On the probabilistic formulation of the replica approach to spin glasses

TL;DR

This paper reviews the replica approach's predictions on overlap distributions in spin glasses, emphasizing replica equivalence, ultrametricity, and comparing with rigorous methods, highlighting a variational principle for ultrametric solutions.

Contribution

It provides a probabilistic formulation of the replica approach, clarifies the role of algebraic properties, and connects ultrametricity with a variational principle.

Findings

Replica equivalence leads to universal relations independent of symmetry breaking.

Ultrametricity can be derived from a variational principle.

Comparison with rigorous methods validates the replica predictions.

Abstract

In this paper we review the predictions of the replica approach on the probability distribution of the overlaps among replicas and on the sample to sample fluctuations of this probability. We stress the role of replica equivalence in obtaining relations which do not depend on the form of replica symmetry breaking. A comparison is done with the results obtained with a different rigorous approach. The role of the ultrametricity and of other algebraic properties in discussed. It is shown that the ultrametric solution can be obtained from a variational principle.