Deformations of OP ensembles in a bulk critical scaling

TL;DR

This paper investigates how deformations of exponential weights in orthogonal polynomial ensembles influence local fluctuations around bulk points, revealing a new limiting kernel linked to integrable differential equations and finite temperature effects.

Contribution

It introduces a novel limiting kernel for deformed OP ensembles, connecting it to non-local differential equations and finite temperature deformations of the Sine kernel.

Findings

Identified the limiting kernel as a solution to an integrable non-local differential equation.

Connected the kernel to a conditional thinned process starting from the Sine point process.

Showed that deformation causes oscillatory behavior in recurrence coefficients even in regular cases.

Abstract

We study orthogonal polynomial ensembles whose weights are deformations of exponential weights, in the limit of a large number of particles. The deformation symbols we consider affect local fluctuations of the ensemble around a bulk point of the limiting spectrum. We identify the limiting kernel in terms of a solution to an integrable non-local differential equation. This novel kernel is the correlation kernel of a conditional thinned process starting from the Sine point process, and it is also related to a finite temperature deformation of the Sine kernel as recently studied by Claeys and Tarricone. We also unravel the effect of the deformation on the recurrence coefficients of the associated orthogonal polynomials, which display oscillatory behavior even in a one-cut regular situation for the limiting spectrum.