ColumbiaPrinceton Probability Day 2013
Friday, March 6, 2015 • Princeton University 
Title: Dissipation and High Disorder Abstract: The main goal of this talk is to describe when, and how, typical families of infinitelymany interacting diffusions are finite systems in the usual of particle systems. As it turns out, such systems are typically finite if and only if the system is “highly disordered” in a sense that will be made precise. This is joint work with Le Chen, Michael Cranston, and Kunwoo Kim.

Title: Generalized Smoluchowski Equations and Scalar Conservation
Laws Abstract: By a classical result of Bertoin, if initially a solution to Burgers' equation is a Levy process without positive jumps, then this property persists at later times. According to a theorem of Groeneboom, a white noise initial data also leads to a Levy process at positive times. Menon and Srinivasan observed that in both aforementioned results the evolving Levy measure satisfies a Smoluchowski–type equation. They also conjectured that a similar phenomenon would occur if instead of Burgers' equation, we solve a general scalar conservation law with a convex flux function. Though a Levy process may evolve to a Markov process that in most cases is not Levy. The corresponding jump kernel would satisfy a generalized Smoluchowski equation. Along with Dave Kaspar, we show that a variant of this conjecture is true for monotone solutions to scalar conservation laws.

Title: Displacement convexity of entropy and curvature in discrete
settings Abstract: Inspired by exciting developments in optimal transport and Riemannian geometry (due to the work of LottVillani and Sturm), several independent groups have formulated a (discrete) notion of curvature in graphs and finite Markov chains. I will describe some of these approaches briefly, mention some surprising byproducts (such as a tight Cheegertype inequality on abelian Cayley graphs), and several open problems of potential independent interest.

Title: The maximal particle of branching random walk in random
environment Abstract: We consider onedimensional branching random walk in a random branching environment and show that after recentering and scaling certain quantities related to the maximal particle exhibit normal fluctuations.

Title: Dyson's spike and spectral measure of groups Abstract: Consider the graph of the integers with iid edge weights. In 1953 Dyson showed that for exactly solvable cases even a small amount of randomness results in a logarithmic spike in the spectral measure. With Marcin Kotowski, we prove that this phenomenon holds in great generality. As a result, we find that the NovikovSchubin invariant can be zero even in Lie groups.

Title: Eigenfunctions of stochastic integrable particle systems Abstract: I will discuss interacting particle systems on the line which are solvable by the coordinate Bethe ansatz. A classical example is the Asymmetric Simple Exclusion Process whose celebrated Bethe ansatz solution is due to Tracy and Widom. The main focus of the talk is on properties of eigenfunctions and associated Fourierlike transforms coming from such particle systems. The most general eigenfunctions arising in this way are certain rational deformations of the HallLittlewood symmetric polynomials. They are associated with the qHahn particle system, and also with higher spin vertex models on the 2d lattice. The spectral theory allows to obtain explicit formulas for particle systems started from an arbitrary initial configuration. Some of these formulas are useful for establishing KardarParisiZhangtype asymptotics. 