Dimers and imaginary geometry

Abstract

We show that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate, thereby establishing a strong form of universality. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations.The proof relies on a connection to imaginary geometry, where the scaling limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field. In particular, we obtain an explicit construction of the a.s. unique Gaussian free field coupled to a continuum uniform spanning tree in this way, which is of independent interest.

Non-integrable Dimers: Universal Fluctuations of Tilted Height Profiles

“…By using the bounds (6.60) on the weighted L 1 norm of W (h L ) n,m,i,q;ω,r , we obtain (6.138). Similarly, using the corresponding pointwise bound 14 , analogous to (6.134), namely…”

“…143) 14 More precisely, the norm in the left side of (6.143) is a mixed norm, pointwise in y and L 1 in x. In order to obtain (6.143), we proceed as follows: we start from the appropriate analogue of (6.132), which has a factor 2 h(2−m−n/2) instead of 2 h (2−m) , and where h = h L ; next, we keep the factor 2 h L (−n/2) on a side and manipulate the rest of the expression as discussed after (6.132), thus obtaining the right side of (6.143).…”

“…Let us mention also [1], which obtains convergence to the GFF (with interaction-dependent amplitude) for a dimer model with a special nonlocal interaction that makes it integrable, although not determinantal. Finally, a very interesting recent development is [14]: while in this work the convergence to the GFF is proven only for the non-interacting dimer model, the method of proof, that goes through Temperley's bijection and Wilson's algorithm rather than via Kasteleyn's theory, might prove robust enough to allow for extensions to some non-determinantal situations.…”

(1+𝜀) moments suffice to characterise the GFF

“…The Gaussian free field (GFF) is a universal object believed (and in many cases proved) to govern the fluctuation statistics of many natural random surface models [10,18,17,12,6,3,2,7,16] (see, e.g., [1,20] for an introduction and survey of some recent developments). Although the GFF can be defined in any dimension, this article is concerned with the planar continuum version, which satisfies two special properties; namely, conformal invariance and a domain Markov property.…”

Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces

“…Since the seminal works of Kenyon [11,12] and Kenyon, Okounkov and Sheffield [16,17] it is either rigorously known or predicted that in many setups the fluctuations of the height function become Gaussian in the socalled scaling limit δ → 0 when a sequence of subgraphs G δ of a periodic grid (e.g., square or honeycomb) of mesh size δ approximate a given planar domain Ω; e.g., see [2,3,8,10,13,14,[19][20][21][22][23] and references therein. It is worth emphasizing that the identification of the two-point correlation is often nontrivial: it should be thought of as the Green function of the Laplacian (with Dirichlet boundary conditions) in a certain non-trivial metric in Ω, which depends on the concrete setup; see [16,17] or [10,14] for details.…”