Lectures on the Ising and Potts Models on the Hypercubic Lattice

Abstract

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 2 + √ 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. These notes are largely inspired by [39,41,42].

Emergent planarity in two-dimensional Ising models with finite-range Interactions

“…. , x 2n } as, under this assumption, the expression (1.8) and the bound (1.14) yield 15) which is the claimed statement. Two different, but not unrelated, mechanisms may make the probability of avoided intersections (for sets of widely separated source points) small for the Ising model at its critical point.…”

Sharp threshold phenomena in statistical physics

“…The generalization of Theorem 3.3 requires the development of a new RSW theory enabled to tackle percolation models with dependency. This theory led to a number of new applications on these models, including a precise description of the critical behavior (see [20,14,11] and [12,13] for reviews). We chose not to discuss this here, and focus on the generalizations of Theorems 2.2 and 2.5.…”

Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

“…The original proof of the OSSS inequality, see [20] or [6] for more probabilistic version, uses the product structure of the space considered (the Bernoulli percolation model). But P z,β Λ,ω Λ c is not a product measure and we need a more elaborate method.…”

“…This is done using the OSSS inequality (4.7) to the wired measure µ z n for the Boolean function f (ω) = 1 0←→ r ∂Λn (ω) to the well chosen algorithms from [7,8]. This proposition uses the now standard algorithms used in [6,7,9]. The proof of Proposition 4.2 is done in the Annex Section 6.…”