June 9 - 2000
On June 9 2000, the Mark Kac seminar will meet again. These will be the last
talks in the Mark Kac seminar in the academic year 1999-2000.
The schedule for these talks is
11:15-13:00: Marek Biskup (Microsoft Research, Redmond):
Rigorous theory of partition function zeros in models with
a convergent contour expansion.
14:15-16:00: Janos Engländer (EURANDOM):
Superdiffusions--construction and long-term behaviour.
The meeting will take place in room 220 of the James Boswell Institute, Bijlhouwerstraat 6, Utrecht.
ABSTRACT MAREK BISKUP
In 1952, Lee and Yang proved that the complex zeros of the Ising
partition function (i.e., moment generating function of the
energy) lie all on the unit circle. They also proposed an
ambitious program to analyze phase transitions in terms of these
zeros. Until recently, their outline could not seriously be
pursued for the lack of explicit and/or rigorous information on
the positions of the zeros.
In my talk I will describe the recently developed technique which
provides the missing information in models amenable to
Pirogov-Sinai analysis. The result clarifies and, in many cases,
invalidates numerous non-rigorous findings that have spawned in the
last couple of years (mostly) as a result of the increased power
of computer simulations. As a warm-up, I will briefly review the
Lee-Yang circle theorem and also some semi-rigorous applications
to, e.g., Griffiths' singularities.
Concerning the classic results, sufficient background provide the
original Lee-Yang papers
Phys. Rev. 87 (1952) 404--409, 87 (1952) 410-419,
and also Ruelle's textbook
Statistical Mechanics: Rigorous Results, W.A. Benjamin, Inc.
The new result I will describe has been obtained
jointly with C.Borgs, J.Chayes, L.Kleinwaks and R.Kotecký.
A short announcement has just appeared in Phys. Rev.
Lett. 84 (2000) 4794-4798.
ABSTRACT JANOS ENGLÄNDER
The theory of measure-valued processes ("superprocesses") has
become one of the central topics in probability theory in the last 25
years, partly because of their connection to nonlinear PDE's and partly
because they pop up as rescaled limits of various stochastic models.
In this talk I will give an overview on some recent results I have
obtained with R. Pinsky
and K. Fleischmann concerning the construction and large time behavior
of superdiffusions with a branching mechanism which varies in space. In
this model the motion process is a diffusion which lives on an arbitrary
Euclidean domain and may reach the boundary or infinity in finite time
("explosion"). Also, we allow the branching term to vary in an
unbounded way, and we even consider a non-regular case ("single point
source").
Basic properties of the underlying motion process (such as
transience/recurrence and explosion/non-explosion) are obviously related
to the support behavior of the superprocess.
This relationship however becomes nontrivial for spatially dependent
branching and, surprisingly,
an auxiliary motion process is involved in the results.
In fact, a number of counterexamples reveal that in certain cases the
intuition gained from the approximating branching particle system is
completely wrong.
No prior exposure to super-Brownian motions or superdiffusions will be
assumed.