Functional Central Limit Theorems for Constrained Mittag-Leffler Ensemble in Hard Edge Scaling

TL;DR

This paper establishes functional central limit theorems for radius-dependent additive statistics in the constrained Mittag-Leffler ensemble at the hard edge, revealing their Gaussian fluctuations and hitting times.

Contribution

It introduces the first functional CLTs for radius-dependent statistics in the constrained Mittag-Leffler ensemble, extending to vector limits including hitting times.

Findings

Gaussian fluctuations of radius-dependent statistics

Extension to vector functional CLT including hitting times

Use of coupling and Skorohod's representation in proofs

Abstract

We consider the hard-edge scaling of the Mittag-Leffler ensemble confined to a fixed disk inside the droplet. Our primary emphasis is on fluctuations of rotationally-invariant additive statistics that depend on the radius and thus give rise to radius-dependent stochastic processes. For the statistics originating from bounded measurable functions, we establish a central limit theorem in the appropriate functional space. By assuming further regularity, we are able to extend the result to a vector functional central limit theorem that additionally includes the first hitting "time" of the radius-dependent statistic. The proof of the first theorem involves an approximation by exponential random variables alongside a coupling technique. The proof of the second result rests heavily on Skorohod's almost sure representation theorem and builds upon a result of Galen Shorack (1973).