Ising-like models on arbitrary graphs : The Hadamard way

TL;DR

This paper introduces a universal framework for analyzing Ising-like models on arbitrary graphs using Hadamard transforms, enabling faster spectrum computation especially for regular graphs.

Contribution

It presents a novel Hadamard transform-based formalism for arbitrary graph Ising models, simplifying spectrum calculations and extending to partition functions and transfer matrices.

Findings

Hadamard transform of a sparse coding vector describes the energy spectrum.

Fast Hadamard transform algorithms can accelerate spectrum computation.

Recurrence relations simplify spectrum analysis for regular graphs.

Abstract

We propose a generic framework to describe classical Ising-like models defined on arbitrary graphs. The energy spectrum is shown to be the Hadamard transform of a suitably defined sparse "coding" vector associated with the graph. We expect that the existence of a fast Hadamard transform algorithm (used for instance in image ccompression processes), together with the sparseness of the coding vector, may provide ways to fasten the spectrum computation.. Applying this formalism to regular graphs, such as hypercubic graphs, we obtain a simple recurrence relation for the spectrum, which significantly speeds up its determination. First attempts to analyse partition functions and transfer matrices are also presented.