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Matrix Random Walks and the Lima Bean Law

Bruce K. Driver, Brian C. Hall, Ching Wei Ho, Todd Kemp, Yuriy Nemish, Evangelos A. Nikitopoulos, Felix Parraud

TL;DR

This work establishes a universal framework for the eigenvalue behavior of matrix random walks built from bi-invariant steps. By connecting rescaled products of random matrices to Brown measures in free probability, the authors show that, as the matrix size grows, the eigenvalue distribution converges to the Brown measure of a corresponding free random walk, with a Lima Bean-shaped support when steps share a fixed Hilbert–Schmidt norm. They prove a Wong–Zakai-type convergence for the matrix walk to Brownian motion on GL$(N,\mathbb{C})$, derive strong $L^p$ and, in the free setting, almost-sure convergence results, and establish that the Brown measure of the free random walk $b_k(t)$ converges to that of the free multiplicative Brownian motion $b(t)$ as $k\to\infty$ (the Lima Bean Law). The analysis hinges on a linearization technique, freeness for $\mathscr{R}$-diagonal steps, and delicate control of small singular values via Wegner-type estimates, providing a robust universality statement for a broad class of bi-invariant step distributions.

Abstract

A matrix random walk is a stochastic process of the form $B_k = (I+A_1)\cdots(I+A_k)$ where $A_j$ are independent ``step'' matrices in $\mathrm{M}_N(\mathbb{C})$. With the right entry-covariance, a rescaled matrix random walk converges to Brownian motion $B(t)$ on a matrix Lie group. In this paper, we study the eigenvalues of such rescaled matrix random walks, as $N\to\infty$ and $k\to\infty$. The standard Brownian motion $W(t)$ on $\mathrm{M}_N(\mathbb{C})$ has independent Gaussian entries at each $t$. It is bi-invariant: mutiplying on the left or right by a unitary does not change the distribution. We prove that the empirical eigenvalue distribution of any matrix random walk $B_k$ with bi-invariant steps $A_j$ and initial distribution converges (for fixed $k$ as $N\to\infty$) to a probability measure on $\mathbb{C}$: the Brown measure of the free probability $\ast$-distribution limit $b_k$ of the random walk. If the steps $A_j$ are identically distributed with normalized Hilbert--Schmidt norm $\|A_j\|_2 = t$, the limit law of eigenvalues is supported on a compact ``lima bean'' shaped region. We explicitly compute the limit measure and region, and characterize their phase transitions as $t$ evolves. We prove that the Brown measure of $b_k$ converges as $k\to\infty$, to the Brown measure of the free multiplicative Brownian motion, assuming only that the steps are bi-invariant and normalized in Hilbert--Schmidt norm. Thus the Brownian motion is the universal limit of rescaled matrix random walks, under very general assumptions on the distribution of steps.

Matrix Random Walks and the Lima Bean Law

TL;DR

This work establishes a universal framework for the eigenvalue behavior of matrix random walks built from bi-invariant steps. By connecting rescaled products of random matrices to Brown measures in free probability, the authors show that, as the matrix size grows, the eigenvalue distribution converges to the Brown measure of a corresponding free random walk, with a Lima Bean-shaped support when steps share a fixed Hilbert–Schmidt norm. They prove a Wong–Zakai-type convergence for the matrix walk to Brownian motion on GL, derive strong and, in the free setting, almost-sure convergence results, and establish that the Brown measure of the free random walk converges to that of the free multiplicative Brownian motion as (the Lima Bean Law). The analysis hinges on a linearization technique, freeness for -diagonal steps, and delicate control of small singular values via Wegner-type estimates, providing a robust universality statement for a broad class of bi-invariant step distributions.

Abstract

A matrix random walk is a stochastic process of the form where are independent ``step'' matrices in . With the right entry-covariance, a rescaled matrix random walk converges to Brownian motion on a matrix Lie group. In this paper, we study the eigenvalues of such rescaled matrix random walks, as and . The standard Brownian motion on has independent Gaussian entries at each . It is bi-invariant: mutiplying on the left or right by a unitary does not change the distribution. We prove that the empirical eigenvalue distribution of any matrix random walk with bi-invariant steps and initial distribution converges (for fixed as ) to a probability measure on : the Brown measure of the free probability -distribution limit of the random walk. If the steps are identically distributed with normalized Hilbert--Schmidt norm , the limit law of eigenvalues is supported on a compact ``lima bean'' shaped region. We explicitly compute the limit measure and region, and characterize their phase transitions as evolves. We prove that the Brown measure of converges as , to the Brown measure of the free multiplicative Brownian motion, assuming only that the steps are bi-invariant and normalized in Hilbert--Schmidt norm. Thus the Brownian motion is the universal limit of rescaled matrix random walks, under very general assumptions on the distribution of steps.

Paper Structure

This paper contains 44 sections, 108 theorems, 441 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Random walk approximations to Brownian motion
  5. Convergence of Eigenvalues
  6. Linearization and Freeness
  7. Brown Measures of Free Random Walks
  8. The Lima Bean Law
  9. Background
  10. Noncommutative Probability
  11. $\ast$- and $W^\ast$-Probability Spaces
  12. Noncommutative $L^p$ Spaces
  13. $\ast$-Distributions
  14. Free Probability and Asymptotic Random Matrix Theory
  15. Free Independence and Free Cumulants
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $(X_n)_{n=1}^\infty$ be i.i.d. $L^2$ random vectors in $\mathbb{R}^d$ with mean $\mathbb{E}[X_n]=\mathbf{0}$ and covariance $\mathbb{E}[X_nX_n^\top]=I$. Let $(S(t))_{t\ge 0}$ be the piecewise affine process defined by $S(0)=\mathbf{0}$ and $S(n) = X_1+\cdots+X_n$ for $n\in\mathbb{N}$: Then the diffusion rescaled sequence $W_k(t) = S(kt)/\sqrt{k}$ converges pathwise in distribution (i.e. weakl

Figures (5)

  • Figure 1: The Brown measure of the free random walk $b_k(t)$ with $k=6$ and $t=1$ (left) and $t=2$ (right). Also shown are $1000$ eigenvalues of the associated $B^N_k(t)$. Here the step distribution is circular / Ginibre, corresponding directly to the Brownian motion.
  • Figure 2: Two instances of the Brown measure of $Z+\sqrt{t/k}\,A$, alongside the Brown measure of the free random walk $u_0 b_k(t)$, with $k=3$ and $t=2.2$ (top) and $t=2.6$ (right). Here the initial condition $u_0$ in $Z$ has atoms at $e^{\pm i\pi/3}$ with equal mass $\frac{1}{2}$, and the step distribution is circular / Ginibre. Each image also shows $1000$ eigenvalues of the assocciated matrix model.
  • Figure 3: The Brown measure linearized model $Z+\sqrt{t/k}A$ at $t$ evolves through the phase transitions of its topology; here $k=3$, the initial condition is $u_0=1$, and the step distribution is Haar unitary.
  • Figure 4: The evolution of the region $\Omega_k(t)$ for the linearized model $Z+\sqrt{t/k}A$ with $k=3$, initial condition $u_0=1$, circular / Ginibre steps.
  • Figure 5: The evolution of the region $\Sigma_k(t)$ for the same parameter choices as in Figure \ref{['fig.6.3']}.

Theorems & Definitions (229)

  • Theorem 1.1: Donsker, Donsker
  • Definition 1.2
  • Theorem 1.4: Wong--Zakai, WongZakai1WongZakai2
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7: Berger, Berger
  • Remark 1.8
  • Lemma 1.9
  • Remark 1.10
  • Theorem 1.11
  • ...and 219 more