Stochastic PDE Limit of the Six Vertex Model

Abstract

We study a stochastic PDE limit of the height function of the dynamic asymmetric simple exclusion process (dynamic ASEP). A degeneration of the stochastic Interaction Round-a-Face (IRF) model of [Bor17], dynamic ASEP has a jump parameter q ∈ (0, 1) and a dynamical parameter α > 0. It degenerates to the standard ASEP height function when α goes to 0 or ∞. We consider very weakly asymmetric scaling, i.e., for ε tending to zero we set q = e −ε and look at fluctuations, space and time in the scales ε −1 , ε −2 and ε −4 . We show that under such scaling the height function of the dynamic ASEP converges to the solution of the space-time Ornstein-Uhlenbeck (OU) process. We also introduce the dynamic ASEP on a ring with generalized rate functions. Under the very weakly asymmetric scaling, we show that the dynamic ASEP (with generalized jump rates) on a ring also converges to the solution of the space-time OU process on [0, 1] with periodic boundary conditions.

KPZ Equation Limit of Stochastic Higher Spin Six Vertex Model

“…Referring to Proposition 7.1, each gradient on V (x 1 , x 2 ), (y 1 , y 2 ), t, s gives an extra decay of 1 √ t−s+1 , which helps us to conclude (A). We remark that for demonstrating (A), our argument is actually simpler than that of [CGST18]. Since we assume I ≥ 2, (1.16) holds for all x 1 ≤ x 2 , while in the situation of the stochastic six vertex model (I = 1), (1.16) holds only for x 1 < x 2 , due to the exclusion restriction (i.e.…”

“…In this section, we explain the method used in proving Theorem 1.6. Although initially our methods follow [CGST18], rather quickly, we encounter novel complexities that are not present in [CGST18] which require new ideas.…”

“…As illustrated in Section 1.2, via fusion, to study the fused SHS6V model, it suffices to work with the unfused version. Similar to [CGST18], the first step is to perform a microscopic Hopf-Cole transform of the SHS6V model (5.6). The existence of the microscopic Hopf-Cole transform is guaranteed by one particle version of the duality (3.8) (which goes back to [CP16, Theorem 2.21]).…”

“…two particles can not stay in the same site). In fact, [CGST18] needs both of the duality [CP16, Theorem 2.21] and [Lin19, Theorem 1.5] to prove (A).…”

“…For (B), there are two tasks: Identifying λ and proving the self-averaging. These were done simultaneously for the stochastic six vertex model [CGST18]: Note that by (1.15),…”