Inequalities for the Local Energy of Random Ising Models

TL;DR

This paper establishes rigorous bounds on the average local energy in random Ising models, revealing how randomness and interaction distributions influence energy inequalities and system stability.

Contribution

It introduces new lower bounds for local energy in disordered Ising models, extending understanding beyond ferromagnetic cases and analyzing the Nishimori line.

Findings

Lower bound on average local energy derived

Introduction of interactions increases local energy contrary to ferromagnetic systems

Maximum probability of lower local energy occurs on Nishimori line

Abstract

We derive a rigorous lower bound on the average local energy for the Ising model with quenched randomness. The result is that the lower bound is given by the average local energy calculated in the absence of all interactions other than the one under consideration. The only condition for this statement to hold is that the distribution function of the random interaction under consideration is symmetric. All other interactions can be arbitrarily distributed including non-random cases. A non-trivial fact is that any introduction of other interactions to the isolated case always leads to an increase of the average local energy, which is opposite to ferromagnetic systems where the Griffiths inequality holds. Another inequality is proved for asymmetrically distributed interactions. The probability for the thermal average of the local energy to be lower than that for the isolated case takes a maximum value on the Nishimori line as a function of the temperature. In this sense the system is most stable on the Nishimori line.