMcCarthy \\cite{mcc}, Pariskij et al. \\cite{pari:kop}). \nHigh-$z$ radio galaxies are unique laboratories for investigating the early\nstages of galaxy and AGN evolution at look-back times corresponding to \nmore than 90\\% of the age of the universe derived from\nFriedmann models. \nThey may be used as tracers of the first generation of galaxy clusters\n(Peacock \\& Nicholson \\cite{pea:nic}; Peacock \\cite{peac}) and of the\nphysical state of the intergalactic space\n(Parijskij at al. \\cite{pari:goss}). By high-resolution \none may study important morphological\nfeatures related to e.g. \nmerging activity and ``star burst'' regions. \n\nAt redshift $z>2$, 120 radio galaxies are known at present\n(de Brueck et al. \\cite{debr:br}),\nin comparison with about 250 radio loud quasars\n($S_{5GHz}>$0.03 Jy in Veron-Cetty \\& Veron \\cite{ver:ver}),\n though the former are intrinsically more abundant. \nAccording to the popular unified scheme, both classes\nare the same thing. One can study the host galaxies and \nclose environments of radio galaxies, but this is difficult\nfor QSOs at a similar redshift.\n\n\n\n\nOne aspect, where even a single galaxy may be decisive, is the question of\nhow close in time to the cosmological singularity it is possible\nto find galaxies, with normal stellar population\nand supermassive compact objects in their nuclei.\nThough the use of high-$z$ objects\nin classical cosmological tests is hampered by\nsevere problems, development of such tests is still\none aim of observational cosmology.\nTo identify selection effects and\nevolution, large samples are required.\nOne must increase identifications of very remote galaxies, also\nin view of the new generation ground and space telescopes,\nwhich will allow their study at high resolution.\n\n\n\n\\subsection{\nExtension of identified USS sources to fainter fluxes}\n\nIt has been known since the late 70's \n(Tielens et al., \\cite{tie:mi}; Blumenthal \\& Miley \\cite{blu:mi}) \nthat radio sources with steep \nspectra are optically fainter (and hence probably more distant) \nthan sources with flatter spectra. \nLater it was established \nthat observing faint radio sources\nwith ultra-steep spectra (USS) is an efficient\nway to detect radio galaxies at high redshifts\n(see e.g. McCarthy \\cite{mcc}).\nAs the USS Fanaroff-Riley type II \n(FRII-type; Fanaroff \\& Riley \\cite{fan:ri}) \nradio galaxies are not\ngood ``standard radio candles'' and as \nthe reason for the success of\nthe spectrum criterium is not known \n(see e.g. R\\\"{o}ttgering et al. \\cite{rott:lacy}),\nit is not clear what the outcome will be when USS samples are extended to \na progressively fainter flux limit.\nFainter flux may\nimply 1) larger redshifts, 2) similar redshifts, though weaker\nradio luminosity, or 3) smaller redshifts and still weaker luminosities.\nThe first alternative is most interesting, though cases 2 and 3 \nare also important: extension of the luminosity range will help\none to uncover the influence of radio luminosity on the classical\ncosmological tests (angular size--redshift; \nNilsson et al. \\cite{nils:val}\nand Hubble diagram; Eales et al. \\cite{eal:ra}) and\nto decide whether alignment effect depends primarily\non redshift or luminosity.\n\n\nThe flux range where differential normalized source counts \nshow steepening is generally regarded as the most promising \nhunting place for high redshift objects.\nParijskij et al. (\\cite{pari:bur}) pointed out that the bulk of the\nRATAN-600 sample (see below) has fluxes in the range \nof 10-50 mJy at 3.9 GHz where the\nnormalized counts show a maximum steepening, usually interpreted as\na cosmological effect.\nA similar steepening in the counts is seen separately for steep\nspectrum sources (Fig. 6 in Kellermann \\& Wall 1987).\nIt has been suggested (e.g. R\\\"{o}ttgering et al. \\cite{rott:lacy}) \n that the most effective way to \nfind distant galaxies would be a USS sample with \n$S_{408}\\sim 0.2-1$ Jy.\nIndeed, this has proven to be so\nsince about 50\\% of the R\\\"{o}ttgering et al. (\\cite{rott:lacy})\nUSS objects have $z>2$ (van Ojik et al. \\cite{vanojik:rott}).\nThe bright end of the USS sources is well studied\n(e.g. $4C\/USS, B2\/1Jy, MRC\/1Jy$ McCarthy \\cite{mcc}\nand references therein) and\nrecently fainter flux limits have been reached \n(e.g. $B3\/VLA$ $S_{408}>0.8$ Jy Thompson et al. \\cite{thompson};\nESO\/Key-Project $S_{365}>0.3$ Jy\nR\\\"{o}ttgering at el. \\cite{rott:lacy}). \nHowever, in the R\\\"{o}ttgering et al. (\\cite{rott:lacy})\nsample 365 MHz flux density distribution peaks at about 1 Jy. \n\n\n\n\\subsection{RATAN-600 (RC) and UTRAO catalogues}\n\nThis paper is part of a programme initiated\nat the Special Astrophysical Observatory (Russia) with\nthe aim of searching distant radio galaxies and\ninvestigating the early evolutionary stages of\nthe universe (Goss et al. \\cite{goss:par}).