Free positive multiplicative Brownian motion and the free additive convolution of semicircle and uniform distribution

TL;DR

The paper proves that the free spectral distribution $\nu_t$ of free positive multiplicative Brownian motion equals the exponential image of a free additive convolution of a semicircle and a uniform distribution, i.e., $\nu_t=\exp(\mu_{\operatorname{sc},2\sqrt{t}}\boxplus \operatorname{Unif}_{[-t/2,t/2]})$, by equating moments of $\nu_t$ with those of the log-transifted additive convolution. It provides an explicit moment formula $m_n(t)=n!\sum_{j=\lceil n/2\rceil}^n \frac{t^j}{j!(1+j)!} s(1+j,n+1-j)$ involving signed Stirling numbers $s(\cdot,\cdot)$, and derives integral representations for $\nu_t$ in terms of ${}_1F_1$ and Laguerre polynomials. The approach yields generalizations to free additive convolutions with intervals $\operatorname{Unif}_{[b,c]}$ and offers a new combinatorial identity (Lemma) as a key technical ingredient, contributing both a new proof technique and explicit combinatorial formulas for this class of measures.

Abstract

The free positive multiplicative Brownian motion $(h_t)_{t\geq0}$ is the large $N$ limit in non-commutative distribution of matrix geometric Brownian motion. It can be constructed by setting $h_t:=g_{t/2}g_{t/2}^*$, where $(g_t)_{t\geq0}$ is a free multiplicative Brownian motion, which is the large $N$ limit in non-commutative distribution of the Brownian motion in $\operatorname{Gl}(N,\mathbb{C})$. One key property of $(h_t)_{t\geq0}$ is the fact that the corresponding spectral distributions $(ν_t)_{t\geq0}\subset M^1((0,\infty))$ form a semigroup w.r.t. free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that $ν_t$ can be expressed by the image measure of a free additive convolution of the semicircle and the uniform distribution on an interval under the exponential map. In this paper, we provide a new proof of this result by calculating the moments of the free additive convolution of semicircle and uniform distributions on intervals. As a by-product, we also obtain new integral formulas for $ν_t$ which generalize the corresponding known moment formulas involving Laguerre polynomials.

Theorems & Definitions (10)

• Theorem 1.1: Auer2025
• Theorem 1.2
• Corollary 1.3
• proof : Proof of Theorem \ref{['thm_main']} & Corollary \ref{['cor_main']}
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof : Proof of Theorem \ref{['thm_moment_formula']}
• proof : Proof of Lemma \ref{['lem_stirling_identiy']}