Deformations of biorthogonal ensembles and universality

TL;DR

This paper studies how deformations of biorthogonal ensembles affect their universal limiting behavior, providing conditions under which the deformed ensembles converge to deformed universal processes, with applications to Painlevé-type kernels.

Contribution

It introduces a novel approach based on probability generating functionals to analyze deformations, relaxing regularity assumptions and connecting to Painlevé kernels.

Findings

Deformed biorthogonal ensembles converge to deformed universal processes.

New probabilistic interpretations of Painlevé-type kernels.

Method relies on marking and conditioning, not correlation kernels.

Abstract

We consider a large class of deformations of continuous and discrete biorthogonal ensembles and investigate their behavior in the limit of a large number of particles. We provide sufficient conditions to ensure that if a biorthogonal ensemble converges to a (universal) limiting process, then the deformed biorthogonal ensemble converges to a deformed version of the same limiting process. To construct the deformed version of the limiting process, we rely on a procedure of marking and conditioning. Our approach is based on an analysis of the probability generating functionals of the ensembles and is conceptually different from the traditional approach via correlation kernels. Thanks to this method, our sufficient conditions are rather mild and do not rely on much regularity of the original ensemble and of the deformation. As a consequence of our results, we obtain probabilistic interpretations of several Painlev\'e-type kernels that have been constructed in the literature, as deformations of classical sine and Airy point processes.