Statistical Laboratory
Probability Seminars
Michaelmas Term 2001
All interested are encouraged
to take part to the full by presenting their ideas and discussing
those of others.
Talks will be in Meeting Room 12 of the
Centre for Mathematical Sciences, Pavilion D.
Tuesday 16 October
2pm Geoffrey Grimmett
Rigid interfaces in percolation/Ising/Potts models
Abstract. Dobrushin proved, in a remarkable and influential 1972 paper,
the rigidity of interfaces in the three-dimensional
Ising model, and the existence of non-translation invariant Gibbs states.
Guy Gielis and I can show that such results are valid for all
random-cluster
models, including percolation and all Potts models in three dimensions.
I will try to communicate the importance of
such results and the basic ideas of the proof.
Tuesday 23 October
2pm Malwina Luczak
Building uniform subtrees of a Cayley tree
Abstract: We prove the existence of, and describe, a (random) process
which builds subtrees of a Cayley tree one node at a time, in such a way
that the subtree created at stage $n$ is precisely a uniformly random
subtree of size $n$.
Tuesday 30 October
2pm Christina Goldschmidt
A fluid limit for an urn model
Abstract. We describe a method for proving fluid limit results for Markov jump
processes and give an example of its use.
Consider an urn with balls of N different colours under a particular
scheme of deletion and replacement. We investigate an
infinite-dimensional process arising from this model and prove a fluid
limit result for it.
Tuesday 6 November
2pm Thierry Lévy
Is Yang-Mills measure Gaussian?
Abstract.
The Yang-Mills measure is used in a very successful way in physical
theories such as Quantum Electrodynamics, which describes the interaction
of light and matter. Physicists usually take for granted that, in a
two-dimensional space-time, it is essentially a Gaussian object.
We present a construction of the Yang-Mills measure in two dimensions, on
a lattice and on a continuous space-time, and discuss a result which
indicates in certain cases that the Gaussian point of view is
questionable.
Followed at 3.30pm in MR5 by: Oliver Johnson, Match Lengths and Quantum
Data Compression
Tuesday 13 November
2pm Alan Stacey
An introduction to random k-satisfiability
Tuesday 20 November
2pm Harry Kesten (Cornell)
Greedy lattice animals when negative contributions are
allowed
Abstract.
We consider a family {X(v): v \in Z^d} and define for any
finite subset \xi of Z^d,
S(\xi) = \sum_{v \in \xi} X(v).
We think of X(v) as some reward located at v and S(\xi) as the
total reward in \xi. Various problems lead to the study of the
following quantities
Earlier work by Cox, Griffin, Gandolfi, Kesten and Martin showed that
if the X(v) are nonnegative, then, under mild moment conditions
Tuesday 27 November
2pm Alison Etheridge (Oxford)
Grasshoppers everywhere: evolution in spatially continuous
populations
Abstract. This talk discusses the modelling of populations that evolve
in 2-dimensional continua. Although partially addressed by
classical stepping stone/coalescent models, some of the assumptions
of these models are violated in our context and there is concern
that their predictions might therefore be misleading. The first challenge
is to produce a class of spatially continuous models in which local rules
regulate global population size in such a way as to prevent `clumping'
and extinction. A possible ecological model was proposed by Bolker and
Pacala. We will present conditions under which models of Bolker-Pacala
type predict `stable' population dynamics and conditions under which the
population will die out.
Tuesday 22 January
2pm James Norris
Tuesday 29 January
2pm James Martin
Tuesday 19 February
2pm Martin Barlow (UBC, ENS)
Graduate students especially are urged to attend.
Directions to the Laboratory.
The CMS is reached by a path along the east
side of the Isaac Newton Institute in Clarkson Road.
Abstract. We extend work of Quas to analyse the Grassberger prefixes (the longest
substring not seen anywhere else in a random string) in the case of an
independent non-identical process. We analyse two particular cases, which
have an application in quantum systems with either Bose-Einstein or
Fermi-Dirac statistics, allowing us to extend Schumacher's quantum data
compression theorem to these systems. This talk is based on joint work
with Yuri Suhov, math-ph/0109023.
M_n := \max{S(\xi):|\xi| = n, \xi is a
self-avoiding walk on
Z^d$ starting at the origin},
N_n := \max {S(\xi): |\xi|=n, \xi is a
connected set in Z^d which contains the origin}
G_n:= \max {S(\xi): \xi is a
connected set of Z^d in [-n,n]^d}.
M:= \lim_{n \to \infty} \frac 1n M_n
N:= \lim_{n \to \infty} \frac 1n N_n
exist almost surely and in L^1. We describe
these earlier results for nonnegative X(v) and their application to
so-called \rho-percolation. We further discuss recent work with
Dembo and Gandolfi when X(v) is allowed to be
negative. First we discuss the existence of N =
\lim_{n \to \infty} N_n. Then we show that
(under some moment condition) there is a phase transition for
G_n. If N > 0 then G_n grows like n^d as n \to \infty and if
N <0, then G_n only grows like \log n.
Lent Term
to return to Geoffrey Grimmett's home page.