N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.

Markov Chains and Mixing Times

Convex bodies: the brunn–minkowski theory

The large deviation approach to statistical mechanics

The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results that have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.

/pdf/rigorous-probabilistic-analysis-of-equilibrium-crystal-3frmtauqj2.pdf

Rigorous probabilistic analysis of equilibrium crystal shapes

We consider the massless field with zero boundary conditions outside D

 N
 ≡D∩ (ℤ
 d
 /N) (N∈ℤ+), D a suitable subset of ℝ
 d
 , i.e. the continuous spin Gibbs measure ℙ
 N
 on ℝ
 ℤd/N
 with Hamiltonian given by H(ϕ) = ∑
 x,y:|x−y|=1
 
V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D

 N
 C
 . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D

 N
 . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ
 N
 = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ
 N
 } and of the induced family of gradient fields ∇
 N
 ξ
 N
 as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ
 N
 } and show that the corresponding rate function is given by ∫
 D
 σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ
 N
 under the volume constraint, i.e. the constraint that (1/N

 d
 ) ∑
 x∈DN
 ξ
 N
 (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ
 N
 (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ
 N
 of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇
 N
 ξ
 N
 (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ?
 p
 estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sjostrand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields.

Large deviations and concentration properties for ∇ϕ interface models

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k≥2 is extremal if and only if β is less or equal to the spin glass transition value, given by tanh(β

 c
 SG
 = 1/√k.

On the extremality of the disordered state for the Ising model on the Bethe lattice

We develop a non-perturbative version of the Dobrushin–Kotecký–Shlosman theory of phase separation in the canonical 2D Ising ensemble. The results are valid for all temperatures below critical.

Dobrushin-Kotecký-Shlosman Theorem up to the Critical Temperature

We prove an upper large deviation bound for the block spin magnetization in the 2D Ising model in the phase coexistence region. The precise rate (given by the Wulff construction) is shown to hold true for all β > βc. Combined with the lower bounds derived in [I] those results yield an exact second order large deviation theory up to the critical temperature.

Dmitry Ioffe

Papers

Rigorous probabilistic analysis of equilibrium crystal shapes

Large deviations and concentration properties for ∇ϕ interface models

On the extremality of the disordered state for the Ising model on the Bethe lattice

Dobrushin-Kotecký-Shlosman Theorem up to the Critical Temperature

Exact large deviation bounds up toT c for the Ising model in two dimensions