Combinatorics, topology and statistical mechanics of polymer models II
Organizer and Chair: Nicholas Beaton (University of Saskatchewan) and Andrew Rechnitzer (University of British Columbia)
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Entanglement of Dense Polymer Systems  [PDF]

Entanglement complexity of dense polymer systems has been investigated numerically in a number of articles. We have modelled a set of entangled chains confined to a tube by a system of self- and mutually avoiding walks (SSAW). We have defined a measure of entanglement complexity (the so-called EC) for dense polymer systems modelled by SSAWs. In this talk I will discuss a number of important properties of EC. I will also discuss some questions regarding dense polymer systems with a fixed topological structure such as the Unknot. This talk is based on a joint work with Chris Soteros.

NATHAN CLISBY, The University of Melbourne
Monte Carlo calculation of a new universal amplitude ratio for self-avoiding walks  [PDF]

We describe Monte Carlo simulations of self-avoiding walks and half-space walks which have led to accurate estimates of a new universal amplitude ratio, and the connective constant for various lattices.

Quenched Topology of Branched Polymers  [PDF]

To model adsorption of branched polymers onto a plane (within 3-space), we choose a random tree in the cubic lattice at high temperature (i.e. with uniform probability) and then decrease the temperature, so that the polymer adsorbs onto the plane without changing its topology. This contrasts with the more standard approach that "anneals" the topology instead of "quenching" it. A key question is which topologies are most likely; that is, for a fixed number of vertices, which underlying (abstract) trees have the most embeddings in the cubic lattice?

STEVE MELCZER, University of Waterloo \& ENS Lyon
Enumerating Lattice Paths Through Multivariate Diagonals  [PDF]

We examine the efficacy of encoding generating functions of lattice paths restricted to cones as diagonals of multivariate rational functions. By combining the \textit{kernel method} for lattice path problems with recent results in the field of analytic combinatorics in several variables, this approach allows us to determine general formulas for the dominant asymptotics of counting sequences of certain symmetric models restricted to $d$-dimensional orthants. After giving the results for "symmetric" models -- obtained by studying an algebraic variety related to the generating function -- we outline how they might be extended, and the difficulties involved in a more general analysis.

KOYA SHIMOKAWA, Saitama University
Unknotting operation and growth constant of knots in tube region  [PDF]

We will characterize unknotting operations of lattice knots in the 2x1 tube region. As an application we will discuss the growth constant of lattice knots in the tube.