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

Selecting band combinations with thematic mapper data

A problem arises in making color composite images because there are 210 different possible color presentations of TM three-band images. A method is given for reducing that 210 to a single choice, decided by the statistics of a scene or subscene, and taking into full account any correlations that exist between different bands. Instead of using total variance as the measure for information content of the band triplets, the ellipsoid of maximum volume is selected which discourages selection of bands with high correlation. The band triplet is obtained by computing and ranking in order the determinants of each 3 x 3 principal submatrix of the original matrix M. After selection of the best triplet, the assignment of colors is made by using the actual variances (the diagonal elements of M): green (maximum variance), red (second largest variance), blue (smallest variance)

Does Work Stress Predict the Occurrence of Cold, Flu and Minor Illness Symptoms in Clinical Psychology Trainees?

Objectives: The present study examined the three/four-day lagged relationship between daily work stress and upper respiratory tract infection (URTI) and other minor illness symptoms. Methods: Twenty-four postgraduate clinical psychology trainees completed work stress, cold/flu symptoms and somatic symptoms checklists daily for four weeks. Results: Increases in work stress were observed two days prior to a cold/flu episode but not three or four days preceding a cold/flu episode. Work stress was unrelated to peaks in somatic symptom reporting. Conclusions: There was some evidence of a lagged relationship between work stress and symptoms, but not of the expected duration, suggesting that the relationship between work stress and URTI symptoms was not mediated by the immune system