\nWe wish to extend the steep-spectrum criteria to fainter fluxes\nthan previously.\nThis is accomplished by RC and UTRAO catalogues (see Fig. ~\\ref{fig1})\n\n\n\n\\begin{figure}\n\\begin{center}\n\\hspace*{0.5cm}\n\\epsfig{file=astrds7556f1.ps, height=5.0cm}\n\\end{center}\n\\caption[]{\nFrequency - flux limit diagram with the positions of the RC sample\nand some other major radio catalogues. \nUTRAO (-36$\\degr < \\delta <$ 72$\\degr$) is the optimum \nlow frequency catalogue presently available, \nwhich can be used for calculating\nthe spectral index for a large part of the RC sample \n($\\delta \\sim 5\\degr$).\nNote that the 6C sample has $\\delta >$ 20$\\degr$.\nThe lines correspond to a source with $\\alpha$=1.\n}\n\\label{fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\hspace*{0.5cm}\n\\epsfig{file=astrds7556f2.ps, height=5.0cm}\n\\caption[]{\nHubble diagram in R-band for various radio galaxies \nfrom the literature.\nThe triangles are from the complete Molonglo sample\n(McCarthy et al. \\cite{mcc:kap2}),\nthe open boxes are from \nAllington-Smith et al.(\\cite{all:spi}),\nMaxfield et al. (\\cite{max:tho}),\nMcCarthy et al. (\\cite{mcc:spi}),\nMcCarthy et al. (\\cite{mcc:kap1}),\nMcCarthy et al. (\\cite{mcc:van}),\nThompson et al. (\\cite{thompson}),\nWindhorst et al. (\\cite{wind}).\nFilled dots are from\nCarilli et al. (\\cite{car:ro}),\nChambers et al. (\\cite{cham:mi}),\nDjorgovski et al. (\\cite{djor:spin}),\nDunlop\\&Peacock (\\cite{dun}),\nEales et al. (\\cite{eal:raw}), \nHammer\\&LeFevre (\\cite{ham:le}), \nKristian et al. (\\cite{kri:san}),\nLacy et al. (\\cite{lac:mi}), \nLeFevre et al. (\\cite{lefev:ham1}),\nLeFevre\\&Hammer (\\cite{lefev:ham2}),\nLilly (\\cite{lil1}),\nLilly (\\cite{lil2}),\nOwen\\&Keel (\\cite{owen:keel}),\nMiley et al. (\\cite{mi:ch}), \nSpinrad et al. (\\cite{spin}) \nand filled stars are from the ESO\/Key-Project\n(R\\\"{o}ttgering et al. \\cite{rott:miley}\nR\\\"{o}ttgering et al. \\cite{rott:west}\nvan Ojik et al. \\cite{van:rott}).\nOpen symbols are $r$-magnitudes, which are transformed\nas $R=r-0.4$.\nThe histogram of RC\/USS sources\n$R$-magnitudes is shown above.\nThe magnitudes are from K95b.\nPresent NOT-observations are concerned with $R\\la24$. \n}\n\\label{fig2}\n\\end{figure}\n\n\nOur high frequency\ncatalogue is based on a sample of faint radio sources\noriginally discovered using the RATAN-600 radio telescope\nin the \"Kholod\" (\"Cold\") experiment in 1980-81\n(Parijskij et al. \\cite{pari:bur}; Parijskij et al. \\cite{pari:bur2}\nParijskij \\& Korolkov \\cite{par:kor}).\nIn the experiment, performed at 7.6 cm (3.9 GHz),\nthe strip around the sky at $\\delta$=5$\\degr \\pm$ 20$\\arcmin$\nwas surveyed with a limiting flux of about 4 mJy.\nThe RC catalogue resulted in containing 1145 objects.\nWithin the inner strip of $\\pm$ 5$\\arcmin$\nthe completeness of the catalogue reaches 80\\% at\nthe flux limit S$_{3.9} =7.5$ mJy and is almost\n100\\% at 15mJy (Parijskij et al. \\cite{pari:bur}).\nSuch flux limits are really quite faint and allow one\nto identify a large number of steep spectrum sources,\nif a low frequency catalogue with\nsufficiently faint flux limit is available.\nThe UTRAO (Douglas et al. \\cite{douglas})\nis such a catalogue with a\nflux limit of $\\sim$ 100mJy at 365 MHz (see Fig. ~\\ref{fig1}).\nThe RATAN-600 catalogue (RC) provided the first sample \nwhich allowed one\nto calculate the spectral index for practically all\nUTRAO sources within the region covered by the \"Kholod\"\nexperiment (Soboleva et al. \\cite{sobo:pari}).\nOf the original sample of 840 sources (Parijskij et al. \\cite{pari:bur}), \n491 sources matched those of the UTRAO catalogue. \nSoboleva et al. (\\cite{sobo:pari}) could identify optically from\nPOSS (Palomar Optical Sky Survey) 240 sources at galactic \nlatitude $>$ 20$\\degr$.\n\n\n\n\n\\begin{table*}\n\\caption[]{RC\/USS source parameters. \nThe IAU name is in the first column followed by the\nequatorial, then galactic coordinates and galactic extinction in R-band. \nThe radio spectral index is in the seventh column, followed by\n3.9 GHz flux density and the LAS of the radio source.\nThe results of optical identification are in the last column.