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1. CMB 2018 (vol 61 pp. 518)

Hong, Kyungpyo; Oh, Seungsang
 Bounds on multiple self-avoiding polygons A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding 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 self-avoiding 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 self-avoiding polygons in the $m \times n$ rectangular grid on the square lattice. For $m=2$, $p_{2 \times n} = 2^{n-1}-1$. And, for integers $m,n \geq 3$, $$2^{m+n-3} \left(\tfrac{17}{10} \right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \ 2^{m+n-3} \left(\tfrac{31}{16} \right)^{(m-2)(n-2)}.$$ Keywords:ring polymer, self-avoiding polygonCategories:57M25, 82B20, 82B41, 82D60

2. 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

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, representationsCategories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35

4. CMB 2011 (vol 55 pp. 858)

von Renesse, Max-K.
 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 well-known (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 submersionCategories: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 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

7. CMB 2009 (vol 52 pp. 9)

Chassé, Dominique; Saint-Aubin, 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 physicsCategories: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
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