1. CMB 2018 (vol 61 pp. 518)
 Hong, Kyungpyo; Oh, Seungsang

Bounds on multiple selfavoiding polygons
A selfavoiding polygon is a lattice polygon consisting of a
closed selfavoiding walk on a square lattice.
Surprisingly little is known rigorously about the enumeration
of selfavoiding polygons,
although there are numerous conjectures that are believed to
be true
and strongly supported by numerical simulations.
As an analogous problem of this study,
we consider multiple selfavoiding polygons in a confined region, as a model for multiple ring polymers in physics.
We find rigorous lower and upper bounds of the number $p_{m \times
n}$
of distinct multiple selfavoiding polygons in the $m \times
n$ rectangular grid on the square lattice.
For $m=2$, $p_{2 \times n} = 2^{n1}1$.
And, for integers $m,n \geq 3$,
$$2^{m+n3}
\left(\tfrac{17}{10}
\right)^{(m2)(n2)} \ \leq \ p_{m \times n} \ \leq \
2^{m+n3}
\left(\tfrac{31}{16}
\right)^{(m2)(n2)}.$$
Keywords:ring polymer, selfavoiding polygon Categories:57M25, 82B20, 82B41, 82D60 

2. CMB 2012 (vol 57 pp. 113)
 Madras, Neal

A Lower Bound for the EndtoEnd Distance of SelfAvoiding Walk
For an $N$step selfavoiding walk on the hypercubic lattice ${\bf Z}^d$,
we prove that the meansquare endtoend 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:selfavoiding walk, critical exponent Categories:82B41, 60D05, 60K35 

3. CMB 2011 (vol 56 pp. 13)
 Alon, Gil; Kozma, Gady

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 

4. CMB 2011 (vol 55 pp. 858)
 von Renesse, MaxK.

An Optimal Transport View of SchrÃ¶dinger's Equation
We show that the SchrÃ¶dinger equation is a lift of Newton's third law
of motion $\nabla^\mathcal W_{\dot \mu} \dot \mu = \nabla^\mathcal W F(\mu)$ on
the space of probability measures, where derivatives are taken
with respect to the Wasserstein Riemannian metric. Here the potential
$\mu \to F(\mu)$ is the sum of the total classical potential energy $\langle V,\mu\rangle$
of the extended system
and its Fisher information
$ \frac {\hbar^2} 8 \int \nabla \ln \mu ^2
\,d\mu$. The precise relation is established via a wellknown
(Madelung) transform which is shown to be a symplectic submersion
of the standard symplectic
structure of complex valued functions into the
canonical symplectic space over the Wasserstein space.
All computations are conducted in the framework of Otto's formal
Riemannian calculus for optimal transportation of probability
measures.
Keywords:SchrÃ¶dinger equation, optimal transport, Newton's law, symplectic submersion Categories:81C25, 82C70, 37K05 

5. CMB 2011 (vol 55 pp. 98)
 Glied, Svenja

Similarity and Coincidence Isometries for Modules
The groups of (linear) similarity and coincidence isometries of
certain modules $\varGamma$ in $d$dimensional Euclidean space, which
naturally occur in quasicrystallography, are considered. It is shown
that the structure of the factor group of similarity modulo
coincidence isometries is the direct sum of cyclic groups of prime
power orders that divide $d$. In particular, if the dimension $d$ is a
prime number $p$, the factor group is an elementary abelian
$p$group. This generalizes previous results obtained for lattices to
situations relevant in quasicrystallography.
Categories:20H15, 82D25, 52C23 

6. CMB 2009 (vol 40 pp. 19)
 Derbez, Eric; Slade, Gordon

Lattice trees and superBrownian motion
This article discusses our recent proof that above eight dimensions
the scaling limit of sufficiently spreadout lattice trees is the variant
of superBrownian motion called {\it integrated superBrownian excursion\/}
($\ISE$), as conjectured by Aldous. The same is true for nearestneighbour
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 meanfield 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 highdimensional percolation models at the critical point.
Categories:82B41, 60K35, 60J65 

7. CMB 2009 (vol 52 pp. 9)
 Chassé, Dominique; SaintAubin, Yvan

On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics
Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements
$[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the
symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{1})}$ with
$0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in
\sn$. We give the spectrum of $R_n$ and show that the ratio of the
largest eigenvalue $\lambda_0$ to the second largest one (in absolute
value) increases as a positive power of $n$ as $n\rightarrow \infty$.
Keywords:symmetric group, representation theory, eigenvalue, statistical physics Categories:20B30, 20C30, 15A18, 82B20, 82B28 

8. CMB 1998 (vol 41 pp. 166)
 Hof, A.

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 
