May 12 - 2000:

On May 12 2000, the Mark Kac seminar will meet again. The schedule for these talks is

11:15-13:00: Mario Wüttrich (Nijmegen):

Brownian motion in a scaled Poissonian potential.

14:15-16:00: Rob van den Berg (CWI):

Absence of phase transition for the monomer-dimer model, with application to random sampling.

The meeting will take place in room 220 of the James Boswell Institute, Bijlhouwerstraat 6, Utrecht.

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ABSTRACT MARIO WÜTTRICH

We consider d-dimensional Brownian motion among a scaled soft Poissonian potential \beta (\log t)^{-2/d} V. The scaling (\log t)^{-2/d} is chosen to be of critical order, i.e.it is determined by the typical size of the largest hole of the d-dimensional Poissonian cloud in the box (-t,t)^d. The scaled Poissonian potential has the effect that it is partially absorbing the diffusing particles. We study the quenched path measure of a diffusing particle conditioned on the event that it has not been absorbed up to time t. In the investigation of the large-time behavior of these quenched path measures, one is naturally led to consider the principal Dirichlet eigenvalue of the corresponding Schrödinger operator on the box (-t,t)^d. We prove that this (random) principal Dirichlet eigenvalue (properly rescaled) converges to a deterministic limit I(\beta). I(\beta) can be described by a (deterministic) variational principle. Analyzing this variational principle, we get a phase transition in dimensions d >= 4: There is
a critical constant \beta_c(d)>0 such that
        I(\beta)= \beta  for \beta <= \beta_c(d),
        I(\beta)< \beta  for \beta > \beta_c(d).
For d<4 we prove that this phase transition takes place at \beta_c(d)=0. Further we provide critical exponents and the large-\beta-behavior of I(\beta).

ABSTRACT ROB VAN DEN BERG

The monomer-dimer model originates from Statistical Physics, where it has been used to study the absorption of oxygen molecules on a surface, and the properties of a binary mixture. In the early seventies Heilmann and Lieb proved that, in a certain sense, this model has no phase transition. Their proof has a strong analytical flavour. We prove a very similar result by a probabilistic argument which (in my opinion) is more intuitively appealing. Moreover, this approach, in combination with a result of Jerrum and Sinclair, can be used to obtain random samples for this model in an (asymptotically) more efficient way. Part of the results are based on joint work with Rachel Brouwer.