Local statistics of lattice dimers

By Richard Kenyon. 2001

237 citations recorded by the source corpus.

Abstract We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy μ on the space of tilings of the plane with dominos. We construct a measure ν on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the ν-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for μ and ν, and compute the rate of convergence to mixing (the correlation between distant events). For the measure ν we compute the variance of the height function.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b939e3358997f96a3605ba63deb4059289a15469

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram (2001, background): “Local statistics of random domino tilings have been the subject of intense recent studies, see for example [3, 5, 6, 13] and the survey [14].”
Long-range properties of spanning trees (2000, methodology): “1 as stated in [12] is about perfect matchings.”
Random matrices and determinantal processes (2005, unknown): “To a path π in G(λ) from (−L, 1 − j) to (K, 1 − j), for some j ≥ 1, we can associate points Qi(π) = (i−L, xi), 0 ≤ i ≤ K+L.”
Random planar curves and Schramm-Loewner evolutions (2003, background): “Rick Kenyon [62, 64] had proved that LERW (and UST’s) have conformally invariant features exploiting the relation between UST and dimer models (and some explicit computations).”
Dimers and amoebae (2003, background): “, ek} being in a random matching is the determinant of the corresponding submatrix of K, times the product of the edge weights [10].”
Secure multiparty computation of approximations (2001, unknown): “These include counting problems that arise naturally in physics, such as the number of tilings of certain lattices, or problems concerning the bond strength of molecules [ 25 ].”
Random Skew Plane Partitions and the Pearcey Process (2005, methodology): “in the bulk of the disordered region, the correlation are given by the incomplete beta kernel [12, 15] with the parameters determined by slope of the limit shape (a special case of this is the discrete sine kernel); 2.”
An introduction to the dimer model (2003, background): “We should also point out that versions of Theorems 1 and 2 hold for arbitrary planar graphs, see [9] and [10].”
Lectures on Dimers (2009, background): “5 Local statistics There is an important corollary to Theorem 2: Corollary 3 ([11]).”
A variational principle for domino tilings (2000, background): “By (a straightforward extension of) Theorem 6 of [ Ke1 ], we have”
Probability on Trees and Networks (2017, background): “The expected number of the four 2-neighbors of 0 that lie on the ray in T that starts at 0 is 5/4, as shown by Poghosyan, Priezzhev, and Ruelle (2011) and Kenyon and Wilson (2015). One may consider the uniform spanning tree on 2 embedded in 2. In fact, consider it on ε2 in 2 and let ε → 0. In a”
Non-intersecting paths, random tilings and random matrices (2000, background): “A dimer covering of this lattice can also be thought of as a certain cylindrical rhombus tiling, see [45].”
Zeros of Gaussian Analytic Functions and Determinantal Point Processes (2009, background): “We state without proof the following theorem of Richard Kenyon (46).”
Determinantal point processes (2018, result): “olumns parameterized by the edges of G as follows: K(e,e′) = K−1(w,b′), where wis the white vertex on the edge e, and b′ is the black vertex on the edge e′. The next claim follows from the results of [Ken97]. Theorem 7.1 Consider the uniform measure on the dimer covers of G as a random point process”