Percolation properties of ferromagnetic and frustrated replicated Ising models

Abstract

In this thesis, the percolation properties of the ferromagnetic as well as a disordered and frustrated multi-replica Ising model in two dimensions are considered. The investigated systems can be understood as a collection of non-interacting copies (replicas) at the same temperature. In this setup we define a correlated percolation problem, where we introduce and study two types of clusters, namely the soft and hard constraint clusters. For the ferromagnetic case the 1-, 2-, and 3-replica Ising models have been considered, with some preliminary results concerning the 4-replica case. By means of Monte Carlo simulations on relatively large system sizes and a finite-size scaling analysis we investigate the critical behaviour of the system and provide estimates of the critical exponents. Specifically, for the 1-replica Ising model the critical exponents concerning the percolation strength and average cluster size are determined, by considering the influence on the estimates of the exponents when particular cluster sets are included or excluded in the definition of the observables. Subsequently, for the 2- and 3-replica case the critical behaviour of the system have been discussed in terms of the percolation point, and the critical exponents concerning the correlation length, percolation strength, and average cluster size for the soft and hard constraint clusters have been computed, respectively. The inclusion or exclusion of different cluster sets in the definitions of percolation strength and average cluster size have been also considered. Some preliminary results for the 4-replica Ising model are also given. For the frustrated Ising model, i.e., the Edwards-Anderson spin-glass, the percolation properties of Houdayer’s clusters have been investigated. Such clusters define a percolation process similar to the 2-replica Ising ferromagnet, with the obvious distinction that interactions are now random.

Date of Award	Apr 2023
Original language	English
Awarding Institution	Coventry University
Supervisor	Nikos Fytas (Supervisor), Martin Weigel (Supervisor) & Thierry Platini (Supervisor)