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 ΩN be the space of configurations of an N−spins system, each spin having a finite set Ω of inner states, and let μ:ΩN→0,1 be some probability measure. Here we give an argument to encode μ into a kernel function M:0,12→Ω, and use this notion to reinterpret the assumptions of the Replica Symmetry Breaking ansatz (RSB) of Parisi (1980) and Parisi et al. (1987) without using replicas, nor averaging on the disorder.