Replica Symmetry Breaking without Replicas

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.