Session: Flows and Random Media<br>Organizer:
Friday, August 5, 2005 - 2:45pm - 4:30pm
- Spatial inhomogeneities and large scale behavior of the asymmetric exclusion process
Timo Seppalainen (University of Wisconsin, Madison)
This talk describes some results and open problems for the one-dimensional asymmetric exclusion process in situations where spatial inhomogeneities, either deterministic or random, are added to the model.
- Spectral asymptotics of Laplacians on bond-percolation graphs
Peter Mueller (Georg-August-Universität zu Göttingen)
Bond-percolation graphs are random subgraphs of the d-dimensional integer lattice generated by a standard Bernoulli bond-percolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, self-adjoint, ergodic random operators. They possess almost surely the non-random spectrum [0,4d] and a self-averaging integrated density of states. This integrated density of states is shown to exhibit Lifshits tails at both spectral edges in the non-percolating phase. Depending on the boundary condition and on the spectral edge, the Lifshits tail discriminates between different cluster geometries (linear clusters versus cube-like clusters) which contribute the dominating eigenvalues. Lifshits tails arising from cube-like clusters continue to show up above the percolation threshold. In contrast, the other type of Lifshits tails cannot be observed in the percolating phase any more because they are hidden by van Hove singularities from the percolating cluster.
- The pinning transition for a polymer in the presence of a random potential
Ken Alexander (University of Southern California)
We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume the probability of an excursion of length n from 0, in the absence of the potential, decays like n-c for some c>1. Disorder is introduced by, having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction, called the contact fraction, of its monomers at 0 with high probability. We obtain bounds for the contact fraction near the critical point and examine the effect of the disorder on the specific heat exponent, which describes the approach to 0 of the contact fraction at the critical point. Our results are consistent with predictions in the physics literature that the effect of disorder is quite different in the cases c3/2.