Large deviations and Chernoff bound for certain correlated states on a spin chain

TL;DR

This paper extends large deviation results for correlated states on spin chains, showing that certain factorization properties ensure upper bounds and applying spectral theory to establish full large deviation principles.

Contribution

It demonstrates that a factorization property of reference states suffices for large deviation bounds and applies spectral theory to prove full large deviation principles for finitely correlated states.

Findings

Large deviation upper bounds hold under specific factorization conditions.

Chernoff bounds are established for correlated states with factorization.

Full large deviation principles are proven for ergodic finitely correlated states.

Abstract

In this paper we extend the results of Lenci and Rey-Bellet on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state, in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state the spectral theory of positive maps is applied to prove the full large deviation principle.