Large deviations for the $${\hbox {Sine}}_\beta $$ and $$\hbox {Sch}_\tau $$ processes

Abstract

The \({\hbox {Sine}}_\beta \) process is the bulk point process limit of the Gaussian \(\beta \)-ensemble. For \(\beta = 1, 2\) and 4 this process gives the limit of the GOE, GUE and GSE random matrix models. The \(\hbox {Sch}_\tau \) process is obtained similarly as the bulk scaling limit of the spectrum of certain discrete one-dimensional random Schrodinger operators. Both point processes have asymptotically constant average density, in our chosen normalization one expects close to \(\tfrac{1}{2\pi } \lambda \) points in a large interval of length \(\lambda \). We prove large deviation principles for the average densities of the processes, identifying the rate function in both cases. Our approach is based on the representation of the counting functions using coupled systems of stochastic differential equations. 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 \({\hbox {Sine}}_\beta \) process. They are consistent with the existing rigorous results on large gap probabilities and confirm the physical predictions made using log-gas arguments.