publication . Preprint . 2021

Free boundary dimers: random walk representation and scaling limit

Qian, Wei; Berestycki, Nathanaël; Lis, Marcin;
Open Access English
  • Published: 07 Oct 2021
  • Publisher: HAL CCSD
  • Country: France
Abstract
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 $z>0$ to the total weight of the configuration. A bijection described by Giuliani, Jauslin and Lieb 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 $z>0$, the scaling limit of the height function is the Gaussian free field with Neumann (or free) boundary conditions, thereby answering a question of Giuliani et al.
Comment: 69 pages
Subjects
free text keywords: [MATH]Mathematics [math], [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Mathematics - Probability, Mathematical Physics
Related Organizations

3 In nite volume limit 29 3.1 Construction of the potential kernel in the in nite volume setting . . . . . . . . . . . 29 3.2 Coordinatewise mirror coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Suitability of coupling (proof of (3.2)) . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 A priori estimate on the gradient of the potential kernel . . . . . . . . . . . . . . . . 37 3.5 Convergence of Green's function di erences to gradient of potential kernel . . . . . . 38 Scaling limit of discrete derivative of potential kernel 41 4.1 Proof of Proposition 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Proof of Proposition 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Convergence to the Neumann Gaussian Free eld 51 5.1 In nite volume coupling function and its scaling limit . . . . . . . . . . . . . . . . . 51 5.2 Pfa ans and Kasteleyn theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3 Matchings and permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4 Moments of the height function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.5 Convergence of the height function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2(zi i [1] J. Aru, A. Sepulveda, and W. Werner. On bounded-type thin local sets of the two-dimensional gaussian free eld. Journal of the Institute of Mathematics of Jussieu, 18(3):591{618, 2019.

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