Seminar on Stochastic Processes
March 15-17, 2007
at the Fields Institute
Ted Cox (University of Syracuse)
Survival and coexistence for a stochastic Lotka-Volterra model
In 1999, Neuhauser and Pacala introduced a stochastic spatial version
of the Lotka-Volterra model for interspecific competition. In this
talk I will discuss some recent work with Ed Perkins analyzing this
model. Our approach, which works when the parameters of the process
are "nearly critical," is to (1) show that suitably scaled
Lotka-Volterra models converge to super-Brownian motion, and (2)
"transfer" information from the super-Brownian motion
back to the Lotka-Volterra models. We are able to show survival
and coexistence for certain parameter values this way.
Bob Griffiths (University of Oxford)
Diffusion processes and coalescent trees.
Diffusion process models for evolution of neutral genes have the
coalescent process underlying them. Models are reversible with transition
functions having a diagonal expansion in orthogonal polynomial eigenfunctions
of dimension greater than one, extending classical one-dimensional
diffusion models with Beta stationary distribution and Jacobi polynomial
expansions to models with Dirichlet or Poisson Dirichlet stationary
distributions. Another form of the transition functions is as a
mixture depending on the mutant and non-mutant families represented
in the leaves of the infinite- leaf coalescent tree.
Charles Newman (Courant Institute)
Percolation methods for spin glasses
Percolation methods, e.g., those based on the Fortuin-Kasteleyn
random cluster representation (of vacant and occupied bonds), have
been enormously important in the mathematical analysis of ferromagnetic
Ising models. There exists a Fortuin-Kasteleyn representation for
non-ferromagnetic Ising models (including spin glasses) but to date
that has not been terribly useful in the non-ferromagnetic context.
We will discuss why this is so and the prospects for this to change
in the future. Although our motivation is to study short-range models,
we may also describe the percolation situation in the mean-field
Sherrington-Kirkpatrick spin glass. Much of the talk is joint work
with Jon Machta and Dan Stein.
Kavita Ramanan (Carnegie Mellon)
Measure-valued Process Limits of Some Stochastic Networks
Markovian representations of certain classes of stochastic networks
give rise naturally to measure-valued processes. In the context
of two specific examples, we will describe some techniques that
have proved useful in obtaining limit theorems for such processes.
In particular, we will discuss the role of certain mappings, which
can be viewed as a generalization to the measure-valued setting
of the Skorokhod map that has been used to analyze stochastic networks
admitting a finite-dimensional representation. This talk is mainly
based on various joint works with Haya Kaspi, Lukasz Kruk, John
Lehoczky and Steven Shreve.
Balasz Szegedy (University of Toronto)
Limits of Discrete Structures
Take a family of discrete objects, define a limit notion on them
and take the topological closure of the family. We study the discrete
objects through the analytic properties of their closure. An example
for this strategy is classical ergodic theory by Furstenberg where
the discrete structures are subsets of intervals of the integers
and the limit objects are certain group invariant measures. This
theory, in particular, leads to various strengthenings of the famous
theorem by Szemeredi on arithmetic progressions. We present analogous
theories where the discrete objects are graphs or hypergraphs. Among
the applications we show various results about group invariant random
processes. The talk is based on joint works with Gábor Elek
and László Lovász.