Date of Award

December 2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics

Advisor(s)

A. Alan Middleton

Keywords

Combinatorics, Ising Model, Minimal Spanning Tree, Percolation, Scaling, Shortest Path

Subject Categories

Physical Sciences and Mathematics

Abstract

Disordered systems arise in many physical contexts. Not all matter is uni-

form, and impurities or heterogeneities can be modeled by fixed random disor-

der. Numerous complex networks also possess fixed disorder, leading to appli-

cations in transportation systems [1], telecommunications [2], social networks

[3, 4], and epidemic modeling [5], to name a few.

Due to their random nature and power law critical behavior, disordered

systems are difficult to study analytically. Numerical simulation can help

overcome this hurdle by allowing for the rapid computation of system states.

In order to get precise statistics and extrapolate to the thermodynamic limit,

large systems must be studied over many realizations. Thus, innovative al-

gorithm development is essential in order reduce memory or running time

requirements of simulations.

This thesis presents a review of disordered systems, as well as a thorough

study of two particular systems through numerical simulation, algorithm de-

velopment and optimization, and careful statistical analysis of scaling proper-

ties.

Chapter 1 provides a thorough overview of disordered systems, the his-

tory of their study in the physics community, and the development of tech-

niques used to study them. Topics of quenched disorder, phase transitions, the

renormalization group, criticality, and scale invariance are discussed. Several

prominent models of disordered systems are also explained. Lastly, analysis

techniques used in studying disordered systems are covered.

In Chapter 2, minimal spanning trees on critical percolation clusters are

studied, motivated in part by an analytic perturbation expansion by Jackson

and Read [6] that I check against numerical calculations. This system has a

direct mapping to the ground state of the strongly disordered spin glass [7].

We compute the path length fractal dimension of these trees in dimensions

d = {2, 3, 4, 5} and find our results to be compatible with the analytic results

suggested by Jackson and Read.

In Chapter 3, the random bond Ising ferromagnet is studied, which is es-

pecially useful since it serves as a prototype for more complicated disordered

systems such as the random field Ising model and spin glasses. We investigate

the effect that changing boundary spins has on the locations of domain walls

in the interior of the random ferromagnet system. We provide an analytic

proof that ground state domain walls in the two dimensional system are de-

composable, and we map these domain walls to a shortest paths problem. By

implementing a multiple-source shortest paths algorithm developed by Philip

Klein [8], we are able to efficiently probe domain wall locations for all possible

configurations of boundary spins. We consider lattices with uncorrelated dis-

order, as well as disorder that is spatially correlated according to a power law.

We present numerical results for the scaling exponent governing the probabil-

ity that a domain wall can be induced that passes through a particular location

in the system’s interior, and we compare these results to previous results on

the directed polymer problem.

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Open Access

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