Uniqueness of maximal entropy measure on essential spanning forests

An essential spanning forest of an infinite graph $G$ is a spanning forest of $G$ in which all trees have infinitely many vertices. Let $G_n$ be an increasing sequence of finite connected subgraphs of $G$ for which $\bigcup G_n=G$. Pemantle's arguments imply that the uniform measures on spanning trees of $G_n$ converge weakly to an $\operatorname {Aut}(G)$-invariant measure $\mu_G$ on essential spanning forests of $G$. We show that if $G$ is a connected, amenable graph and $\Gamma \subset \operatorname {Aut}(G)$ acts quasitransitively on $G$, then $\mu_G$ is the unique $\Gamma$-invariant measure on essential spanning forests of $G$ for which the specific entropy is maximal. This result originated with Burton and Pemantle, who gave a short but incorrect proof in the case $\Gamma\cong\mathbb{Z}^d$. Lyons discovered the error and asked about the more general statement that we prove.Comment: Published at http://dx.doi.org/10.1214/009117905000000765 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tricolor percolation and random paths in 3D

We study "tricolor percolation" on the regular tessellation of R^3 by truncated octahedra, which is the three-dimensional analog of the hexagonal tiling of the plane. We independently assign one of three colors to each cell according to a probability vector p = (p_1, p_2, p_3) and define a "tricolor edge" to be an edge incident to one cell of each color. The tricolor edges form disjoint loops and/or infinite paths. These loops and paths have been studied in the physics literature, but little has been proved mathematically. We show that each p belongs to either the compact phase (in which the length of the tricolor loop passing through a fixed edge is a.s. finite, with exponentially decaying law) or the extended phase (in which the probability that an n by n by n box intersects a tricolor path of diameter at least n exceeds a positive constant, independent of n). We show that both phases are non-empty and the extended phase is a closed subset of the probability simplex. We also survey the physics literature and discuss open questions, including the following: Does p=(1/3,1/3,1/3) belong to the extended phase? Is there a.s. an infinite tricolor path for this p? Are there infinitely many? Do they scale to Brownian motion? If p lies on the boundary of the extended phase, do the long paths have a scaling limit analogous to SLE_6 in two dimensions? What can be shown for the higher dimensional analogs of this problem?Comment: 27 pages, 25 figure

The harmonic explorer and its convergence to SLE(4)

The harmonic explorer is a random grid path. Very roughly, at each step the harmonic explorer takes a turn to the right with probability equal to the discrete harmonic measure of the left-hand side of the path from a point near the end of the current path. We prove that the harmonic explorer converges to SLE(4) as the grid gets finer.Comment: Published at http://dx.doi.org/10.1214/009117905000000477 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ribbon Tilings and Multidimensional Height Functions

We fix $n$ and say a square in the two-dimensional grid indexed by $(x,y)$ has color $c$ if $x+y \equiv c \pmod{n}$. A {\it ribbon tile} of order $n$ is a connected polyomino containing exactly one square of each color. We show that the set of order-$n$ ribbon tilings of a simply connected region $R$ is in one-to-one correspondence with a set of {\it height functions} from the vertices of $R$ to $\mathbb Z^{n}$ satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph. Using these facts, we describe a linear (in the area of $R$) algorithm for determining whether $R$ can be tiled with ribbon tiles of order $n$ and producing such a tiling when one exists. We also resolve a conjecture of Pak by showing that any pair of order-$n$ ribbon tilings of $R$ can be connected by a sequence of local replacement moves. Some of our results are generalizations of known results for order-2 ribbon tilings (a.k.a. domino tilings). We also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems.Comment: 25 pages, 7 figures. This version has been slightly revised (new references, a new illustration, and a few cosmetic changes). To appear in Transactions of the American Mathematical Societ

Internal DLA and the Gaussian free field

In previous works, we showed that the internal DLA cluster on \Z^d with t particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and O(\sqrt{\log t}) if d > 2. This paper addresses "average error": in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r^{1-d/2} (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.Comment: 29 pages, minor revisio