Patterned matrices with random walk entries

TL;DR

This paper studies the convergence of patterned random matrices with entries as continuous-time random walks (CTRWs) within a non-commutative probability framework, revealing limits related to free Brownian motion and other complex processes.

Contribution

It introduces new results on the asymptotic behavior of patterned matrices with CTRW entries, including free and classical limits, and extends to time-changed versions.

Findings

Limits are free Brownian motion and time-changed versions for Wigner matrices.

Eigenvalue-based empirical processes converge to classical Brownian motions.

Algebraic limits for iid and elliptic matrices match circular and elliptic variables.

Abstract

It is well known that the weak limit of a suitably scaled continuous-time random walk (CTRW) is the Brownian motion. We investigate the convergence of certain patterned random matrices whose entries are independent CTRWs and their time-changed versions, in a non-commutative probability framework. For the Wigner link function, the limits are free Brownian motion and its time-changed version driven by an inverse stable subordinator. For the symmetric circulant and the circulant with CTRW entries, we use their explicit eigenvalue expressions to define some empirical processes that converge weakly to a Brownian motion and a complex Brownian motion, respectively. For matrices with iid entries, and for elliptic matrices, the algebraic limits are equal in $*$-distribution to processes whose marginals are circular and elliptic variables, respectively. A random time-changed variant of these results is also established.