Richard Kenyon

A variational principle for domino tilings

We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within epsilon (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges.

Local statistics of lattice dimers

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.

The Laplacian and Dirac operators on critical planar graphs

Abstract.On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and Dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of the Dirac operator and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete holomorphic functions which, via convolutions, gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs.

The asymptotic determinant of the discrete Laplacian

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R^2. As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.

Planar dimers and Harnack curves

In this paper we study the connection between dimers and Harnack curves discovered in math-ph/0311005. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the space of Harnack curve of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates. We characterize Harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their Ronkin function with given boundary conditions.

Projecting the one-dimensional Sierpinski gasket

LetS⊂ℝ2 be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1.

Tiling a polygon with rectangles

The authors study the problem of tiling a simple polygon of surface n with rectangles of given types (tiles). They present a linear time algorithm for deciding if a polygon can be tiled with 1 * m and k * 1 tiles (and giving a tiling when it exists), and a quadratic algorithm for the same problem when the tile types are m * k and k * m.<<ETX>>

The construction of self-similar tilings

We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient λɛℂ (satisfying the necessary algebraic condition of being a complex Perron number).For any integerm>1 we show that there exists a self-similar tiling with 2π/m-rotational symmetry group and expansion λ if and only if either λ or λe2π∿/m is a complex Perron number for which e2π∿/m is in ℚ[λ], respectivelyQ[λe2πı/m].

Arithmetic construction of sofic partitions of hyperbolic toral automorphisms

For each irreducible hyperbolic automorphism

Spanning forests and the vector bundle Laplacian

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