10:00 - 10:50 |
Tony Guttmann: Random and self-avoiding walks subject to tension and compression ↓ In recent years there have been important experiments involving the
pulling of polymers from a wall. These are carried out with atomic
force microscopes and other devices to determine properties of
polymers, including biological polymers such as DNA. We have studied a
simple model of this system, comprising two-dimensional self-avoiding
walks, anchored to a wall at one end and then pulled from the wall at
the other end. In addition, we allow for binding of monomers in
contact with the wall. The geometry is shown in the following figure:
There are two parameters in the model, the strength of the interaction
of monomers with the surface (wall), and the force, normal to the
wall, pulling the polymer. We have constructed (numerically) the
complete phase diagram, and can prove the locus of certain phase
boundaries in that phase diagram, and also the order of certain phase
transitions as the phase boundaries are crossed. A schematic of the
phase diagram is shown below. Most earlier work focussed on simpler
models of random, directed and partially directed walk models. There
has been little numerical work on the more realistic SAW model. A
recent rigorous treatment by van Rensburg and Whittington established
the existence of a phase boundary between an adsorbed phase and a
ballistic phase when the force is applied normal to the surface.
We give the first proof that this phase transition is first-order. As
well as finding the phase boundary very precisely, we also estimate
various critical points and exponents to high precision, or, in some
cases exactly (conjecturally). We use exact enumeration and series
analysis techniques to identify this phase boundary for SAWs on the
square lattice. Our results are derived from a combination of three
ingredients: (i) Rigorous results.
(ii) Faster algorithms giving extended series data.
(iii) New numerical techniques to extract information from the data.
A second calculation considers polymers squeezed towards a surface by
a second wall parallel to the surface wall. In this problem we ignore
the interaction between surface monomers and the wall. We find,
remarkably, that in this geometry there arises an unexpected stretched
exponential term in the asymptotic expression for the number of
configurations. We show explicitly that this can occur even if one
uses simple random walks as the polymer model, rather than the more
realistic self-avoiding walks. Aspects of this work have been carried
out with Nick Beaton, Iwan Jensen, Greg Lawler and Stu Whittington. (Conference Room San Felipe) |
15:00 - 15:50 |
Tom Kennedy: The first order correction to the exit distribution for some random walks ↓ We consider three random walk models on several
two-dimensional lattices - the usual nearest neighbor random walk, the
nearest neighbor random walk without backtracking and the smart
kinetic walk (a type of self-avoiding walk). For all these models the
distribution of the point where the walk exits a simply connected
domain in the plane converges weakly to the harmonic measure on the
boundary as the lattice spacing goes to zero. We study the first order
correction, i.e., the limit of the difference divided by the lattice
spacing. Monte Carlo simulations lead us to conjecture that this
measure has density cf(z) where the function f(z) only depends on the
domain and the constant c only depends on the model and the
lattice. So there is a form of universality for this first order
correction. For a particular random walk model with continuously
distributed steps we can prove the conjecture. (Conference Room San Felipe) |