Large deviations for quantum spin systems

TL;DR

This paper establishes large deviation principles and a central limit theorem for empirical averages in high temperature quantum spin systems, enhancing understanding of their statistical behavior.

Contribution

It proves level one and level two large deviation principles and derives a central limit theorem for quantum spin systems at high temperature.

Findings

Large deviation principle for empirical averages of quantum spins.

Central limit theorem derived from analyticity of the generating function.

Extension to a level two large deviation principle for empirical measures.

Abstract

We consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages $\frac{1}{|\Lambda|} \sum_{i\in\Lambda} X_i$, where the $X_i$'s are copies of a self-adjoint element $X$ (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of $\frac{1}{|\Lambda|}\sum_{i\in\Lambda} \delta_{X_i}$.