R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

We are delighted to announce that Professor Martin Evans of University of Edinburgh has been appointed as the new Editor-in-Chief of Journal of Physics A: Mathematical and Theoretical. Martin Evans has been Editor of the Statistical Physics section of the journal since 2009. Prior to this, he served as a Board Member for the journal. His areas of research include statistical mechanics of nonequilibrium systems, phase transitions and scaling regimes in nonequilibrium statistical physics, glassy dynamics, phase transitions and ordering in driven diffusive systems, mass transport models, condensation models, zero range processes and exclusion processes. We very much look forward to working with Martin to continue to improve the journal's quality and interest to the readership. We would like to thank our outgoing Editor-in-Chief, Professor Murray Batchelor. Murray has worked hard and provided excellent guidance in improving the quality of the journal and the service that the journal provides to authors, referees and readers. During the last five years, we have raised the quality threshold for acceptance in the journal and currently reject over 70% of submissions. As a result, papers published in Journal of Physics A: Mathematical and Theoretical are amongst the best in the field. We have also maintained and improved on our excellent receipt-to-first-decision times, which now average under 40 days for papers. With the help of Martin Evans and our distinguished Editorial Board, we will be working to further improve the quality of the journal whilst continuing to offer excellent services to our readers, authors and referees. We hope that you benefit from reading the journal. If you have any comments or questions, please do not hesitate to contact us at jphysa@iop.org. Rebecca Gillan Publisher

New Editor-in-Chief for Journal of Physics A: Mathematical and Theoretical

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Principles Of Random Walk

The statistical mechanics of lattice gases. Vol. 1. B. Simon, Princeton University Press, Princeton, New Jersey, 1993

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proofs that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by $\sqrt{2+\sqrt 2}$. In higher dimensions, the understanding also progresses with the proof that the phase transition of Potts models is sharp, and that the magnetization of the three-dimensional Ising model vanishes at the critical point.

Lectures on the Ising and Potts models on the hypercubic lattice

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2-n}}}$. For $0<n\le 2$, the transition line has been further conjectured to separate a regime with short loops when $x<x_c(n)$ from a regime with macroscopic loops when $x\ge x_c(n)$. 
In this paper, we prove that for $n\in [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. This is the first instance in which a loop $O(n)$ model with $n\neq 1$ is shown to exhibit such behaviour. A main tool in the proof is a new positive association (FKG) property shown to hold when $n \ge 1$ and $0<x\le\frac{1}{\sqrt{n}}$. This property implies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box-crossing property). We develop a 'domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when $x=x_c(n)$ and $n\in[1,2]$.

Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point

The symmetric six-vertex model with parameters~$a,b,c>0$ is expected to exhibit different behavior in the regimes $a+b c$ (ferroelectric). In this work, we study the way in which the transition between the regimes $a+b=c$ and $a+b<c$ manifests in the thermodynamic limit. 
It is shown that the height function of the six-vertex model delocalizes with logarithmic variance when $a+b=c$ while remaining localized when $a+b<c$. In the latter regime, the extremal translation-invariant Gibbs states of the height function are described. Qualitative differences between the two regimes are further exhibited for the Gibbs states of the six-vertex model itself and for the Gibbs states of its associated spin representation. 
Via a coupling, our results further allow to study the self-dual Ashkin--Teller model on $\mathbb{Z}^2$. It is proved that on the portion of the self-dual curve~$\sinh 2J = e^{-2U}$ where $J<U$ each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered. This is in contrast to the case $J=U$ (the critical 4-state Potts model) where it is known that there is power-law decay for both Ising configurations and for their product. 
The proofs rely on the recently established order of the phase transition in the random-cluster model, which relates to the six-vertex model via the Baxter--Kelland--Wu coupling. Additional ingredients include the introduction of a random-cluster model with modified weight for boundary clusters, analysis of a spin (mixed Ashkin--Teller model) and bond representation (of the FK-Ising type) for the six-vertex model, and the introduction of triangular lattice contours and associated bijection for the analysis of the Gibbs states of the height function.

On the transition between the disordered and antiferroelectric phases of the 6-vertex model

Uniform integer-valued Lipschitz functions on a domain of size $N$ of the triangular lattice are shown to have variations of order $\sqrt{\log N}$. The level lines of such functions form a loop $O(2)$ model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop $O(2)$ model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.

Uniform Lipschitz functions on the triangular lattice have logarithmic variations

We study the loop $O(n)$ model on the honeycomb lattice. By means of local non-planar deformations of the lattice, we construct a discrete stress-energy tensor. For $n\in [-2,2]$, it gives a new observable satisfying a part of Cauchy-Riemann equations. We conjecture that it is approximately discrete-holomorphic and converges to the stress-energy tensor in the continuum, which is known to be a holomorphic function with the Schwarzian conformal covariance. In support of this conjecture, we prove it for the case of $n=1$ which corresponds to the Ising model. Moreover, in this case, we show that the correlations of the discrete stress-energy tensor with primary fields converge to their continuous counterparts, which satisfy the OPEs given by the CFT with central charge $c=1/2$. 
Proving the conjecture for other values of $n$ remains a challenge. In particular, this would open a road to establishing the convergence of the interface to the corresponding $\mathrm{SLE}_\kappa$ in the scaling limit.

Alexander Glazman

Papers

Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point

Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point

On the transition between the disordered and antiferroelectric phases of the 6-vertex model

Uniform Lipschitz functions on the triangular lattice have logarithmic variations

Discrete stress-energy tensor in the loop O(n) model