Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

Abstract

We study a reversible continuous-time Markov dynamics of a discrete (2+1)-dimensional interface. This can be alternatively viewed as a dynamics of lozenge tilings of the L × L torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in [15]: in terms of interlaced particles, a particle jump of length n that preserves the interlacement constraints has rate 1/(2n). This dynamics presents special features: the average mutual volume between two interface configurations decreases with time [15] and a certain one-dimensional projection of the dynamics is described by the heat equation [21].In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as L → ∞ to the solution of a non-linear parabolic PDE. The initial profile is assumed to be C 2 differentiable and to contain no "frozen region". The explicit form of the PDE was recently conjectured [13] on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg-Landau model [7,16], here the mobility coefficient turns out to be a non-trivial function of the interface slope.2010 Mathematics Subject Classification: 60K35, 82C20, 52C20

(2 + 1)-Dimensional Interface Dynamics: Mixing Time, Hydrodynamic Limit and Anisotropic KPZ Growth

“…8 • A fact that plays a crucial role in the proof of the hydrodynamic limit is that the PDE (3.3) contracts the L 2 distance D 2 (t) = dx(φ (1) (x, t)− φ (2) (x, t)) 2 between solutions. I believe this is not a trivial or general fact: in fact, to prove contraction [28], we use the specific form (3.7) of µ and the explicit expression of σ i,j for the dimer model. (Note that if the mobility were constant, as it is for the Ginzburg-Landau model, L 2 contraction would just be a consequence of convexity of the surface tension σ).…”

“…(Note that if the mobility were constant, as it is for the Ginzburg-Landau model, L 2 contraction would just be a consequence of convexity of the surface tension σ). I think it is an intriguing question to understand whether the identities (see [28,Eqs. (6.19)-(6.22)]) leading to dD 2 (t)/dt ≤ 0 have any thermodynamic interpretation.…”

Hydrodynamic Limit Equation for a Lozenge Tiling Glauber Dynamics

“…In the following of this section, we will implicitly assume that the domain U and the initial condition ψ 0 (·) are regular enough that (3.3) admits a unique classical solution ψ(u, t) that is C 1 in U × [0, ∞) where we recall that the domain U is closed. In the forthcoming [12] we explain how to extract such existence, uniqueness and smoothness statements from the existing literature (e.g. [13, Chap.…”

“…Under the equilibrium Gibbs measures π ρ the random variable k(η, v) is known to have exponential moments of all orders, so the situation looks promising. However, justifying the replacement of (5.31) with (5.32) looks definitely harder: following the H −1 method of [6], it appears that one needs some form of uniform integrability bound on k(η, v), uniformly in L. In the forthcoming work [12], in the case of periodic boundary conditions we manage to bypass this difficulty.…”

“…(3.13)) that allows to show (see Section 3.1.2) that the L 2 distance between solutions contracts with time. This is an important point in the program of rigorously proving the convergence towards the hydrodynamic limit equation, and we use this property crucially in our forthcoming work [12] in the periodic boundary condition setting. In fact, the idea of the H −1 method is to prove that the L 2 distance between the deterministic PDE and the randomly evolving interface stays close to zero at all times.…”