Large Deviations and Sum Rules for Spectral Theory - A Pedagogical Approach

TL;DR

This paper provides a pedagogical overview of using large deviation principles to derive sum rules in spectral theory, specifically explaining how to recover classical theorems like Szegő and Killip-Simon.

Contribution

It introduces a clear, accessible approach to applying large deviations in spectral theory, bridging the gap for researchers unfamiliar with large deviation techniques.

Findings

Demonstrates how large deviations can be used to derive sum rules

Provides a step-by-step pedagogical explanation

Connects large deviation principles with classical spectral theorems

Abstract

This is a pedagogical exposition of the large deviation approach to sum rules pioneered by Gamboa, Nagel and Rouault. We'll explain how to use their ideas to recover the Szeg}o and Killip{ Simon Theorems. The primary audience is spectral theorists and people working on orthogonal polynomials who have limited familiarity with the theory of large deviations.