\nThe data have been taken from \nKopylov et al. \\cite{kop:goss1}, \\cite{kop:goss2} \nand Parijskij et al. \\cite{pari:goss}. }\n\\begin{flushleft}\n\\begin{center}\n\\begin{tabular}{lllrrrlrrr}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nName& R.A. &Decl &$l$&$b$&A$_{R}$& $\\alpha^{365}_{3900}$ & S$_{3900}$ & LAS & m$_{R}$\\\\\n & B1950 &B1950& \\degr &\\degr & & &mJy & $[\\arcsec]$ & \\\\ \n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\object{J0406+0453}&04 03 48.22&4 39 49.7&186 &-33&0.41 &1.02 & 79 & 21.8 & 24.9\\\\\n\\object{J0444+0501}&04 41 38.68&4 55 55.79&192&-25&0.41 &1.09 & 69 & 10.8 &23.0 \\\\\n\\object{J0457+0452}&04 55 15.09&4 49 13.77&194&-22&0.23 &1.12 & 56 & 34 &19.4 \\\\\n\\object{J0459+0456}&04 56 25.51&4 51 30.45&194&-22&0.23 &0.95 & 76 & 63.8 &22.1 \\\\\n\\object{J0506+0508}&05 03 45.56&5 04 21.1 &195&-20&0.27 &0.88 & 70 & 0.8 &21.6 \\\\\n\\object{J0552+0451}&05 50 16.92&4 46 49.9 &201&-10&1.51 &1.18 & 65 & 1.6 &$>$25.5 \\\\\n\\object{J0743+0455}&07 40 36.54&5 03 02.88&214& 13&0.12 &1.07 & 37 & 20.5 &23.5 \\\\\n\\object{J0756+0450}&07 53 31.2 &4 47 17.1 &216& 16&0.07 &1.16 & 14 & $<$1&$>$25.0 \\\\\n\\object{J0837+0446}&08 34 51.28&4 54 51.82&221& 25&0.07 &1.0 & 54 & 3.9 &22.4 \\\\\n\\object{J0845+0444}&08 42 53.28&4 53 52.9 &222& 27&0.09 &1.14 & 135 & 4.6 &21.4 \\\\\n\\object{J0909+0445}&09 07 13.51&4 56 37.0 &225& 32&0.08 &1.0 & 64 & 1 &20.6 \\\\\n\\object{J0934+0505}&09 31 48.21&5 17 10.76&229& 38&0.06 &1.07 & 36 & 5 &24.4 \\\\\n\\object{J1031+0443}&10 28 43.01&4 58 33.53&240& 49&0.06 &1.2 & 191 & 33 &22.5 \\\\\n\\object{J1043+0443}&10 41 10.27&4 56 12.62&243& 52&0.06 &1.14 & 37 & 48 &23.0 \\\\\n\\object{J1113+0436}&11 11 24.05&4 54 20.12&252& 57&0.10 &0.98 & 52 & 29 &22.4 \\\\\n\\object{J1152+0449}&11 49 49.71&5 04 56.75&268& 63&0.07 &1.0 & 29 & 7 &$>$24.0 \\\\\n\\object{J1155+0444}&11 52 45.43&5 00 13.86&269& 63&0.07 &1.0 & 54 & 13 &18.6 \\\\\n\\object{J1219+0446}&12 17 06.94&5 04 02.84&282& 66&0.06 &1.23 & 23 & 118 &22.0 \\\\\n\\object{J1235+0435}&12 33 16.52&4 49 26.7 &292& 67&0.06 &0.98 & 45 & 7 &21.5 \\\\\n\\object{J1322+0449}&13 19 31.84&5 04 28.13&322& 66&0.06 &0.96 & 47 & 7 &20.4 \\\\\n\\object{J1333+0451}&13 30 32.35&5 07 08.5 &328& 65&0.05 &1.3 & 11 & 1 &23.4 \\\\\n\\object{J1333+0452}&13 30 54.66&5 07 21.17&328& 65&0.05 &1.4 & 16 & 54 &23.3 \\\\\n\\object{J1339+0445}&13 37 06.5 &5 10 15.85&332& 64&0.06 &1.07 & 41 & 34 &22.7 \\\\\n\\object{J1347+0441}&13 44 37.58&4 57 16.48&336& 63&0.06 &0.98 & 43 & 1.4 &23.5 \\\\\n\\object{J1429+0501}&14 26 45.73&5 14 43.41&353& 57&0.06 &0.92 & 82 &11.1 &$>$24.0\\\\\n\\object{J1436+0501}&14 34 04.66&5 15 10.8 &356& 56&0.06 &1.25 & 48 & 15 &22.9 \\\\\n\\object{J1439+0455}&14 37 15.64&5 08 38.68&357& 55&0.06 &1.15 & 40 & 17.9&$>$24.0 \\\\\n\\object{J1510+0438}&15 07 43.00&4 50 51.72& 4& 50&0.06 &0.9 & 67 & 3.4 &22.1 \\\\\n\\object{J1609+0456}&16 06 54.69&5 07 50.48& 16& 38&0.14 &1.15 & 30 & 6.3 &$>$24.5 \\\\\n\\object{J1626+0448}&16 24 21.72&4 55 33.4 & 19& 34&0.18 &1.26 & 39 & 2.4 &22.9 \\\\\n\\object{J1646+0501}&16 44 24.94&5 06 28.92& 22& 29&0.27 &0.92 & 54 & 15.7 &21.2 \\\\\n\\object{J1658+0454}&16 55 43.34&4 58 04.9 & 23& 27&0.27 &1.25 & 31&$<$0.3&$>$24.5\\\\\n\\object{J1703+0502}&17 01 01.3 &5 06 20.0 & 24& 26&0.27 &1.18 & 175 & 1.8 &23.6 \\\\\n\\object{J1720+0455}&17 17 36.0 &4 56 48.0 & 26& 22&0.27 &1.22 & 19 & $<$0.5&20.6 \\\\\n\\object{J1725+0457}&17 23 04.58&5 00 05.0 & 27& 21&0.27 &1.26 & 27 & 1 &$>$24.0 \\\\\n\\object{J1735+0454}&17 33 13.52&4 57 07.37& 28& 19&0.39 &1.0 & 30 & 4 &23.5 \\\\\n\\object{J1740+0502}&17 38 06.03&5 04 11.1 & 29& 18&0.41 &1.2 & 32 & 4 &22.5 \\\\\n\\object{J2013+0508}&20 10 54.69&5 01 24.78& 47&-15&0.46 &0.96 & 51 & 10 &21.1 \\\\\n\\object{J2036+0451}&20 34 27.46&4 39 22.7 & 50&-20&0.27 &1.02 & 75 & 56 &19.0 \\\\\n\\object{J2144+0513}&21 41 56.65&4 57 26.1 & 61&-34&0.18 &1.06 & 72 & $<$5.5&18.8 \\\\\n\\noalign{\\smallskip}\t\t\t\t\t\t \n\\hline\t\t\t\t\t\t\t\t \n\\end{tabular}\t\t\t\t\t\t\t \n\\end{center}\n\\end{flushleft}\n\\end{table*}\t\t\t\t\t\t\t \n\t\t\t\t\t\t\t\t \n\t\t\t\t\t\t\t\t \n\n\n\\subsection{\nConstruction and properties of the RC\/USS sample}\n\n\nThe present study is concerned with\nsources in the range 4$^{h}$ 0.9, \\, $f_{\\nu}\\propto\\nu^{-\\alpha}$),\ndouble or triple FRII sources,\nand optically fainter than the POSS limit. The radio morphology\ncomes from observations with the VLA \n(Kopylov et al. \\cite{kop:goss1}).\nThe largest angular size of the radio source (LAS) \nwas not used as a criterion, because\nonly eight sources had LAS larger than 30\".\nThe median LAS of the sample is 7$\\arcsec$. \nThe median 365 MHz flux density is 0.5 Jy (average 0.7 Jy) \nranging from 0.2 Jy to 3 Jy.\n\n\nOptical identifications were made from deep observations at\nthe 6 m telescope, down to about $m_{R}$=24. These results\nand the optical fields around the sources have been reported\nby Kopylov et al. (\\cite{kop:goss2}, here after K95b). \nTable 1 contains information on the basic RC\/USS sample:\nsource name, equatorial and galactic coordinates, spectral index,\nflux, LAS and $m_{R}$. \nFrom this list we selected\nobjects which are not unreasonably faint\n($m_{R} <$ 24 mag) for a medium sized telescope.\n\n\n\nFig.