[Paper Review] A note on the folklore of free independence

Arijit Chakrabarty, Sukrit Chakraborty+1 more|arXiv (Cornell University)|Feb 3, 2018

Random Matrices and Applications8 references3 citations

TL;DR

This paper establishes asymptotic free independence between a Wishart matrix of i.i.d. complex Gaussian entries and an independent random matrix under minimal spectral assumptions. Using truncation and spectral distribution techniques, it proves convergence in law of joint spectral measures under two definitions: normalized expected trace and limiting spectral distribution, with key results holding even when the limiting spectral measure of the independent matrix is not compactly supported.

ABSTRACT

It is shown that a Wishart matrix of standard complex normal random variables is asymptotically freely independent of an independent random matrix, under minimal conditions, in two different sense of asymptotic free independence.

Motivation & Objective

• To resolve ambiguities in the folklore of asymptotic free independence in random matrix theory.
• To clarify the meaning of 'asymptotically freely independent' by distinguishing two definitions: normalized expected trace and limiting spectral distribution.
• To establish conditions under which a Wishart matrix and an independent random matrix become asymptotically free, minimizing assumptions on the latter.
• To provide rigorous proofs for asymptotic free independence in two distinct senses, using truncation and convergence arguments.
• To show that the results hold with minimal assumptions on the limiting spectral distribution of the independent matrix, even when it is not compactly supported.

Proposed method

• Define a Wishart matrix $ W_N = \frac{1}{M_N} X_N^* X_N $, where $ X_N $ has i.i.d. complex standard normal entries and $ \lim_{N\to\infty} \frac{N}{M_N} = \lambda \in (0,\infty) $.
• Use the empirical spectral distribution (ESD) and expected empirical spectral distribution (EESD) to analyze spectral convergence.
• Apply a truncation procedure: define $ Y_N' = P_N T_N' P_N^* $, where $ T_N' $ is a truncated version of the triangularized matrix $ T_N = P_N^* Y_N P_N $, with $ P_N $ from a functional calculus of $ Y_N $.
• Prove convergence of EESD of $ p(W_N, Y_N') $ to the free product distribution $ \mathcal{L}(p(w,y)) $ via moment convergence and boundedness.
• Establish $ \lim_{N\to\infty} \| \text{EESD}(p(W_N, Y_N)) - \text{EESD}(p(W_N, Y_N')) \| = 0 $ using rank control and convergence of spectral measures.
• Use weak convergence of $ \mu_M = \mu|_{[-M,M]} + \mu([-M,M]^c)\delta_0 $ to $ \mu $ as $ M \to \infty $, and free convolution stability under weak limits.

Experimental results

Research questions

• RQ1 Under what conditions is a Wishart matrix asymptotically freely independent of an independent random matrix in the sense of normalized expected trace?
• RQ2 Can asymptotic free independence be established in the weaker sense of limiting spectral distribution, even when the limiting spectral measure is not compactly supported?
• RQ3 How does truncation of the eigenvalues of the independent matrix affect the joint spectral distribution of the pair $ (W_N, Y_N) $?
• RQ4 What is the minimal assumption on the limiting spectral distribution of $ Y_N $ for asymptotic free independence to hold?
• RQ5 Can the convergence of joint spectral measures be established via moment matching and bounded convergence under truncation?

Key findings

• Theorem 2.1 establishes asymptotic free independence in the sense of normalized expected trace: for any polynomial $ p $, $ \text{EESD}(p(W_N, Y_N)) \to \mathcal{L}(p(w,y)) $, provided $ \mu $ is compactly supported.
• Theorem 2.2 proves convergence in law of the EESD of $ p(W_N, Y_N) $ to the free product distribution under the same compact support assumption, using truncation and rank control.
• Theorem 2.3 extends the result to non-compactly supported $ \mu $, showing that $ \text{EESD}(Y_N + W_N) \to \mu \boxplus \nu_\lambda $ and $ \text{EESD}(Y_N W_N) \to \mu \boxtimes \nu_\lambda $ weakly in probability as $ N \to \infty $, with convergence in $ M $.
• The truncation error satisfies $ \lim_{M\to\infty} \limsup_{N\to\infty} \frac{1}{N} \mathbb{E}[\text{Rank}(Y_N - Y_N')] = 0 $, ensuring asymptotic equivalence of $ Y_N $ and $ Y_N' $ in distribution.
• The proof relies on the fact that $ \text{EESD}(Y_N') \to \mu $ weakly in probability and that moments of $ Y_N' $ converge to those of $ \mu $, enabling moment matching.
• The results are robust under minimal assumptions: the only requirement is that $ \text{ESD}(Y_N) \to \mu $ weakly in probability, with no need for uniform integrability or boundedness beyond truncation.