Large deviations for the Sine_beta and Sch_tau processes

TL;DR

This paper establishes large deviation principles for the Sine_beta and Sch_tau point processes, which are connected to random matrix spectra, providing new insights into their rare density fluctuations across all parameter ranges.

Contribution

It introduces novel large deviation results for these processes, extending understanding even in classical cases and confirming physical predictions with rigorous proofs.

Findings

Large deviation principles for average densities established

Path level large deviations for associated diffusions proved

Results consistent with and extend existing gap probability theories

Abstract

We study two one-parameter families of point processes connected to random matrices: the Sine_beta and Sch_tau processes. The first one is the bulk point process limit for the Gaussian beta-ensemble. For beta=1, 2 and 4 it gives the limit of the GOE, GUE and GSE models of random matrix theory. In particular, for beta=2 it is a determinantal point process conjectured to have similar behavior to the critical zeros of the Riemann zeta-function. The second process can be obtained as the bulk scaling limit of the spectrum of certain discrete one-dimensional random Schr\"odinger operators. Both processes have asymptotically constant average density, in our normalization one expects close to lambda/(2pi) points in a large interval of length lambda. Our main results are large deviation principles for the average densities of the processes, essentially we compute the asymptotic probability of seeing an unusual average density in a large interval. Our approach is based on the representation of the counting functions of these processes using stochastic differential equations. We also prove path level large deviation principles for the arising diffusions. Our techniques work for the full range of parameter values. The results are novel even in the classical beta=1, 2 and 4 cases for the Sine_beta process. They are consistent with the existing rigorous results on large gap probabilities and confirm the physical predictions made using log-gas arguments.