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 $\mu :{\Omega }^N\to \left[0,1\right]$ be some probability measure. Here we give an argument to encode $\mu$ into a kernel function $M:{\left[0,1\right]}^2\to \Omega$, 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.

57 pages, 14 figures

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• A simplified Parisi Ansatz II: REM universality
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