Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models

TL;DR

This paper derives the exact distribution of occupation times in a non-Markovian sequence, revealing non-Gaussian tails and connecting it to spin glass models, with explicit results for models like Sherrington-Kirkpatrick.

Contribution

It provides the first exact computation of occupation time distribution in a non-Markovian sequence and links it to spin glass models through a gauge transformation.

Findings

Distribution has non-Gaussian tails with explicit large deviation function.

Mapping to spin glass models shows nontrivial disorder distribution at finite and infinite temperatures.

Explicit results obtained for the Sherrington-Kirkpatrick model with Gaussian disorder.

Abstract

We compute exactly the distribution of the occupation time in a discrete {\em non-Markovian} toy sequence which appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function which is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting new question for a generic finite sized spin glass model: at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.