Matrix Random Walks and the Lima Bean Law
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arXiv
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| Autori principali: | , , , , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866911206398230528 |
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| author | Driver, Bruce K. Hall, Brian C. Ho, Ching Wei Kemp, Todd Nemish, Yuriy Nikitopoulos, Evangelos A. Parraud, Felix |
| author_facet | Driver, Bruce K. Hall, Brian C. Ho, Ching Wei Kemp, Todd Nemish, Yuriy Nikitopoulos, Evangelos A. Parraud, Felix |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix Random Walks and the Lima Bean Law Driver, Bruce K. Hall, Brian C. Ho, Ching Wei Kemp, Todd Nemish, Yuriy Nikitopoulos, Evangelos A. Parraud, Felix Probability 60B20, 46L54 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. |
| title | Matrix Random Walks and the Lima Bean Law |
| topic | Probability 60B20, 46L54 |
| url | https://arxiv.org/abs/2510.10712 |