~\\ref{fig2} gives a representative\n m$_{R}$-$z$ Hubble diagram for radio galaxies\ncollected from the literature, together with the magnitude distribution\nof the RC\/USS objects. \nThe Hubble diagram allows one to estimate\na lower limit to redshift, because of the rather sharp lower envelope,\nespecially above $m_{R}$=21. Where the bulk of the RC\/USS galaxies \nare situated, redshift is expected to be $\\ga$0.7 as shown\nin Fig. 2. \nSoboleva et al. (\\cite{sobo:pari}) estimated the maximum \nphotometric redshifts for the RC\/USS objects from \nthe requirement that radio\nluminosity is not higher than optical luminosity: when radio flux\nis known, the minimum optical magnitude may be calculated,\nhence the rough maximum $z_{ph}$, which is usually large, $>$1.\n\n\nIt should be mentioned that one optically\nbright ($m_{R}$=19) object RC2036+0451 was measured at the\n6 m telescope to have $z$=2.95 (Pariskij et al. \\cite{pari:sobo}).\nThough for a quasar, this large $z$ also supports the view that\npresent selection criteria lead to high average redshift.\n\nThe aim of the NOT imaging\nwas to study the morphology\nof the RC\/USS sources with high resolution\nand confirm the optical identifications.\nThis paper is organised as follows. \nIn Sect. 2 we describe our observations\nand data reduction. Morphology of individual galaxies \nis discussed\nin Sect. 3. The results are summarised in Sect. 4.\n\n\n\n\n\\section{Observations and reductions}\n\n\\subsection{Observations}\n\n\nOptical images were obtained with the 2.56 m\nNordic Optical Telescope (NOT)\nat La Palma during three observing runs in March,\nMay and December 1994.\nTable 2 summarises the instrumentation used. \nIn addition, we have\nsome supplementary observations from other observing runs.\nI-band observations of RC1510+0438 were made with \n``Stockholm'' CCD in July 1994 and\nRC2013+0508 was observed with Brocam1 in September 1994.\nThe complete log of observations is given in Table 3.\nFor each observed object it contains the filter used, \nnumber of separate images,\ntotal integration time, seeing, and date. \nCalibration stars from Landolt (\\cite{landolt}) were\nobserved several times each night at a range of air masses.\n\n\n\n\\begin{table*}\n\\caption[]{Instruments}\n\\begin{flushleft}\n\\begin{tabular}{lllll}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nCCD & Date & Field size (pixels) & Field size ($\\arcmin$)&\nPixel size ($\\arcsec$) \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nAstromed &&&&\\\\\nEEV P88200 & March 1994 & 1152x770 & 3.1x2.1& 0.163\\\\\nIAC CCD &&&&\\\\\nTHX31156 & May 1994 & 1024x1024 & 2.4x2.4 &\n0.14\\\\\nBrocam 1 &&&&\\\\\nTK1024A& December 1994 & 1024x1024& 3x3 & 0.176\\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{flushleft}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\caption[]{Journal of observations. }\n\\begin{flushleft}\n\\begin{tabular}{cccrll|cccrll}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nObject & Band & No. of & T$_{int}$ & Seeing & Date & \nObject & Band & No. of & T$_{int}$ & Seeing & Date \\\\\n & & images & $[sec]$ & $[\\arcsec]$ & 1994 & \n& & images & $[sec]$ & $[\\arcsec]$&1994 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nRC0444+0501 & R& 5 & 3300 & 2.0 &3.12. &RC1219+0446 & R &5 & 3000 & 0.5&18.5 \\\\\n & I &3 & 2700 & 2.0 & 4.12.&RC1235+0435 & V &5 & 3000 & 3 & 14.3 \\\\\nRC0457+0452 & B &3 & 3200 & 3.2-3.9& 3.\\&5.12.& & V &3 & 2100 & 0.6 &17.5 \\\\\n & V &2 & 1800 & 0.7 & 2.12.& \t & R &3 & 1800 & 3 & 14.3\\\\ \n & R &4 & 3600 & 0.8 & 2.12.&\t& R &3 & 1800 & 0.6&17.5\\\\ \n & I &4 & 2100 & 2.0-2.9 & 4.\\& 5.12. & RC1322+0449 &V&6 &3600 & 1.8 & 14.3\\\\\nRC0459+0456 & V &2 & 1800 & 0.7& 2.12.&\t & R &3 & 1800 & 1.8 & 14.3\\\\\n & R &2 & 1800 & 0.6 & 2.12.& RC1333+0451 & V &1 & 600 &0.6 & 18.5\\\\\n & I &3 & 2700 & 2.0 & 3.12.& \t & R &3 & 2100 & 0.5&18.5 \\\\\nRC0506+0558 & V &2 & 1200 & 0.8& 2.12.&\tRC1339+0445 &V &5 & 3000 & 1.4 & 15.3\\\\\n & R &6 & 1800 & 0.8 & 2.12.&\t & R &3 & 1800 & 1.4 & 15.3\\\\ \n & I &2 & 1200 & 1.5 & 4.12.& RC1347+0441 & V &5 &3300 & 1.1 &16.\\\\\nRC0743+0455 & V &3 & 1800 & 1.3 & 15.3&\t & R &3 & 2400 & 0.7 &16.5 \\\\\n & R &3 & 1800 & 1.1 & 15.3&\t RC1510+0438& V &3 & 2100 & 0.6 &17.5\\\\\n & R &1 & 600 & 0.8 & 2.12.&\t& R &7 & 4100 & 0.5&17\\&18.5 \\\\\nRC0837+0446 & V &3 & 1800 & 0.9 & 14.3&\tRC1609+0456 & V &1& 600 & 0.6 &18.5 \\\\\n & R &4 & 2400 & 1.0 & 14.3&\t & R &2 & 600 & 0.5&18.5 \\\\\n & I &3 & 2400 & 1.6 & 4.12.&RC1626+0448& V &2 & 1200 & 1.7 & 15.3\\\\\nRC0845+0444 & V &1 & 600 & 1.2 & 15.3&\t & R &2 & 1200 & 1.7 & 15.3\\\\\n & R &1 & 600 & 1.1 & 15.3&\tRC1646+0501&V&3&1800 & 1.5-2.0 & 13.3\\\\\n & I &2 & 2800 & 1.2 & 4.12.& & R &3 & 1800 & 1.5-2.0 & 13.3\\\\ \nRC0909+0445 & B &3 & 2700 & 1.8& 3.12.&\tRC1703+0502 &R &4 & 2400 & 0.6 & 17.5\\\\\n & B &4 & 4500 & 3.0 & 5.12.&RC1720+0455& V &2 & 1200 & 1.5-2.0 & 13.3\\\\ \n & V &2 & 1200 & 1.7 & 14.3 & & V &1 & 600 &0.7 &16.5 \\\\ \n & V &2 & 1800 & 1.6 & 3.12.