Invariance Principle for a Potts Interface Along a Wall

Abstract

We consider nearest-neighbour two-dimensional Potts models, with boundary conditions leading to the presence of an interface along the bottom wall of the box. We show that, after a suitable diffusive scaling, the interface weakly converges to the standard Brownian excursion.

Critical prewetting in the 2d Ising model

“…Therefore many references to previous papers are to be expected. In particular, while not necessary to follow the proofs, some familiarity with (a subset of) [27,29,28,31,23] will definitely smoothen the reading. First note that any realization of the interface γ partitions the box B N into two (not necessarily connected) pieces: the part located "above the interface", denoted…”

“…In such a box, the effect of the magnetic field λ/N is weak enough to be ignored. This allows us to deduce the claim by importing a recent entropic repulsion result from [23] that applies to the model without magnetic field. Claim (1.7) then follows from a union bound (over translates of the small box).…”

“…Finally, tightness can be proved by a straightforward adaptation of the approach we used in [29], but we provide an alternative approach relying on a suitable timechange, which reduces the analysis to a fixed number of steps; this allows us to conclude as in [23] (this is explained in Section 6.6).…”

“…The width of this layer is of order √ n (thus mesoscopic). The statistical properties of the interface in this regime are now well understood: after a diffusive rescaling, the latter has been proved to converge weakly to a Brownian excursion [23].…”

Entropic repulsion of 3D Ising interfaces

“…Without any floor (i.e., under µ ∓ n as opposed to µ h n ), the Ising interface in a strip of side-length n with Dobrushin's boundary conditions is known to have √ n height fluctuations [14], and converge to a Brownian bridge under a diffusive rescaling at all β > β c (d) [24,26]. In the presence of a floor at height zero, either by means of plus boundary conditions in the entire lower half-space or by means of conditioning on the interface being restricted to the upper halfspace, one could easily deduce that the fluctuations remain O( √ n); convergence to a Brownian excursion was recently shown in [28]. In particular, no matter the conditioning on a floor (at whatever negative height) the entropic repulsion effect does not change the order of the typical height of the interface.…”