UVA Probability Seminar
Wednesdays, 3:30 - 4:30pm
Kerchof 326
 Organizers: Christian Gromoll & Tai Melcher
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# Fall 2012

29 Aug Organizational meeting Jonathon Peterson (Purdue) Large deviations for random walks in a random environment on a strip Dan Dobbs Two limit laws for an infinite-dimensional process Ajay Chandra* (joint with Math Physics, meets at 2p) Proving Scale Invariance of a Massless QFT over the p-adics (II) Ajay Chandra* (joint with Math Physics, meets at 2p) Proving Scale Invariance of a Massless QFT over the p-adics (III) Ajay Chandra* (joint with Math Physics, meets at 2p) Proving Scale Invariance of a Massless QFT over the p-adics (IV) Rick Durrett (Duke) Evolving voter model Weining Kang (UMBC) Diffusion approximation of an input-queued switch operating under a maximum weight matching policy Ivan Corwin* (Clay Mathematics Institute, MIT, and Microsoft Research -- joint with Math Physics, meets at 1p in Ker 317) From duality to determinants for ASEP and q-TASEP no seminar (Thanksgiving holiday) Jim Nolen (Duke) Homogenization and normal fluctuations of a random conductor
* please note time, date, and/or location change

# Abstracts

 Large deviations for random walks in a random environment on a strip Jonathon Peterson (Purdue) We consider large deviations of random walks in a random environment on the strip $\mathbb{Z} \times \{1,2,\ldots,d\}$. Large deviations for random walks in random environments have been studied in a variety of different types of graphs, but only in the one-dimensional nearest-neighbor case is there a known variational formula relating the quenched and averaged rate functions. We will generalize the argument for the one-dimensional case to that of a strip of finite width and prove quenched and averaged large deviation principles with a variational formula relating the two rate functions. The main novelty in our approach will be to use an idea of Furstenburg and Kesten to obtain probabilistic formulas for the limits of certain products of random matrices.top of page Two limit laws for an infinite-dimensional process Dan Dobbs The most well-known limit laws in probability are the law of large numbers and the central limit theorem. I will present two other limit laws: Chung's law of the iterated logarithm and the functional law of the iterated logarithm as well as their analogues for a process on certain Heisenberg-like groups.top of page Proving Scale Invariance of a Massless QFT over the p-adics (II) Ajay Chandra* (joint with Math Physics, meets at 2p) In a sequence of talks I will show how one can leverage correlation inequalities and the theory of Gibbs measures to prove scale invariance of a measure corresponding to a certain Renormalization Group fixed point. Necessary tools include Griffiths' inequalities for ferromagnetic Ising models, a result of Aizenman, Barsky, and Fernandez on the sharpness of certain phase transitions, and techniques developed by Ruelle along with Lebowitz and Presutti for "superstable" interactions in statistical mechanics. I will give short overviews of these tools and how we apply them. This is joint work with Abdelmalek Abdesselam and Gianluca Guadagni.top of page Evolving voter model Rick Durrett (Duke) In the evolving voter model we choose oriented edges (x,y) at random. If the two individuals have the same opinion, nothing happens. If not, x imitates y with probability 1-\alpha, and otherwise (i.e., with probability \alpha severs the connection with y and picks a new neighbor at random (i) from the graph, or (ii) from those with the same opinion as x. One model has a discontinuous transition, the other a continuous one. We will explain why this is true and describe some remarkable conjectures about the multi-opinion case.Most of the talk comes from "Graph fission in an evolving voter model" Proc. Nat'l. Acad. Sci. 109 (2012), 3682-3687, which is joint work with James Gleeson, Alun Lloyd, Peter Mucha, Feng Shi, David Sivakoff, Josh Socolar and Chris Varghese.top of page Diffusion approximation of an input-queued switch operating under a maximum weight matching policy Weining Kang (UMBC) In this talk we consider an N by N input-queued switch operating under a maximum weight matching policy. To analyze the system performance in the heavy traffic, we establish a diffusion approximation for the (2N-1)-dimensional workload process associated with this switch model when all input ports and output ports are heavily loaded. The diffusion process is a semimartingale reflecting Brownian motion living in a polyhedral cone with N^2 boundary faces, each of which has an associated constant direction of reflection. Its stationary distribution in terms of the solution to a second order PDE with boundary condition is discussed.top of page From duality to determinants for ASEP and q-TASEP Ivan Corwin* (Clay Mathematics Institute, MIT, and Microsoft Research -- joint with Math Physics, meets at 1p in Ker 317) I will show how Markov process dualities for certain integrable probabilistic systems (ASEP and q-TASEP) imply that moments solve variants of the delta Bose gas. For special initial data, these Bose gas equations can be solved explicitly via a nested contour integral ansatz originating in the theory of Macdonald processes. Taking a generating series of these moment formulas leads to Fredholm determinants characterizing one-point fluctuations in the original systems. This provides a rigorous discrete deformation of the physicist's (famously non-rigorous) polymer replica trick (for KPZ equation). This is based on joint work with Alexei Borodin and Tomohiro Sasamoto.top of page Homogenization and normal fluctuations of a random conductor Jim Nolen (Duke) I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. For example, if an electric potential is imposed at the boundary, some current will flow through the material. What is the net current? For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance, when the domain is large. I'll give an error estimate for this approximation in total variation; the estimate scales optimally with the domain size.top of page
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