Models in Statistical Physics on Infinite Graphs

Abstract

Physicists and mathematicians love regularity, mainly for the reason that it makes things simple, clear and, primely, computable; nature, on the other hand, has its own way of deciding what’s the right way to behave, and although it presents the attentive observer with beautifully ordered crystals, impressive fractals and perfectly spherical soap bubbles, the vast majority of matter we interact with every day has no definite structural order: wood, plastic, cloth, all are microscopically complex structures which do not enjoy neither translational nor scale invariance, nor any other form of symmetry.

However, even when the spatial disposition of the constituents of matter is highly irregular, what determines the physical properties of the system are not so much the details of the distribution, as the topological structure of their interactions, which can usually be safely reduced to nearest neighbour or next-nearest neighbour relations.

As a consequence, both regular and complex matter are well described by means of graphs, which represent in a compact form the interactions among their constituents.

When paired with an underlying mathematical model, a graph is able to describe many physical systems in an approximate yet effective way. Furthermore, a graph can equally well model a network of interacting agents, as one often finds in economic literature, or computers - the Internet is the prime example - or abstract data structures in IT, or a deluge of other possibilities.

Given their broad applicability, it is not surprising that a great wealth of scientific literature in many fields regards physical and mathematical models defined on graphs. In particular, during my doctoral period I focused on two different general purpose models, which enjoy a wide diffusion: discrete random walks and the Ising model.

Random Walks.

Regarding random walks, my current research focuses on the statistics of multiple agents simultaneously travelling on an infinite graph, and their collective properties.

Several degrees of technical difficulty accompany this endeavour, associated with keeping track of multiple positions, on the one hand, and with the non-linearity of collective properties, on the other. A simplifying picture can be sometimes achieved by means of the distance graph, each vertex of which corresponds to a relative position of the particles, so that, for example, simultaneous encounters of all the particles correspond to a return to the origin in the distance graph.

Constructing the distance graph proves however impossible in general, so that case specific methods need to be used to calculate the same quantities.

While further work is in progress, the results I obtained up to date regard three particles moving on an infinite line: owing to the homogeneity of the structure, a distance graph exists, which corresponds to the triangular 2-d lattice, so that the probability of finding the particles at given relative distances can be straightforwardly calculated by means of a standard Fourier transform and saddle point approximation.

As a further step, we computed the probability that the minimum and maximum distances among the particles are lower than some constant d, and found that asymptotically they scale as a function of d^2/t, implying that the surfaces of fixed probability undergo a regular diffusion.

The results have been submitted for publication.

The Ising Model.

The second area of research I investigated revolves around the magnetization properties of the ferromagnetic Ising model on arbitrary graphs. More specifically, my aim was to find a relationship between the long range topology of a graph and its spontaneous magnetization.

Several results are present in literature for regular lattices, fractals and a few more general graphs, and an extension of the Mermin-Wagner theorem states that transient on the average graphs show indeed spontaneous magnetization; however, a very wide set of graphs was still uncovered, and no sufficiently general criterion for assessing the existence of long range correlation existed. The approximate direction of exploration was clear since the beginning: on the one side, the number of different paths present in a graph is a first measure of the quality of the correlation between far away points; in turn, the transmission of information from one vertex to another is strongly inhibited when all the paths connecting them must cross just a small number of edges of the graph.

These considerations formed the basis from which we proceeded to the construction of two theorems: a necessary condition for the presence of spontaneous magnetization, in zero external field and no boundary conditions, and a sufficient condition for its absence. These two theorems represent the most encompassing criteria to date for what regards the magnetizability of graphs: all the known structures enter the hypothesis of either of the theorems, even though no sufficient and necessary condition is known.

The gap left between the hypotheses of the two theorems regards a very small class of pathological graphs whose existence is not even acknowledged.

The theorems have been published as two different research papers.

A part of my study was cast into an analysis of the different ways in which spontaneous magnetization can be defined, as either a spin expectation value or a long range order, for a single spin or averaged over the graph, under an external field or boundary conditions or neither. A chapter has been devoted to a thorough discussion of the matter.