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Search: MSC category 60K35 ( Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] )

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1. CMB 2012 (vol 57 pp. 113)

 A Lower Bound for the End-to-End Distance of Self-Avoiding Walk For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$, we prove that the mean-square end-to-end distance is at least $N^{4/(3d)}$ times a constant. This implies that the associated critical exponent $\nu$ is at least $2/(3d)$, assuming that $\nu$ exists. Keywords:self-avoiding walk, critical exponentCategories:82B41, 60D05, 60K35

2. CMB 2011 (vol 56 pp. 13)

 Ordering the Representations of $S_n$ Using the Interchange Process Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec$ on the irreducible representations of $S_n$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the Aldous order'' completely is a generalized question. We show a few additional entries for this order. Keywords:Aldous' conjecture, interchange process, symmetric group, representationsCategories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35
 Percolation on Penrose tilings In Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely~0 or~1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of $\hbox{\Bbbvii Z}^d$, and to other percolation processes, including Bernoulli bond percolation. Categories:60K35, 82B43
 Lattice trees and super-Brownian motion This article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion called {\it integrated super-Brownian excursion\/} ($\ISE$), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is $\ISE$, in all dimensions. A connection is drawn between $\ISE$ and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point. Categories:82B41, 60K35, 60J65