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 …

The classical spin O (n) model is a model on ad-dimensional lattice in which a vector on the
(n-1)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites …

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) …

In this paper we study the large N limit of the O (N)-invariant linear sigma model, which is a
vector-valued generalization of the Φ 4 quantum field theory, on the three dimensional torus …

Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are
shown to have variations of order\log N log N. The level lines of such functions form a loop O …

Our first main result is that correlations between monomers in the dimer model in do not
decay to 0 when. This is the first rigorous result about correlations in the dimer model in …

We prove the existence of macroscopic loops in the loop O (2) model with $\frac12\leq x^
2\leq 1$ or, equivalently, delocalisation of the associated integer-valued Lipschitz function …

We provide a uniformly-positive point-wise lower bound for the two-point function of the
classical spin O (N) model on the torus of Z^ d Z d, d ≥ 3 d≥ 3, when N ∈ N _> 0 N∈ N> 0 …

In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at
weak coupling, we prove exponential decay of correlations for a wide class of gauge …

In the loop O(n) model a collection of mutually-disjoint self-avoiding loops is drawn at
random on a finite domain of a lattice with probability proportional to λ^\#edgesn^\#loops …