Lack of self-average in weakly disordered one dimensional systems

TL;DR

This paper analyzes a one-dimensional disordered Ising model revealing a non-self-averaging overlap distribution at zero temperature, with an analytical form for any disorder realization, and shows how it becomes self-averaging at finite impurity concentration.

Contribution

It introduces an exactly solvable disordered Ising model exhibiting non-self-averaging behavior at zero temperature and describes the transition to self-averaging at finite impurity levels.

Findings

Non-self-averaging overlap distribution at zero temperature

Analytical calculation of the distribution for any disorder realization

Transition to self-averaging distribution at non-zero impurity concentration

Abstract

We introduce a one dimensional disordered Ising model which at zero temperature is characterized by a non-trivial, non-self-averaging, overlap probability distribution when the impurity concentration vanishes in the thermodynamic limit. The form of the distribution can be calculated analytically for any realization of disorder. For non-zero impurity concentration the distribution becomes a self-averaging delta function centered on a value which can be estimated by the product of appropriate transfer matrices.