Maximum and Shape of Interfaces in 3D Ising Crystals

Abstract

Dobrushin in 1972 showed that the interface of a 3D Ising model with minus boundary conditions above the xy‐plane and plus below is rigid (has O(1) fluctuations) at every sufficiently low temperature. Since then, basic features of this interface—such as the asymptotics of its maximum—were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D solid‐on‐solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side length n with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for Mn, its maximum: if the inverse temperature β is large enough, then Mn/logn → 2/αβ as n → ∞, in probability, where αβ is given by a large‐deviation rate in infinite volume. We further show that, on the large‐deviation event that the interface connects the origin to height h, it consists of a 1D spine that behaves like a random walk in that it decomposes into a linear (in h) number of asymptotically stationary, weakly dependent increments that have exponential tails. As the number T of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in T. These results generalize to every dimension d ≥ 3. © 2020 Wiley Periodicals LLC.

Entropic repulsion of 3D Ising interfaces

“…(The actual bound available to us-see (2.9)-has a few small differences (error terms, a restriction on h) in addition to one major proviso we will soon describe.) This estimate is quite intuitive in light of the fact ( [22,23]) that the µ ∓ n -probability of a local oscillation of height at least h below a site in the bulk is approximately e −α h ; indeed, if the oscillations in every site of S were mutually independent, we would get the bound (1 − e −α h ) |S| ≈ exp(−|S|e −α h ). In the event that a ceiling C as described above exists, we are guaranteed to have M ↓ S < h * n − k by the implicit conditioning on I fl h in µ h n .…”

“…Remark 1.2. Theorem 1.1 extends to treat h = h * n −1 whenever the determinstic quantity λ n := log n−α h * n , which is known to belong to [−2β − ε β , 2β] for all n (the upper bound is by definition (1.3) whereas the lower bound is by results in [22]) does not fall in a certain ε β -fraction of this interval. For instance, (1.5) extends to h = h * n − 1 for λ n ≥ 2 log β, while (1.6) extends to h = h * n − 1 for λ n ≤ log β (see Remarks 5.8 and 6.6).…”

“…Remark 1.3. In [22], the authors derived a law of large numbers for the maximum height M n of I ∼ µ ∓ n : as n → ∞ one has M n / log n → 2/α in µ ∓ n -probability, for the quantity α = α(β) defined there as…”

“…Recall that without any floor conditioning, the interface is rigid about its ground state, i.e., the interface is mostly flat at height zero, with O(1) height oscillations about that with exponential tails [16,38]. In [22,23] the authors studied more refined features regarding the typical shape of this interface near its high points, and used that to deduce that the maximum height is tight and has uniform Gumbel tails about its median m * n = ( 2 α + o(1)) log n. When introducing a hard floor, either via conditioning or via a set of plus spins in the lower half-space, far less is known, though this problem has been discussed in the physics literature at least since [6]. That work proposed using the solid-on-solid (SOS) model [1], a distribution over ϕ :…”

“…A more detailed formulation of (1.5) can be found in Theorem 5.1, and a more detailed formulation of (1.6) can be found in Theorem 6.1. Unlike [16,[21][22][23] the proofs in this paper are primarily probabilistic in nature, with all combinatorial objects we use having already appeared in the previous work [21].…”