论文标题
叶片瓷砖的混合时间Glauber动力学
The mixing time of the lozenge tiling Glauber dynamics
论文作者
论文摘要
这项工作的广泛动机是对可逆的，局部马尔可夫的动力学的严格理解，尤其是通过混合时间$ t_ {mix} $测量的界面速度。在$（d+1）$ - 尺寸设置中，$ d \ ge2 $，这在很大程度上是数学上未开发的领域，尤其是对于离散接口。另一方面，根据均值曲率运动的启发式和模拟，人们期望在任何维度上在订单$ \ time $ \ abor $ \ time $ \ timensive上均衡，并在任何维度上发生平衡，并在$δ\ to to $δ\ to to to lattice nattice nattice网上发生。 我们研究了平面有限域的块状块状的单翼Glauber动力学，被视为$（2+1）$ - 尺寸表面。固定措施是可允许的瓷砖的统一度量。在平衡下，按极限形状定理，高度函数集中为$δ\ to0 $，围绕确定性的轮廓$ ϕ $，这是表面张力功能的独特最小化器。尽管有一些部分数学结果，但猜想$ t_ {mix} =δ^{ - 2+o（1）} $已经证明了，仅在$ ϕ $是仿射函数的情况下，才证明了这一点。在这项工作中，我们证明了唯一假设下的猜想，即极限形状$ ϕ $不包含冷冻区域（方面）。
The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time $T_{mix}$. In the $(d+1)$-dimensional setting, $d\ge2$, this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics and simulations, one expects convergence to equilibrium to occur on time-scales of order $\approx δ^{-2}$ in any dimension, with $δ\to0$ the lattice mesh. We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as $(2+1)$-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem, the height function concentrates as $δ\to0$ around a deterministic profile $ϕ$, the unique minimizer of a surface tension functional. Despite some partial mathematical results, the conjecture $T_{mix}=δ^{-2+o(1)}$ has been proven, so far, only in the situation where $ϕ$ is an affine function. In this work, we prove the conjecture under the sole assumption that the limit shape $ϕ$ contains no frozen regions (facets).
