Diffusive Scaling limit of stochastic Box-Ball systems and PushTASEP

Abstract: We introduce the Stochastic Box-Ball System (SBBS), a probabilistic cellular automaton that generalizes the classic Takahashi-Satsuma Box-Ball System. In SBBS, particles are transported by a carrier with a fixed capacity that may fail to pick up any given particle with a fixed probability $\epsilon$. This model interpolates between two known integrable systems: the Box-Ball System (as $\epsilon\rightarrow 0$) and the PushTASEP (as $\epsilon\rightarrow 1$). We show that the long-term behavior of SBBS is governed by isolated particles and the occasional emergence of short solitons, which can form longer solitons but are more likely to fall apart. More precisely, we first show that all particles are isolated except for a $1/\sqrt{n}$-fraction of times in any given $n$ steps and solitons keep forming for this fraction of times. We then establish the diffusive scaling limit of the associated "gap process," which tracks the distances between consecutive particles. Namely, under diffusive scaling, the gap processes for both SBBS (for any carrier capacity) and PushTASEP converge weakly to Semimartingale Reflecting Brownian Motions (SRBMs) on the non-negative orthant with explicit covariance and reflection matrices consistent with the microscale relations between these systems. Moreover, the reflection matrix for SBBS is determined by how 2-solitons behave. Our proof relies on a new, extended SRBM invariance principle that we develop in this work. This principle can handle processes with complex boundary behavior that can be written as "overdetermined" Skorokhod decompositions involving rectangular reflection matrices, which is crucial for analyzing the complex solitonic boundary behavior of SBBS. We believe this tool may be of independent interest.