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

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

“Of Pure European Descent and of the White Race”: Recruitment Policy and Aboriginal Canadians, 1939–1945

According to the records of the Indian Affairs Branch, just over 3,000 Status Indians voluntarily enlisted in the military forces of Canada during the Second World War. Of these, 213 were killed. These and an unknown number of other non-status Indian, Métis and Inuit men served in all three military branches, and in every theatre where Canadian ground, sea and air forces fought. However, virtually nothing is known of the military service performed by Canada’s Native population. In part, this reflects of the paucity of records available on Native soldiers. Personnel files did not include any mention of ethnicity and thus it will never be known exactly how many Aboriginal men served. The figures of the Indian Affairs Branch are suspect, only partial, and do not account for Métis, Non-Status Indians, or Inuit; nor do they include those conscripted under the National Resources Mobilization Act for service in Canada. Historians have tended to focus either on the operational side of the conflict, or on the political, social and economic upheaval of the home front. The recruitment and military service of the Aboriginal population fits somewhere in between, and has been nearly forgotten

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

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure