Small-world hypergraphs on a bond-disordered Bethe lattice

TL;DR

This paper investigates the thermodynamic behavior of spin systems with bond-disorder on small-world hypergraphs, deriving equations for order parameters and phase transitions, and validating results with simulations.

Contribution

It introduces a novel analysis of bond-disordered spin systems on small-world hypergraphs using transfer-matrix methods and replica symmetry breaking.

Findings

Identified static and dynamic ferromagnetic transitions.

Determined the spin-glass transition within replica symmetry.

Validated theoretical results with Monte-Carlo simulations.

Abstract

We study the thermodynamic properties of spin systems with bond-disorder on small-world hypergraphs, obtained by superimposing a one-dimensional Ising chain onto a random Bethe graph with p-spin interactions. Using transfer-matrix techniques, we derive fixed-point equations describing the relevant order parameters and the free energy, both in the replica symmetric and one step replica symmetry breaking approximation. We determine the static and dynamic ferromagnetic transition and the spinglass transition within replica symmetry for all temperatures, and demonstrate corrections to these results when one step replica symmetry breaking is taken into account. The results obtained are in agreement with Monte-Carlo simulations.