Research

1. Short time large deviations of the KPZ equation (with Li-Cheng Tsai), [arXiv:2009.10787] Accepted by Communications in Mathematical Physics (2021)

Abstract We establish the Freidlin–Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a 5/2 law for the deep lower tail for the KPZ equation under the short time scaling.

2. Lyapunov exponents of the half-line SHE, 2020, [arXiv:2007.10212] Submitted

Abstract We prove an upper tail large deviation of the half-line KPZ equation, with rate function to be 2/3s3/2 .

3. Lyapunov exponents of the SHE for general initial data (with Promit Ghosal), 2020 [arxiv:2007.06505] Submitted

Abstract We consider the (1+1)-dimensional stochastic heat equation (SHE) with multiplicative white noise and the Cole-Hopf solution of the Kardar-Parisi-Zhang (KPZ) equation. We show an exact way of computing the Lyapunov exponents of the SHE for a large class of initial data which includes any bounded deterministic positive initial data and the stationary initial data. As a consequence, we derive exact formulas for the upper tail large deviation rate functions of the KPZ equation for general initial data.

4. The stochastic telegraph equation limit of the stochastic higher spin six vertex model [arXiv:2005.00620] [journal version] Electronic Journal of Probability (2020), Vol. 25, no. 148, 1-30.

Abstract In this paper, we prove that the stochastic telegraph equation arises as a scaling limit of the stochastic higher spin six vertex (SHS6V) model with general spin I/2,J/2. This extends results of Borodin and Gorin which focused on the I=J=1 six vertex case and demonstrates the universality of the stochastic telegraph equation in this context. We also provide a functional extension of the central limit theorem obtained in [Borodin and Gorin 2019, Theorem 6.1].

5. KPZ equation limit of stochastic higher spin six vertex model. Mathematical Physics, Analysis and Geometry (2020), Vol 23, no. 1, 1-118 [arXiv:1905.11155] [journal version]

Abstract We consider the stochastic higher spin six vertex (SHS6V) model introduced in [Corwin-Petrov, 2016] with general integer spin parameters I,J . Starting from near stationary initial condition, we prove that the SHS6V model converges to the KPZ equation under weakly asymmetric scaling.

6. Markov duality for stochastic six vertex model Electronic Communications in Probability (2019), Vol 24, no. 67, 1-17 [arXiv:1901.00764] [journal version]

Abstract We prove that Schütz’s ASEP Markov duality functional is also a Markov duality functional for the stochastic six vertex model. We introduce a new method that uses induction on the number of particles to prove the Markov duality.

7. Second order behavior of the block counting process of beta coalescents (with Bastien Mallein) Electronic Communications in Probability (2017), Vol 22, no. 61, 1-8 [arXiv:1606.06998] [journal version]

Abstract The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg proved a law of large numbers for this quantity. Recently, Limic and Talarczyk proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-time branching processes.