A Note on Schramm’s Locality Conjecture for Random-Cluster Models

Abstract

In this note, we discuss a generalization of Schramm's locality conjecture to the case of random-cluster models. We give some partial (modest) answers, and present several related open questions. Our main result is to show that the critical inverse temperature of the Potts model on Z r × (Z 2nZ) d−r (with r ≥ 3) converges to the critical inverse temperature of the model on Z d as n tends to infinity. Our proof relies on the infrared bound and, contrary to the corresponding statement for Bernoulli percolation, does not involve renormalization arguments.

Exponential Decay of Truncated Correlations for the Ising Model in any Dimension for all but the Critical Temperature

“…We refer to the paper for details on open questions and conjectures. Bodineau's result being the key to our argument (not to mention the heavy use of the random-current representation, which itself is not available for the Potts model), we believe that the exponential decay of correlations would be even harder to obtain than the open problems mentioned in [DT17].…”

“…While the β < β c was recently treated in every dimension in [DRT17], the study of the β > β c regime is still very limited. In [DT17], a partial result going in the direction of the equivalent of [Bod05] for Potts model was obtained. We refer to the paper for details on open questions and conjectures.…”

Magnetization in the zig-zag layered Ising model and orthogonal polynomials

“…From the perspective of the "free fermion algebra" solution [55] of the planar Ising model, our derivations can be viewed as its translation to the language of discrete fermionic observables, see [30] for a discussion of such a correspondence. The latter viewpoint was advertised by Smirnov in his celebrated work on the critical Ising model (e.g., see [24] and references therein). We refer the interested reader to [15,Section 3] for a discussion of equivalences between various combinatorial formalisms used to study the planar Ising model, see also [43] and [18,Section 3.2].…”

“…The planar Ising (or Lenz-Ising) model, introduced by Lenz almost a century ago, has an extremely rich history which is impossible to overview in a short introduction, instead we refer the interested reader to the monographs [6,24,42,48] as well as the papers [15,38,45,46,47] and references therein for more information on various facets of this history. From the "classical analysis" viewpoint, one of the particularly remarkable aspects is a fruitful interplay between the explicit computations for the planar Ising model and the theory of Toeplitz determinants.…”