Brian C. Hall

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math.PR2026

Repeated differentiation of deterministic polynomials with asymptotically radial root distributions

Brian C. Hall, Daniel Perales

Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form $P (z) = p (z^{m})$ , where $p$ is a deterministic p…

math.PR2025

Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators

Brian C. Hall, Ching-Wei Ho, Jonas Jalowy +1

We start with a random polynomial $P^{N} (z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $⌈ N t ⌉$ applications o…

math.PR2025

Support of the Brown measure of a family of free multiplicative Brownian motions with non-negative initial condition

Brian C. Hall, Sorawit Eaknipitsari

We consider a family $b_{s, I ¨ }$ of free multiplicative Brownian motions labeled by a real variance parameter $s$ and a complex covariance parameter $I ¨ $ . We then consider the eleme…

math.PR2025

Matrix Random Walks and the Lima Bean Law

Bruce K. Driver, Brian C. Hall, Ching Wei Ho +4

A matrix random walk is a stochastic process of the form $B_{k} = (I + A_{1}) \dots (I + A_{k})$ where $A_{j}$ are independent ``step'' matrices in $M_{N} (C)$ . With the right en…