& & R &1 & 600 & 1.5-2.0 & 13.3\\\\ \n & R &2 & 1200 & 1.7 & 14.3&\t & R &2 & 1200 & 0.7 &16.5 \\\\ \n & R &4 & 3600 & 2.0 & 3.12.& RC1735+0454&R&12& 3080& 0.6&17\\&18.5\\\\\n & I &1 & 900 & 1.5 & 4.12. & RC1740+0502&V &3 & 1800 & 0.6 &16.5\\\\\n & I &3 & 2700 & 3.0 & 5.12.& & R &2& 1200 & 0.6 &16.5 \\\\\nRC1031+0443 & V &6 & 3600 &1.5-2.0&13.3&RC2013+0508 & V &2&1800&1.5&3.\\&4.12.\\\\\n & V &3 & 1800 & 1.5 & 15.3 & & R &1 & 900 & 1.2 & 4.12.\\\\\n & R &3 & 1800 & 1.6 & 13.3 & & R &1 & 600 & 0.6 & 4.9. \\\\\n & R &3 & 1800 & 0.9 & 15.3 &RC2036+0451&B&2 &1800 & 1.4 & 2.12.\\\\\n & I &2 & 1800 & 1.4 & 4.12.& & V &2 & 1800 & 1.5 & 3.\\& 4.12.\\\\ \nRC1043+0443 & V &5 & 3000 & 1.7 & 14.3 & & R &1 & 900 & 1.1 & 4.12.\\\\ \n & R &4 & 2400 & 1.7 & 14.3 & & I &1 & 900 & 3.0 & 5.12.\\\\ \nRC1113+0436 & V &5 & 3000 & 1.6 & 15.3 &RC2144+0513 & B &2 & 1800&1.8 & 3.12.\\\\\n & R &3 & 1800 & 1.6 & 15.3 & & V &2 & 1800 & 1.0 & 2.12.\\\\ \nRC1152+0449 & V &5 & 3000 & 1.0 &16.5& & R &3 & 1920 & 1.0 & 2.12.\\\\ \n & R &3 & 1800 & 0.8 &16.5& & I &2 & 1800 & 1.2 & 4.12.\\\\ \nRC1155+0444 & V &1 & 600 & 0.6 &17.5 \\\\\n & R &1 & 600 & 0.6 &17.5 \\\\ \n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{flushleft}\n\\end{table*}\n\n\n\nIn this paper,\nwe shall restrict the discussion to observations\nmade under excellent or good seeing (FWHM $\\la 1\\farcs$1) \nconditions, totaling 22 objects.\nWe present only R-band images except for a few cases where\nthe morphology has a strong wavelength dependence and the\nS\/N-ratio in other passbands is high enough.\nAll observations presented were made under\nphotometric conditions. \n\n``Blooming'' of the CCD was a serious problem with the IAC CCD.\nSome objects were close to bright stars which limited\nthe longest possible exposure time, or a bright star had to be\nmoved outside the CCD.\nRC1735+0454 lies close to the galactic plane, hence\nthe field is crowded with bright stars. \nWe could not obtain long exposures of this faint object \nand had to move it close to the edge of the CCD.\nNote that the exposure time of the greyscale image is \n900 seconds and for the contour image 3080 seconds.\nThe bright star northeast from the centre of gravity of RC1219+0446\nhampered the observations and the northern part of the radio\nsource was not observed.\n\n\\subsection{Reductions}\n\nThe reductions were carried out using standard IRAF routines\n(bias subtraction, trimming, flat fielding).\nThe average bias frame was constructed for each night.\nThe flatfielding was made by twilight flats\nobtained each evening and morning.\nAll the scientific frames were flattened at better than a 1\\%-level.\nThe exposures of each object were registered in position\nusing several stars in the field and then averaged. The number of\nreference stars varied from three up to a dozen. \n\n\nThe astrometric calibration was carried out using the\nAPM Catalogue (Irwin et al. \\cite{irwin}) whenever possible.\nFor the objects near the galactic plane\nthe Guide Star Catalog (GSC) (Lasker et al. \\cite{lasker})\nwas used. \nDue to the small field of view of the CCDs there\nwere typically only a few reference stars in the frame.\nThe number of stars and hence the accuracy\nof the astrometry strongly depends on\ngalactic latitude. We estimate the accuracy\nof the astrometric calibration to be typically better than\n1 second of arc. This is enough for the current study,\nbecause the typical resolution of the radio map is\nabout 1$\\farcs$5\nand most of the radio sources are so compact that the optical\nidentification is straightforward. \n\n\nAs a check of our photometry in the March and May 1994 run\nwe measured comparison stars\nof OJ287 (Fiorucci \\& Tosti \\cite{fiorucci}). \nThe derived brightnesses\nwere consistent with each other within 0.1 magnitudes.\n\n\n\n\n\n\\begin{table*}\n\\caption[]{NOT imaging data. The diameter of the aperture is \nindicated in arcseconds. The magnitudes are without \ncorrection of galactic extinction. Ellipticity\nand position angle of resolved sources is measured\nwith the same aperture as the magnitudes.\nThe radio position angles are measured from Kopylov et al.\n(\\cite{kop:goss1}).\n}\n\\begin{flushleft}\n\\begin{center}\n\\begin{tabular}{llllllcrr}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nName & Aperture & $m_{R}$ & merr&$m_{V-R}$&merr&$e$&PA$_{opt}$ & PA$_{radio}$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip }\nRC0457+0452 &7 &19.72 &0.01 &0.93 &0.05 &0.13 &13 & 58 \\\\\nRC0506+0558 &3.5 &21.54 &0.03 &1.29 &0.13 &unresolved & .. & -75 \\\\\nRC0837+0446 &6.5 &22.19 &0.08 &0.74 &0.13 &0.23 & 76 & -67 \\\\\nRC0845+0444 &8.2 &20.75 &0.08 &1.15 &0.17 &0.25 & -78 &9 \\\\\nRC1031+0443 &6.5 &22.32 &0.12 &1.12 &0.23 &0.57 & 88 &-36 \\\\\nRC1152+0449 &3 &22.23 &0.07 & 0.84 &0.15 &0.11 & -33 &-15 \\\\\nRC1155+0444 &8.4 &18.81 &0.02 & 1.09 &0.05 &0.28 & -54 &-76 \\\\\nRC1235+0453 &4.2 &21.70 &0.06 & 0.96 &0.17 &0.26 & 43 &-50 \\\\\nRC1347+0441 &2 &23.99 &0.19 & 0.75 &0.43 &0.32 & -29 &-49 \\\\\nRC1510+0438 &3 &22.20 &0.04 & 1.57 &0.25 &0.08 & -79 &62 \\\\\nRC1703+0502 &3 &23.69 &0.18 & .. & .. &0.49 & -86 &-82 \\\\\nRC1720+0455 &4.2 &20.34 &0.02 & 1.15 &0.07 &unresolved & ..