In this chapter, we describe a few discrete probability models to which we will come back repeatedly throughout the book. While there exists a vast array of well-studied random combinatorial structures (permutations, partitions, urn models, Boolean functions, polytopes, etc.), our focus is primarily on a limited number of graph-based processes, namely percolation, random graphs, Ising models, and random walks on networks. We will not attempt to derive the theory of these models exhaustively here. Instead we will employ them to illustrate some essential techniques from discrete probability. Note that the toolkit developed in this book is meant to apply to other probabilistic models of interest as well, and in fact many more will be encountered along the way. After a brief review of graph basics and Markov chains theory, we formally introduce our main models. We also formulate various key questions about these models that will be answered (at least partially) later on. We assume that the reader is familiar with the measure-theoretic foundations of probability. A refresher of all required concepts and results is provided in the appendix.

Exact solutions for infinite Ising systems are rare, specific in terms of the interactions allowed, and limited to one and two dimensions. To study a wider range of models we must resort to various approximation techniques. One of the simplest and most comprehensive of these is the mean-field approximation, the subject of this chapter. Some versions of this approximation rely on a self-consistent requirement, and in this respect the mean-field method for the Ising model is similar to a number of other self-consistent approximation methods in physics, including the Hartree–Fock approximation for atomic and molecular orbitals, the BCS theory of superconductivity, and the relaxation method for determining electric potentials. We will also introduce a somewhat different mean-field approach, the Landau–Ginzburg approximation, which is based on a series expansion of the free energy. One of the drawbacks of all of the mean-field theories, however, is that they predict the same mean-field critical exponents, which, unfortunately, are at odds with the results of exact solutions and experiments.

A useful way to solve a complex problem – whether in physics, mathematics, or life in general – is to break it down into smaller pieces that can be handled more easily. This is especially true of the Ising model. In this chapter, we investigate various partial-summation techniques in which a subset of Ising spins is summed over to produce new, effective couplings among the remaining spins. These methods are useful in their own right and are even more important when used as a part of position-space renormalization-group techniques.

In the chapters so far, we have studied a number of exact methods of calculation for Ising models. These studies culminated in the exact solution for an infinite one-dimensional Ising model, as well as the corresponding solution on a 2 × ∞ lattice. Neither of these systems shows a phase transition, however. In this chapter, we start with Onsager’s exact solution for the two-dimensional lattice, which quite famously does have a phase transition. Next, we explore exact series expansions from low and high temperature, and show how these results can be combined, via the concept of duality, to give the exact location of the phase transition in two dimensions.

In Chapter 3 we explored transformations where a finite group of Ising spins is summed to produce effective interactions among the remaining spins. In all of these cases a finite sum of Boltzmann factors is sufficient to solve the problem. We turn now to infinite systems, where a straightforward, brute-force summation is not possible. Instead, we develop a number of new techniques that allow us to evaluate an infinite summation in full detail.

Kenneth Wilson introduced the renormalization-group (RG) approach in 1971. This approach gave new life to the study of the Ising model. The implications of this breakthrough were immediately recognized by researchers in the field, and Wilson and the RG technique were awarded the Nobel Prize in Physics soon thereafter. One of the distinguishing features of RG methods is that they explicitly include the effects of fluctuations. In addition, the RG approach gives a natural understanding of the universality that is seen in critical phenomena in general, and in critical exponents in particular. In many respects, the RG approach gives a deeper understanding not only of the Ising model itself, but of all aspects of critical phenomena. The original version of the renormalization-group method was implemented in momentum space – which is a bit like studying a system with Fourier transforms. It is beyond the scope of this presentation. Following that, various investigators extended the approach to position space, which is more intuitive in many ways and is certainly much easier to visualize. We present the basics of position-space renormalization group methods in this chapter. We will also explain the origin of the terms “renormalization” and “group” in the RG part of the name.

In this chapter, we explore Ising systems that consist of just one or a few spins. We define a Hamiltonian for each system and then carry out straightforward summations over all the spin states to obtain the partition function. No phase transitions occur in these systems – in fact, an infinite system is needed to produce the singularities that characterize phase transitions. Even so, our study of finite systems yields a number of results and insights that are important to the study of infinite systems.

Few models in theoretical physics have been studied for as long, or in as much detail, as the Ising model. It’s the simplest model to display a nontrivial phase transition, and as such it plays a unique role in theoretical physics. In addition, the Ising model can be applied to a wide range of physical systems, from magnets and binary liquid mixtures, to adsorbed monolayers and superfluids, to name just a few. In this chapter, we present some of the background material that sets the stage for a detailed study of the Ising model in the chapters to come.

Chapter 6 explains the CUDA random number generators provided by the cuRAND library. The CUDA XORWOW generator was found to be the fastest generator in the cuRAND library. The classic calculation of pi by generating random numbers inside a square is used as a test case for the various possibilities on both host CPU and the GPU. A kernel using separate generators for each thread is able to generate about1012 random numbers per second and is about 20 000 times faster than the simplest host CPU version running on a single core. The inverse transform method for generating random numbers from any distribution is explained. A 3D Ising model calculation is presented as a more interesting application of random numbers.The Ising example has a simple interactive GUI based on OpenCV.

Many different phases of matter can be characterized by the symmetries that they break.The Ising model for interacting spins illustrates this idea.In the absence of a magnetic field, there is a critical temperature, below which there is ferromagnetic ordering, and above which there is not.The magnetization is the order parameter for this transition: it is non-zero only when there is ferromagnetic ordering.The ferromagnetic phase transition in the Ising model is explored using the approximate method of mean field theory.Exact solutions are known for the Ising model in one and two dimensions and are discussed, along with numerical solutions using Monte Carlo simulations.Finally, the ideas of broken symmetry and their relationship to phase transitions are placed in the general framework of Landau theory and compared to results from mean field theory.