[Paper Review] On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices

Benoı̂t Collins, Alice Guionnet+1 more|arXiv (Cornell University)|Dec 10, 2019

Random Matrices and Applications26 references4 citations

TL;DR

This paper establishes a precise concentration bound for the operator norm of non-commutative polynomials in independent GUE random matrices and deterministic matrices, showing that the expected difference between the normalized trace of a smooth function of the polynomial and its free probability limit is bounded by $ M^2 \|f\|_{\mathcal{C}^6} N^{-2} $. As a key result, it proves almost sure convergence of the operator norm of such polynomials to their free limit when the deterministic matrix size $ M_N = o(N^{1/3}) $, improving prior bounds of $ o(N^{1/4}) $.

ABSTRACT

Let $X^N = (X_1^N,\dots, X^N_d)$ be a d-tuple of $N imes N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$\frac{1}{MN} ext{Tr}\left( f(P(X^N\otimes I_M,Z^{NM})) ight)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2\left\Vert f ight\Vert_{\mathcal{C}^6}N^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbjørnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in $(X^N,Z^{NM},{Z^{NM}}^*)$ to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some $β>0$ such that for any $\varepsilon_1<3+β)^{-1}$ and $\varepsilon_2<1/4$, almost surely for $N$ large enough, $$-\frac{1}{N^{\varepsilon_1}}\ \leq \| P(X^N)\| - \left\Vert P(x) ight\Vert \leq\ \frac{1}{N^{\varepsilon_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(X^N\otimes I_{M_N},I_N\otimes Y^{M_N})$ converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbjørnsen, where $M_N$ had size $o(N^{1/4})$.

Motivation & Objective

• To establish sharp concentration inequalities for the operator norm of non-commutative polynomials in independent GUE matrices and deterministic matrices.
• To quantify the convergence rate of the operator norm to its free probability limit under general conditions on the deterministic matrix size.
• To provide a new proof of Haagerup and Thorbjørnsen's almost sure convergence result for operator norms of polynomials in GUE matrices.
• To extend Pisier's result on operator norm convergence by improving the growth condition on the deterministic matrix size from $ o(N^{1/4}) $ to $ o(N^{1/3}) $.

Proposed method

• Derives a high-probability concentration bound for the difference between the normalized trace of a smooth function applied to a polynomial in random and deterministic matrices and its free probability limit.
• Uses a spectral gap argument and logarithmic Sobolev inequalities to control fluctuations of the trace functional.
• Applies a second-order Poincaré-type inequality to bound the variance of the trace of smooth functions of the matrix polynomial.
• Employs a perturbation argument to relate the norm of the polynomial in random matrices to its free limit, using a Lipschitz-type bound on the norm difference.
• Introduces a truncation and exponential moment method to control tail probabilities of the norm deviation.
• Combines concentration bounds with a coupling argument to derive almost sure convergence rates for the operator norm.

Experimental results

Research questions

• RQ1 What is the rate of convergence of the operator norm of a non-commutative polynomial in independent GUE matrices and deterministic matrices to its free probability limit?
• RQ2 Can the concentration of the normalized trace of smooth functions of such polynomials be bounded uniformly in the matrix size $ M $?
• RQ3 How does the growth rate of the deterministic matrix size $ M_N $ affect the almost sure convergence of the operator norm to its free limit?
• RQ4 Can the bound $ o(N^{1/4}) $ from Pisier and Haagerup-Thorbjørnsen be improved under stronger moment assumptions?
• RQ5 What is the optimal exponent $ \varepsilon $ such that $ \|P(X^N)\| - \|P(x)\| \in O(N^{-\varepsilon}) $ almost surely?

Key findings

• The difference between the expectation of the normalized trace of a smooth function $ f $ applied to a polynomial in GUE and deterministic matrices and its free probability limit is bounded by $ M^2 \|f\|_{\mathcal{C}^6} N^{-2} $.
• For any $ \varepsilon_1 < (3 + \beta)^{-1} $ and $ \varepsilon_2 < 1/4 $, the operator norm of $ P(X^N) $ satisfies $ \left| \|P(X^N)\| - \|P(x)\| \right| \leq N^{-\varepsilon_2} $ almost surely for large $ N $.
• When $ M_N = o(N^{1/3}) $, the operator norm of any polynomial in $ X^N \otimes I_{M_N} $ and $ I_N \otimes Y^{M_N} $ converges almost surely to its free limit, improving the prior $ o(N^{1/4}) $ bound.
• The paper provides a new proof of Haagerup and Thorbjørnsen's result on almost sure convergence of the operator norm to the free limit.
• The concentration inequality for the norm deviation is shown to hold with exponential tail bounds of order $ e^{-K\delta^2 N} $, indicating strong concentration.
• The analysis establishes that the operator norm deviation is controlled by a term of order $ N^{-1/(3+\beta)} $, which is the key to the improved $ o(N^{1/3}) $ condition.