&point \\\\\nRC1740+0502 &3 &22.19 &0.08 & 0.74 &0.12 &0.05 & 66 &64 \\\\\nRC2013+0508 &5.3 &20.70 &0.04 & 0.22 &0.06 &unresolved& .. &-55 \\\\\nRC2036+0451 &4.2 &19.06 &0.02 & 0.35 &0.04 &unresolved & .. &-2 \\\\\nRC2144+0513 &3.5 &18.89 &0.02 &0.24 &0.03 &unresolved& .. &point \\\\\n\\noalign{\\smallskip}\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{flushleft}\n\\end{table*}\n\n\\section{Reduced images: Overview of the morphology}\n\n\n\nIn this section \nwe give a greyscale image and a contour map\nfor each identified object (Fig.~\\ref{fig4}) \nThe close environment and faint features can be \nstudied from the greyscale image and\nthe confidence level of the features and light distribution\nfrom the contour map. \nThe field of view of the greyscale image is indicated in the\nupper left hand corner. \nThe images are slightly smoothed with a Gaussian function\n($FWHM=\\frac{1}{2}seeing$) in order to enhance the low \nsurface brightness features and maintain the resolution\nof the original images.\nThe centre of gravity of the radio source and\nthe positions of the radio lobes are indicated with a cross\nand circles, respectively. The images are presented in\nlinear scale from 0$\\sigma$ to 10$\\sigma$ above\nthe background of the image.\nIn contour maps the object is in the origin and \nthe numbers on both the vertical and horizontal\naxes refer to distance in arcsecond.\nThe contour interval is 0.5 mag arcsec$^{-2}$ and\nthe surface brightness of the lowest\ncontour is indicated in the upper right hand corner. \nThe limiting surface brightness at which objects can be detected\nis typically between 25 and 26 mag arcsec$^{-2}$.\nFor RC1510+0438 we also give V and I band images.\nUncertain identifications are presented in Fig. \\ref{fig5}\nand ``faint objects'' in Fig. ~\\ref{fig6}.\n\n\nThe magnitudes are based on aperture photometry using\nDAOPHOT. The size of the aperture was selected\nin such a way that 1) as much light as possible was included\nin the aperture while keeping the errors reasonable but \n2) the companions were excluded.\nThe image shapes are determined by the moments of the \nbrightness distribution\nusing IMEXAMINE (eq. 4 in Valdes et al. \\cite{valdes})\nwith the size of the aperture the same as in photometry. \nThe estimation is vague for small \nellipticity (e.g. RC1152+0449) or when the inner regions \nhave a different position angle than the outer regions\n(e.g. RC1031+0443). The results of photometry and\n image shape analysis are given in Table 4.\nThe magnitudes between this work and K95b\nare generally in agreement.\nThe differences are primarily caused by\nthe better seeing conditions at NOT, as compared with the 6 m-telescope\nand the different size of the aperture.\n\n\n\\subsection{Identified objects}\n\nIn this section we give notes on individual objects (Fig. ~\\ref{fig4}).\nFor the strongest sources alternative names are given in brackets.\n\n\\noindent\n{\\bf \\object{RC0457+0452}}\\\\\nThis is one of the brightest objects in this sample.\nThe new VLA map shows that the radio source has FRI \nstructure (an unpublished radio map).\nThis agrees with a high optical to\nradio luminosity ratio (Parijskij et al. \\cite{pari:goss}).\nThere is a near companion $\\sim$2.5$\\arcsec$ northwest from the nucleus.\nIn addition there are four faint companions in the southeast (5$\\arcsec$).\nThe position angle of the galaxy is not uniform,\nwhich is possibly due the close companions.\nThe inner isophotes are roughly perpendicular to the radio axis\nand the outermost fuzz is roughly aligned with the radio axis. \nThe same trend can be seen in both R and V-band images.\nThis object is possibly situated in a cluster of galaxies and\nthere are a few companions with similar brightness. The \ncompanion 20$\\arcsec$ to the east has a double nucleus\n($m_{R}=19.26, m_{V-R}=0.69$), hence it is \napparently a merger, and the companion 22$\\arcsec$ to the \nsouth has distorted morphology ($m_{R}=19.94, m_{V-R}=0.93$) .\n\n\n\\noindent\n{\\bf \\object{RC0506+0508}}\\\\\nThis is a faint point-like source, even under excellent seeing\nconditions. \nIt might be a high redshift ($z>1, M_{R}< -23$) quasar\nbecause\nin the quasar catalogue by Veron-Cetty \\& Veron (\\cite{ver:ver})\nthere are only a few quasars fainter than \n21 magnitudes with $z<1$.\nThe companions 14$\\arcsec$ to the northwest have a multicomponent\nstructure with an extended diffuse emission. \n\n\\noindent\n{\\bf \\object{RC0837+0446}}\\\\\nThe galaxy is marginally resolved and lies\nin or behind a galaxy cluster.