Replica Symmetry Breaking without Replicas

TL;DR

This paper presents a new combinatorial framework called Kernel Representation that enables analysis of spin glass problems and reinterprets the Replica Symmetry Breaking assumptions without using replicas or disorder averaging.

Contribution

It introduces a novel Kernel Representation framework to analyze spin glasses and reformulates RSB assumptions without replicas or disorder averaging.

Findings

Kernel Representation encodes probability measures into kernel functions.

Reinterpretation of RSB assumptions without replicas.

Framework applicable to spin glass problems.

Abstract

We introduce a mathematical framework based on simple combinatorial arguments (Kernel Representation) that allows to deal successfully with spin glass problems, among others. Let $\Omega^{N}$ be the space of configurations of an $N-$ spins system, each spin having a finite set $\Omega$ of inner states, and let $\mu:\Omega^{N}\rightarrow\left[0,1\right]$ be some probability measure. Here we give an argument to encode $\mu$ into a kernel function $M:\left[0,1\right]^{2}\rightarrow\Omega$, and use this notion to reinterpret the assumptions of the Replica Symmetry Breaking ansatz (RSB) of Parisi et Al. [1, 2], without using replicas, nor averaging on the disorder.