Past Probability Seminars Spring 2020: Difference between revisions
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We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed $v_0 \neq 0$). In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not ``average out'' and so the specific instance of the environment chosen actually matters. | |||
The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps.'' This talk is based on joint work with Milton Jara. | The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps.'' This talk is based on joint work with Milton Jara. | ||
Revision as of 16:59, 4 September 2014
Fall 2014
Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.
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Thursday, September 11, Van Vleck B105, Melanie Matchett Wood, UW-Madison
Please note the non-standard room.
Title: The distribution of sandpile groups of random graphs
Abstract:
The sandpile group is an abelian group associated to a graph, given as
the cokernel of the graph Laplacian. An Erdős–Rényi random graph
then gives some distribution of random abelian groups. We will give
an introduction to various models of random finite abelian groups
arising in number theory and the connections to the distribution
conjectured by Payne et. al. for sandpile groups. We will talk about
the moments of random finite abelian groups, and how in practice these
are often more accessible than the distributions themselves, but
frustratingly are not a priori guaranteed to determine the
distribution. In this case however, we have found the moments of the
sandpile groups of random graphs, and proved they determine the
measure, and have proven Payne's conjecture.
Thursday, September 18, Jonathon Peterson, Purdue University
Title: Hydrodynamic limits for directed traps and systems of independent RWRE
Abstract:
We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed $v_0 \neq 0$). In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not ``average out and so the specific instance of the environment chosen actually matters.
The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps. This talk is based on joint work with Milton Jara.
Thursday, September 25, Sean O'Rourke, University of Colorado Boulder
Title: TBA
Abstract:
Thursday, October 2, Jun Yin, UW-Madison
Title: TBA
Abstract:
Thursday, October 9, No seminar due to Midwest Probability Colloquium
No seminar due to Midwest Probability Colloquium.
Thursday, October 16, TBA
Title: TBA
Abstract:
Thursday, November 6, Vadim Gorin, MIT
Title: TBA
Abstract:
Friday, November 7, Tim Chumley, Iowa State University
Title: TBA
Abstract:
Thursday, November 13, TBA
Title: TBA
Abstract: