Large deviations upper bounds for the laws of matrix-valued processes and non-commutative entropies. (English) Zbl 1022.60026

Cabanal Duvillard, T.; Guionnet, A.

Ann. Probab. 29, No. 3, 1205-1261 (2001).

In this very interesting article the authors study the large deviations properties of non-commutative laws of large random matrices and related non-commutative entropies. The results obtained in this paper can be explained, in an easy way, considering one of the studied examples: the Hermitian Brownian motion matrix. This is a process \((H_N(t))_{t\in R^{+}}\) taking values in the space \({\mathcal H}_N\) of Hermitian matrices such that the entries of \((H_N(t))\) are complex Brownian motions verifying \(E[H_N^{i,j}(t)H_N^{k,l}(s)]=\frac{t\wedge s}N\delta_i^l\delta_k^j\). In a given fixed time \(t\), \(H_N(t)\) is a Wigner’s matrix of the Gaussian unitary ensemble (GUE). Let \(\hat{\mu}_t\equiv\frac{1}N\sum_{i=1}^{N}\delta_{\lambda_i^{(N)}}(t)\) be the spectral empirical process, where \((\lambda_i^{(N)}(t))_{1\leq i\leq N}\) are the real eigenvalues of \((H_N(t))\). In Theorem 1.1 it is established that this measure satisfies a large deviation upper bound in the scale \(N^2\), with rate function \({\mathbf S}(\nu)\) defined, if \(\nu_0=\delta_0\), equal to \[ S^{0,1}(\nu)=\sup_{f\in{\mathcal C}_b^{2,1}(R,[0,1])}\left(S^{0,1}(\nu,f)-\frac{1}{2}\int_0^1(\partial_xf(x,u))^2d\nu_u(x)du\right) \] where \(\nu\in {\mathcal C}([0,1], {\mathcal P}(R))\) is a continuous \({\mathcal P}(R)\)-valued process and \({\mathcal C}_b^{2,1}(R,[0,1])\) is the set of continuously differentiable functions on \(R\) with values in the set of \(2\)-times bounded continuously differentiable functions on \(R\). Moreover \[ S^{0,1}(\nu,f)=\int f(x,t)d\nu_t(x)-\int f(x,s)d\nu_s(x)-\int_s^t\int\partial_uf(x,u)d\nu_u(x)du \]
\[ -\frac{1}{2}\int_s^t\int\int\frac{\partial_xf(x,u) -\partial_xf(y,u)}{x-y}d\nu_u(x)d\nu_u(y)du. \] If \(\nu_0\neq \delta_0\), the authors define \({\mathbf S}(\nu)=\infty\). It is shown that \[ \limsup_{N\to\infty}\frac 1{N^2}\ln P(\hat{\mu}^{(N)})\leq-\inf_{\nu\in F}{\mathbf S}(\nu) \] for every closed subset \( F\in {\mathcal C}([0,1],{\mathcal P}(R))\). In the same theorem a large deviation lower bound is also proved but the authors give evidence that it could be not so sharp. By using these large deviation inequalities and the contraction principle they obtain in Corollary 1.1 a large deviation principle of the GUE (i.e. \(t=1\)). Then in Corollary 1.2 they also get the exponentially fast convergence of \(\mu^{(N)}\) toward the semi-circular process. The method of proof allows also considering several independent Hermitian Brownian motions. Thus the authors establish a large deviation upper bound for the process of the law (in a non-commutative sense) of the time marginal of these processes. By using the contraction principle they get a large deviation upper bound of several independent Wigner’s matrices. Since the obtained rate function defines a natural entropy for the non-commutative law of several variables this quantity is then compared with the one defined by D. Voiculescu.
The key ingredient in the proofs of this paper is that \(\mu^{(N)}\) satisfies a stochastic differential equation with generator and martingale bracket which only depends on \(\mu^{(N)}\). Itô calculus and Markov property lead naturally to use the techniques that have been developed before to obtain large deviations bounds for the empirical processes in the field of hydrodynamics.

Reviewer: J.R.León (Caracas)

MSC:

60F10	Large deviations
60J65	Brownian motion
46L53	Noncommutative probability and statistics

Keywords:

large deviations; random matrices; non-commutative measure; integration

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