Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems

TL;DR

This paper introduces a new combinatorial interpolation method to establish bounds on free energy in diluted spin systems, simplifying previous proofs and extending results to general degree distributions, with applications to independent set sizes.

Contribution

It provides a novel combinatorial interpolation proof for free energy bounds in diluted spin systems with general degree distributions, bypassing complex SK model arguments.

Findings

New bounds for free energy in diluted spin systems.

Bounds for the size of the largest independent set in random regular graphs.

Simplified proof technique using biased random walks.

Abstract

In two papers Franz, Leone and Toninelli proved bounds for the free energy of diluted random constraints satisfaction problems, for a Poisson degree distribution [5] and a general distribution [6]. Panchenko and Talagrand [16] simplified the proof and generalized the result of [5] for the Poisson case. We provide a new proof for the general degree distribution case and as a corollary, we obtain new bounds for the size of the largest independent set (also known as hard core model) in a large random regular graph. Our proof uses a combinatorial interpolation based on biased random walks [21] and allows to bypass the arguments in [6] based on the study of the Sherrington-Kirkpatrick (SK) model.