# Leonid Petrov : Spectral theory for interacting particle systems

I plan to discuss spectral theory-type results for several stochastic interacting particle systems solvable by the coordinate Bethe ansatz. These results include Plancherel type isomorphism theorems which imply completeness and biorthogonality statements for the corresponding Bethe ansatz eigenfunctions. These constructions yield explicit solutions (in terms of multiple contour integrals) for backward and forward Kolmogorov equations with arbitrary initial data. Some of the formulas produced in this way are amenable to asymptotic analysis. In particular, I will discuss the (stochastic) q-Hahn zero-range process introduced recently by Povolotsky, and also the Asymmetric Simple Exclusion Process (ASEP). In particular, the spectral theory provides a new proof of the symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration). Another degeneration takes the q-Hahn zero-range process to the stochastic q-Boson particle system dual to q-TASEP studied by Borodin, Corwin et al. Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar-Parisi-Zhang equation / stochastic heat equation, namely, q-TASEP and ASEP.

**Category**: Probability**Duration**: 01:49:49**Date**: September 11, 2014 at 4:20 PM**Views**: 111-
**Tags:**seminar, Probability Seminar

## 0 Comments