AI 中文总结
研究任意连通有限图上离散高斯自由场热浴Glauber动力学的混合时间，证明从超调和初始条件出发时具有强单调性，得到尖锐混合剖面，适用于无极端连通性的图，包括三维及以上网格有限盒子。
AI 中文摘要
我们研究了任意连通有限图上离散高斯自由场的热浴Glauber动力学的收敛速率。我们证明，当从超调和初始条件出发时，该演化具有强单调性。这使得我们能够在图的大小发散时获得尖锐的混合剖面。更精确地，我们证明混合发生在时间 $\frac{1}{2λ}\log(\mathcal{E})$，窗口为 $\mathcal{O}(1/λ)$，其中 $λ$ 是图拉普拉斯算子的谱隙，$\mathcal{E}$ 是超调和初始条件的能量。该结果适用于任意不具有极端连通性（无论哪一方面）的图。特别地，它适用于 $d\geq 3$ 维的 $\mathbb{Z}^d$ 网格的有限盒子。
英文摘要
We study the convergence rate of the heat-bath Glauber dynamics for the Discrete Gaussian Free Field on arbitrary connected finite graphs. We show that, when starting from super-harmonic initial conditions, the evolution enjoys a strong form of monotonicity. This allows us to get a sharp mixing profile as the size of the graphs diverges. More precisely, we show that mixing occurs at time $\frac{1}{2λ}\log(\mathcal{E})$ with window $\mathcal{O}(1/λ)$, where $λ$ is the spectral gap of the graph Laplacian and $\mathcal{E}$ is the energy of the super-harmonic initial condition. This result holds for arbitrary graphs that do not exhibit extreme connectivity properties (one way or the other). In particular, it holds for finite boxes of the grid $\mathbb{Z}^d$, in dimension $d\geq 3$.
Comments21 pages, 7 figures, corrected author name
