Berestycki Nathanaël, Lis Marcin, Qian Wei
Universität Wien, Vienna, Austria.
Technische Universität Wien, Vienna, Austria.
Probab Theory Relat Fields. 2023;186(3-4):735-812. doi: 10.1007/s00440-023-01203-x. Epub 2023 May 16.
We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight to the total weight of the configuration. A bijection described by Giuliani et al. (J Stat Phys 163(2):211-238, 2016) relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a walk with transition weights that are negative along the free boundary. Yet under certain assumptions, which are in particular satisfied in the infinite volume limit in the upper half-plane, we prove an effective, true random walk representation for the inverse Kasteleyn matrix. In this case we further show that, independently of the value of , the scaling limit of the centered height function is the Gaussian free field with Neumann (or free) boundary conditions. It is the first example of a discrete model where such boundary conditions arise in the continuum scaling limit.
我们研究正方形晶格子图上的二聚体模型，其中边界规定部分（自由边界）上的顶点可能未匹配。每个这样未匹配的顶点称为单体，并对构型的总权重贡献一个固定的乘性权重。朱利亚尼等人（《统计物理杂志》163(2):211 - 238, 2016）描述的一种双射将此模型与一个标准二聚体模型相关联，但该标准二聚体模型是在一个非二分图上。这个二聚体模型的卡斯泰莱因矩阵描述了一种游走，其转移权重在自由边界上为负。然而，在某些假设下，特别是在上半平面的无限体积极限中满足这些假设时，我们证明了逆卡斯泰莱因矩阵的一种有效的、真正的随机游走表示。在这种情况下，我们进一步表明，与 的值无关，中心化高度函数的标度极限是具有诺伊曼（或自由）边界条件的高斯自由场。这是离散模型在连续统标度极限中出现此类边界条件的第一个例子。