\nIn the lower left hand corner of the grey scale image is a\ntrail of a solar system object.\n\n\\noindent\n{\\bf \\object{RC0845+0444}}\\\\\nThe optical counterpart of this radio source\ncoincides with the western radio component.\nThis object is optically extended \nand there is a \nfaint extension towards the southwest. \n\n\\noindent\n{\\bf \\object{RC1031+0443}}\\\\\n(\\object{4C +05.43} \\& \\object{PKSB1028+049})\nThe galaxy is near the centre of gravity of the radio source. \nThe object is extended and has a multicomponent structure. \nThe strongest optical emission is aligned with the radio source,\nbut the position angle of the outermost isophotes is\nalmost perpendicular to the radio axis.\n\n\\noindent\n{\\bf \\object{RC1152+0449}}\\\\\nThis galaxy \nhas two or possibly three components.\nThe outer isophotes of the galaxy are box-like,\nbut the separate components are roughly aligned with the\nradio axis. \nThe faint companion 4$\\arcsec$ to the west is near to the \nwestern radio lobe. This blue companion may be related \nto the radio source ($m_{R}=23.0, m_{V-R}\\sim$0.3).\n\nThis is the only source where the astrometry of the present work\nis not consistent with the result of K95b, but coincides with \ncurrent identification in Parijskij et al. (\\cite{pari:goss}).\n\n\n\\noindent\n{\\bf \\object{RC1155+0444}}\\\\\nThis is the brightest galaxy in our sample.\nThe galaxy is elliptical and it is clearly aligned \nwith the radio source.\nThe two neighbouring galaxies are possibly interacting\nforeground galaxies.\n\n\n\\noindent\n{\\bf \\object{RC1235+0453}}\\\\\nThis faint galaxy is spatially extended in our R-band images. \nThe fuzz $\\sim 1 \\arcsec$ northwest from the nucleus \nhas a clumpy structure.\nOur deep images show faint low surface brightness \ncompanions near the object $\\sim 10 \\arcsec$ to the east, north and west.\nIn the V-band images only the \ncore of the galaxy is detected.\n\n \n\\noindent\n{\\bf \\object{RC1347+0441}}\\\\\nThis galaxy is the faintest of our sample. \nThe object is elongated and \nthe size of the optical object is roughly the same as the\nradio source. \n\n\\noindent\n{\\bf \\object{RC1510+0438}}\\\\\nThis is the most spectacular object in our sample,\nlying in a group of galaxies and having apparently\nwavelength dependent properties.\nIn the R- and I-band image the object is almost round\n(Table 4) but in the V-band the object is weakly elongated with\nthe same position angle as the radio source.\nThe redshift estimation from BVRI colours suggest\n$z\\sim0.6$ (Pariskij et al. \\cite{pari:sobo}).\nIf this is the case, the strong emission lines $[$\\ion{O}{ii}$]$\n3727 and $[$\\ion{O}{iii}$]$ 5007 would be shifted into R and I band,\nrespectively, possibly causing the wavelngth dependence of morphology.\nHowever, new colours from the 6 m-telescope\ndo not agree with strong line contribution. \nAnother consequence of such redshift would be that\nthis would be one of the faintest\n(intrinsically) radio galaxies in the Hubble diagram (Fig. ~\\ref{fig2}).\nThere are three relatively bright galaxies and one faint companion galaxy\nwithin 5$\\arcsec$ of the object. \nAll the companions are bluer than the object ($m_{R-I}=1.52$).\nThe objects towards the east, \nC1 ($m_{R}=22.54, m_{V-R}=0.51, m_{R-I}=0.27$), \nnorth, C2 ($m_{R}=22.34, m_{V-R}=0.81, m_{R-I}=0.87$) \nand west,C3 ($m_{R}=23.24, m_{R-I}=0.83$) could be foreground galaxies. \nIn addition there is a faint companion 1$\\farcs$5 north from the \nobject.\n\n\\noindent\n{\\bf \\object{RC1703+0502}}\\\\\n(\\object{PKS B1701+051})\nThis is one of the strongest and most compact radio sources\nof the present sample.\nThe optical and radio axes are clearly aligned and the sizes are\nalmost the same.\nThere are a few faint galaxies in the field, but no\nclose companions. \nThis object is possibly located behind \na foreground galaxy cluster although some of the\nfield objects might be faint galactic stars.\n\n\n\n\n\\noindent\n{\\bf \\object{RC1720+0455}}\\\\\nThis object is compact in radio and optical.\nThis suggests that it is a QSO and\nthe same conclusion may be drawn from radio-optical luminosity\nconsideration (Parijskij et al. 1996a).\nThe extension towards the southeast, seen in K95b, was an artifact. \nThere are a few companions close to the object.\nThe southern companion either has a double nucleus or a dust lane. \nThe wide field image shows several faint companions,\nhence this galaxy is either in a cluster of galaxies or behind one.\n\n\n\\noindent\n{\\bf \\object{RC1740+0502}}\\\\\nThis source was identified by K95b \nand it is marginally resolved in the R-band image. \nIn the V-band image the object has an extension to the\nsouth west in contrast to almost round morphology in R-band\n(see Table 4).