Dominos and the Gaussian Free Field

By Richard Kenyon. 2000

266 citations recorded by the source corpus.

We define a scaling limit of the height function on the domino tiling model (dimer model) on simply connected regions in $\mathbf{Z}^2$ and show that it is the “massless free field,” a Gaussian process with independent coefficients when expanded in the eigenbasis of the Laplacian.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/229c6c96ac72f39669672d24870ee08b0c64930a

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Lectures on Dimers (2009, result): “Similar results were obtained [9] for uniform dimers on the square lattice, and (harder) uniform weights on these lattices for other EGMs [12].”
Contour lines of the two-dimensional discrete Gaussian free field (2006, background): “... the conformal invariance of the GFF, physicists and mathematicians have hypothesized that discrete random surface models that are believed or known to converge to the Gaussian free field (e.g., the discrete Gaussian free field, the height function of the oriented O(n) loop model with n = 2, heig”
An introduction to the dimer model (2003, background): “Theorem 5 ([12]) As ǫ tends to 0, h0 tends weakly in distribution to 4/ √ π times the massless free field GFF on U in the sense that for any smooth function ψ on U , the random variable ǫ(2) ∑ x∈Uǫ ψ(x)h0(x) tends in distribution to 4 √ π GFF (ψ).”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “Suppose that D ⊆ C is (a) SLE8 (b) SLE16 Time Figure 1.15: Simulations of SLEκ′ processes in [−1, 1]2 from i to −i for the indicated values of κ′.”
Universality in the 2D Ising model and conformal invariance of fermionic observables (2009, background): “[25, 27], were originally obtained on the square lattice.”
Non-intersecting paths, random tilings and random matrices (2000, result): “This type of results have been proved in other tiling models by Kenyon, see [45], [46], [ 47 ].”
Towards conformal invariance of 2D lattice models (2007, background): “Rick Kenyon [16], [17] established conformal invariance of many observables related to dimer models (domino tilings), in particular to uniform spanning tree and loop erased random walk, but stopped short of constructing the limiting curves.”
Imaginary geometry I: interacting SLEs (2012, background): “Just as many random walk models have Brownian motion as a scaling limit, many random (real or integer valued) functions on two dimensional lattices have the GFF as a scaling limit [1,11,18,23,25].”
SLE and the free field: partition functions and couplings (2007, methodology): “In the double domino model, the field jumps by ±4 at the marked point and, according to [17], converges to 4 √ 2 √ π times a standard free field.”
Gaussian free fields for mathematicians (2003, background): “(To be precise, [20] studies random discrete height functions h —for which the lattice spacing is —and shows that for smooth density functions ρ, (h , ρ) converges in law to (h, ρ) = (h,− −1ρ)∇ .”
Localization and delocalization of random interfaces (2005, background): “This turns out to be quite delicate, and the only rigorous works related to this problem, of which I am aware are [73, 74, 57].”