Spectral Bounds for the Ising Ferromagnet on an Arbitrary Given Graph

TL;DR

This paper develops efficient spectral bounds for the Ising ferromagnet on any finite graph, enabling rigorous upper bounds on key quantities like the partition function, magnetizations, and correlations using high-temperature expansions and spectral operators.

Contribution

It introduces a novel method to use classical bounds with spectral operators for arbitrary graphs, improving the analysis of the Ising model in high-temperature regimes.

Findings

Spectral bounds effectively upper-bound the partition function and correlations.

Susceptibility propagation converges and bounds correlations in high-temperature regions.

Method applies to graphs with arbitrary topology and positive couplings.

Abstract

We revisit classical bounds of M. E. Fisher on the ferromagnetic Ising model, and show how to efficiently use them on an arbitrary given graph to rigorously upper-bound the partition function, magnetizations, and correlations. The results are valid on any finite graph, with arbitrary topology and arbitrary positive couplings and fields. Our results are based on high temperature expansions of the aforementioned quantities, and are expressed in terms of two related linear operators: the non-backtracking operator and the Bethe Hessian. As a by-product, we show that in a well-defined high-temperature region, the susceptibility propagation algorithm converges and provides an upper bound on the true spin-spin correlations.