Phase separation in random cluster models II: The droplet at equilibrium, and local deviation lower bounds

Abstract

We study the droplet that results from conditioning the planar subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Γ 0 encircling the origin and enclosing an area of at least (or exactly) n 2 . We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR Γ 0 , this being the maximum distance from a point in the circuit Γ 0 to the boundary ∂conv Γ 0 of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, namely, the length of the longest line segment of which the polygon ∂conv Γ 0 is formed. We prove that there exists a constant c > 0 such that the conditional probability that the normalised quantity n −1/3 log n −2/3 MLR Γ 0 exceeds c tends to 1 in the high n-limit; and that the same statement holds for n −2/3 log n −1/3 MFL Γ 0 . To obtain these bounds, we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a Markov chain that resamples sections of the circuit boundary. We analyse the chain at equilibrium to prove the local roughness lower bounds. Alongside complementary upper bounds provided in [14], the fluctuations MLR Γ 0 and MFL Γ 0 are determined up to a constant factor.

Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation

“…Proof of Lemma 6.1. This follows immediately using (11) and from Lemmas 6.4 and 6.5 and a union bound.…”

“…In this way, we identify exponent pairs of (2/3, 1/3) and (1/3, 2/3) in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [10,11,9] the fluctuation of the boundary separating two phases in subcritical planar random cluster models. Contents arXiv:1804.07843v1 [math.PR] 20 Apr 2018 2 ALAN HAMMOND AND SOURAV SARKAR lim sup t 0 sup 0≤z≤1−t t −2/3 log t −1 −1/3 |ρ ← * (z + t) − ρ ← * (z)| ≤ C .The same result holds for the rightmost polymer.…”

The Competition of Roughness and Curvature in Area-Constrained Polymer Models

“…It may be summarised by an exponent triple (1/2, 1/3, 2/3) representing local interface fluctuation, local roughness (or inward deviation) and convex hull facet length. The three effects arise, for example, in droplets in planar Ising models [20,21,19,2]. In this article, we offer a new perspective on this phenomenon.…”

“…The typical or maximum local roughness along the circuit is a latitudinal counterpart to facet length. Alexander [2], and Hammond [20,21,19] analysed such conditioned circuit models and determined that when the area contained in the circuit is of order n 2 , so that the circuit has diameter of order n, facet length and local roughness scale as n 2/3 and n 1/3 . A similar situation is witnessed when a parabola x → t −1 x 2 is subtracted from a two-sided Brownian motion B : R → R. When t > 0 is large, facets of the motion's convex hull have length Θ(t 2/3 ) and inward deviation Θ(t 1/3 ).…”

“…Finally, we discuss briefly the key inputs used in our paper and how it contrasts with the other examples of fluctuation results already mentioned. The results in [20,21,19] crucially used the refined understanding and geometric estimates for percolation clusters while a study of area trapping planar Brownian loop [22] used well known estimates for Brownian motion. Exact expressions involving Brownian motion conditioned to stay over a parabola was also the key ingredient in the proofs in [17].…”