On this page, you can find the latest preprint version of our introductory book on statistical mechanics. As there are already many such books, both for physicists and mathematicians, we believe it is important to describe what we aimed to achieve when we started with this project. Our goal was to write a book that is

• of limited scope: although we cover many important topics, we decided to limit the scope of the book. Among the most important topics that are not discussed (except, occasionally, in short notes), let us list: quantum models, non-lattice models, disordered systems, critical phenomena, stochastic dynamics.
• student-friendly: we tried to make the book as readable as possible, accessible to advanced undergraduates (say, what corresponds to master students in Europe), both in mathematics and physics. To achieve this end, we have refrained from stating and proving the most general claims possible, but discussed instead in detail particularly important (classes of) examples (among which the Ising model, the Curie-Weiss model, the Blume-Capel model, the Gaussian Free Field, the XY model, the Heisenberg model, etc). We also wanted the book to be more or less self-contained, so that we have included a (rather long) appendix introducing the required mathematical tools, and provided solutions to most of the exercises.
• based on the probabilistic approach to (rigorous) equilibrium statistical mechanics, even though we have not refrained from using other types of tools when needed. In particular, there is no discussion of explicit solutions for integrable models (apart from brief comments).

We keep on this page an up-to-date list of corrections to the book. So, please, do not hesitate to communicate to us any typo or error you spot. All these should be sent to us at introsmbook@gmail.com.

## Reference to the published version

Friedli, S. and Velenik, Y.
Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction
Cambridge: Cambridge University Press, 2017.
ISBN: 978-1-107-18482-4
DOI: 10.1017/9781316882603

## Media

2016: Lectures based on Chapters 3, 8, 9 and 10, given by one of us (YV), have been uploaded on YouTube. They are in English (but with a heavy French accent ☺) . Alternatively, they can be found here (together with other lectures in the same program).

2015: Lectures based on Chapter 3, given by Claudio Landim (IMPA), can be found here. They are in Portuguese.

## Files

Here, you can find the final draft of the book. Its content should be nearly identical to the version published by Cambridge University Press, although the latter may have fewer grammatical and stylistic errors, thanks to their editorial work. We have also implemented, in the draft, the minor corrections listed in the errata. The numbering of the chapters, sections, equations, theorems, etc, is identical, but the page numbering is not.

In this chapter, we briefly (and somewhat heuristically) introduce equilibrium thermodynamics and equilibrium statistical mechanics, explaining the links between these two theories. We also provide an informal discussion of the van der Waals–Maxwell theory of condensation and of the paramagnet/ferromagnet phase transition. Finally, we make some general remarks on the role of the thermodynamic limit and the use of simple models. The chapter concludes with a roadmap to the book and an overview of the literature.

Mean-field models play a useful role, both from the physical and mathematical point of view, as first approximation to more realistic ones. This chapter provides a detailed account of the Curie–Weiss model, which can be seen as the mean-field version of the Ising model. The advantage is that this model exhibits a phase transition between paramagnetic and ferromagnetic behaviors that can be described with elementary tools. The approach is mostly combinatorial, but alternatives are described in complements.

The Ising model is possibly the simplest "realistic" model which exhibits a non-trivial collective behavior. As such, it has played, and continues to play, a central role in statistical mechanics. This chapter uses it to introduce several central notions (e.g., the thermodynamic limit and infinite-volume Gibbs states) and precise definitions (e.g., the analytic and the probabilistic definitions of first-order phase transitions and their equivalence). The complete phase diagram of the model is then constructed, in all dimensions. The analysis makes use of convexity properties of the pressure, correlation inequalities, low- and high-temperature expansions, Peierls' argument and the Lee–Yang theorem, all of which are discussed in detail. The chapter concludes with an extensive list of complements covering a wide range of topics.

Historically, the liquid-vapor equilibrium played a central role in the first theoretical studies of phase transitions. In this chapter, the mathematical description of the lattice gas is exposed in detail, as well as its mean-field and nearest-neighbor (Ising) versions. The mean-field (Kać) limit is also studied in a simple case, providing a rigorous justification of the van der Waals–Maxwell theory of condensation. The chapter concludes with several complements discussing more advanced questions.

The cluster expansion remains the most important perturbative technique in mathematical statistical mechanics. In this chapter, it is presented in a simple fashion and several applications to the Ising model and the lattice gas are presented. This tool is also used several times later in the book and plays, in particular, a central role in the implementation of the Pirogov–Sinai theory of Chapter 7.

In this chapter, we present a probabilistic description of infinite systems of particles at equilibrium, which is known nowadays as the theory of Gibbs measures or as the DLR (Dobrushin–Lanford–Ruelle) formalism. The theory is developed from scratch, restricting attention to models with finite single-spin-space to keep the exposition as simple as possible. The Ising model serves as a guiding example throughout the chapter. Several important aspects, such as Dobrushin's Uniqueness Theorem, spontaneous symmetry breaking, the properties of extremal measures and the extremal decomposition, are exposed in detail. At the end of the chapter, the variational principle is also introduced; the latter is closely linked with the basic concepts of equilibrium thermodynamics.

The Pirogov–Sinai theory is one of the very few general approaches to the rigorous study of first-order phase transitions. It yields, under rather weak assumptions, a sharp description of such phase transitions in perturbative regimes. In this chapter, this theory is first introduced in a rather general setting and then implemented in detail on one specific three-phase model: the Blume–Capel model.

In this chapter, the lattice version of the Gaussian Free Field is analyzed. Several features related to the non-compactness of its single-spin-space are discussed, exploiting the Gaussian nature of the model. We also derive the random walk representation that characterizes its mean and covariance structure. The recurrence properties of this random walk turn out to be crucial to analyze the thermodynamic limit.

An important class of models with a continuous symmetry, including the XY and Heisenberg models, is studied in this chapter. The emphasis is on the implications of the presence of the continuous symmetry on long-range order in these models in low dimensions. In particular, a strong form of the celebrated Mermin–Wagner theorem is proved in an simple way. We then complement that with a discussion of correlations decay.

Reflection positivity is another tool that plays a central role in the rigorous study of phase transitions. We first expose it in detail, proving its two central estimates: the infrared bound and the chessboard estimate. We then apply the latter to obtain several results of importance. In particular, we prove the existence of a phase transition in the anisotropic XY model in dimensions $d\geq 2$, as well as in the (isotropic) $O(N)$ model in dimensions $d\geq 3$. Combined with the results of Chapter 9, this provides a detailed description of this type of systems in the thermodynamic limit.

This appendix regroups short notes that are sometimes referred to in the text.

In this appendix, we introduce various mathematical topics used throughout the book, which might not be part of all undergraduate curricula. For example: elementary properties of convex functions, some aspects of complex analysis, measure theory, conditional expectation, random walks, etc., are briefly introduced, not always in a self-contained manner, often without proofs, but with references to the literature.

Hints or detailed solutions for most of the exercises of the book can be found in this appendix.