Large Deviation For Outlying Coordinates in Beta Ensembles

TL;DR

This paper establishes a large deviation principle for the probability measures associated with beta ensembles in the complex plane, extending previous results to broader settings with Bernstein-Markov conditions.

Contribution

It extends large deviation results for beta ensembles to subsets of the complex plane under Bernstein-Markov conditions, generalizing prior work.

Findings

Proves a large deviation principle for beta ensembles in the complex plane.

Extends Borot-Guionnet's results to new settings with broader restrictions.

Incorporates Bernstein-Markov conditions into the analysis.

Abstract

For Y a subset of the complex plane,a beta ensemble is a sequence of probability measures on Y^n for n=1,2,3...depending on a real-valued continuous function Q and a real positive parameter beta.We consider the associated sequence of probability measures on Y where the probability of a subset W is given by the probability that at least one coordinate of Y^n belongs to W. With appropriate restrictions on Y,Q we prove a large deviation principle for this sequence of measures. This extends a result of Borot-Guionnet to subsets of the complex plane and to beta ensembles defined with measures using a Bernstein-Markov condition.