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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 exponent Categories: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, representations Categories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35 |
3. CMB 1998 (vol 41 pp. 166)
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 |
4. CMB 1997 (vol 40 pp. 19)
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 |