Probability theory and Mathematical Physics:
KPZ universality class, interacting particle systems, random directed polymers, random walks in random environment, random matrix theory.
Facilitated exclusion process and Pfaffian Schur processes, with Jinho Baik, Ivan Corwin and Toufic Suidan, preprint. [Arxiv:1605.00525]
We study the Facilitated TASEP, an interacting particle system on the one dimensional integer lattice. We prove that starting from step initial condition, the position of the rightmost particle has Tracy Widom GSE statistics on a cube root time scale, while the statistics in the bulk of the rarefaction fan are GUE. This uses a mapping with last-passage percolation in a half-quadrant, which we study using the formalism of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either Tracy-Widom GSE distributed, Tracy-Widom GOE distributed, or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy-Widom distribution. We also obtain a two-dimensional crossover between GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. In terms of interacting particle systems, our results further probe the question of how first particles fluctuate for exclusion processes with downward jump discontinuities in their limiting density profiles. Through the Facilitated TASEP and a previously studied MADM exclusion process we deduce that cube-root time fluctuations seem to be a common feature of such systems. However, the statistics which arise are shown to be model dependent (here they are GSE, whereas for the MADM exclusion process they are GUE).
Fluctuations of the first particle in exclusion processes slides, video .
Random walks in Beta-distributed random environment slides.
KPZ scaling theory for integrable exclusion processes english, français.