On global fluctuations for non-colliding processes

TL;DR

This paper investigates the global fluctuations in determinantal point processes derived from non-colliding systems, establishing connections between recurrence matrix asymptotics and fluctuation behavior, including Gaussian Free Field limits.

Contribution

It introduces a framework linking recurrence matrix asymptotics to global fluctuation behavior in non-colliding processes, extending CLTs and GFF results to new models.

Findings

Recurrence matrices determine fluctuation behavior

Shared asymptotics imply shared fluctuation limits

Gaussian Free Field fluctuations are proven for certain models

Abstract

We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models which share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove Central Limit Theorems for the linear statistics. We then show that these results prove Gaussian Free Field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.