\n\n\n\\noindent\n{\\bf \\object{RC2013+0508}}\\\\\nThis unresolved object could possibly be a quasar.\nBecause of its low galactic latitude ($b\\sim$-15),\nthe field is crowded with stars. \n\n\n\\noindent\n{\\bf \\object{RC2036+0451}}\\\\\n(\\object{MRC 2034+046}).\nThis is the second of the two triple radio sources in this sample. \nA point source coincides with the central component fairly well. \nThis is the only object with known redshift ($z$=2.95\nPariskij et al. \\cite{pari:goss}).\nThis indicates that the\nabsolute magnitude of this quasar is M$_{R}\\approx$-29.\n\n\n\\noindent\n{\\bf \\object{RC2144+0513}}\\\\\nThis object is unresolved. \nThe companion 3$\\arcsec$ to the southwest is most likely a\ngalactic star ($m_{R}=20.90, m_{V-R}=1.1$). \nThe profile of this object matches \nperfectly with the average stellar profile from the same field.\n\n\n\n\\subsection{Uncertain identifications}\n\n\\noindent\n{\\bf \\object{RC0459+0456}}\\\\\n(\\object{MRC 0456+048})\nThis source has two candidates for optical identification in\nK95b. Id1 ($m_{R}=22.08$) is an elongated galaxy with roughly the same \nposition angle as the radio source. This object has a \ncompanion to the west (Id2). This is a marginally resolved \npoint-like source, hence it might be\na quasar with a host galaxy ($m_{R}=21.12$).\nFWHM of the id2 0$\\farcs$66 compares with FWHM of a field star\n0$\\farcs$62.\n\n\n\\noindent\n{\\bf \\object{RC1219+0446}}\\\\\nThis is the largest radio source in our sample\n(118$\\arcsec$). The nature of the source remains unclear\nand it is possible that there are indeed two independent radio sources.\nIf this is one source, then a possible identification would be a\nfaint, rather round galaxy (Id1) 5$\\arcsec$ from the centre of \nradio source ($m_{R}=21.9$).\nOn the other hand if the southern radio lobe is an independent\nobject, the identification could be an unresolved \nobject (Id2) 2$\\arcsec$ southeast from the radio source ($m_{R}=17.88$). \n\n\n\\noindent\n{\\bf \\object{RC1735+0454}}\\\\\nThe possible optical counterpart is $\\sim 3 \\arcsec$ to the east of \nthe radio source. The galaxy has several components and it is elongated\nin a north south direction. \nThe identification should be confirmed by future observations.\n\n\n\\subsection{Faint objects}\n\n\n\\noindent\n{\\bf \\object{RC0743+0455}}\\\\\nThis object is very faint and hardly visible in the 30 min. exposure. \n\n\\noindent\n{\\bf \\object{RC1333+0451}}\\\\\nThe radio source is compact. There is a faint \nextended emission exactly at the position of the radio source. \n\n\n\\noindent\n{\\bf \\object{RC1609+0456}}\\\\\nNew 6-m telescope measurements find an object with $m_{R}\\sim25.5$ \nexactly at the position of the radio source.\nOur 600 second exposures are not deep enough to detect \nthis object.\nThe bright nearby object is unresolved and BVRI photometry by K95b\nsuggests it to be a star.\n\n\n\\section{Concluding remarks}\n\nExcepting the quasar RC2036+0451,\nthe present galaxies do not have measured redshifts as yet. Hence,\nit is interesting to ask whether optical\nmorphology\nprovides information on the redshift\ndistribution of the sample.\nOther properties, for example\nthe Hubble diagram of the RC\/USS sample, suggest (Sec. 1.3) \nthat it contains galaxies with $z\\ga$0.7.\n\n\nOf the 22 observed objects, three are extremely faint and \nthree others are either faint or have several possible \nidentifications. Of the remaining 16 objects, 5 are unresolved.\nThis is roughly the same fraction of point sources \nwhich R\\\"{o}ttgering et al. (\\cite{rott:miley})\nfound, but slightly more than \nLu et al. (\\cite{lu:hof}) found from their \n``distant'' sample ($S_{1.4GHz}>35$mJy), which does not have \nradio spectral index selection criteria.\nTypically RC\/USS objects have a multicomponent\nstructure with extended emission.\nThis compares with\n$HST$ images which have shown that about 30\\% of intermediate\nredshift 3CR galaxies have distorted morphology\n(De Koff et al. \\cite{dekoff}).\nMore distant ($z\\sim1$) 3CR galaxies have typically \nmulticomponent structure with diffuse extended emission \n(Best et al. \\cite{best:long}).\nThe ellipticity ($e$) of the current sample (11 objects) ranges from 0.05\nto 0.57, with a mean of 0.25.\nTaking into account the measurement errors\nthese values agree with studies\nby Rigler et al. (\\cite{rigler}) for 3C galaxies (e=0.19)\n and by R\\\"{o}ttgering et al. (\\cite{rott:miley}) USS sample (e=0.33).\n\n\n\nVisual inspection of the images in Fig. ~\\ref{fig4} suggests \nthat about half of the objects have a companion \nwith comparable brightness within 10$\\arcsec$.\nWe examined the excess of companion galaxies \nalong the radio axis suggested by R\\\"{o}ttgering et al.\n(\\cite{rott:west}). Our sample has 7 resolved objects with \n3$\\arcsec