Diffusive Scaling limit of stochastic Box-Ball systems and PushTASEP

TL;DR

This work introduces the Stochastic Box-Ball System (SBBS), a capacity-$c$ carrier with pickup failure probability $\varepsilon$, unifying the classic Box-Ball System (SBBS at $\varepsilon=0$) with PushTASEP (SBBS at $\varepsilon\uparrow1$). It establishes two diffusion-scale limits: (i) SBBS converges to a semimartingale reflecting Brownian motion (SRBM) on the Weyl chamber with covariance $\varepsilon I_d$ and a capacity-dependent reflection matrix $\hat{R}^{c,\varepsilon}$ under time scaling $1/(1-\varepsilon)$, and (ii) PushTASEP converges to an SRBM with covariance $I_d$ and reflection $R_{PT}$; these limits are connected by a commutative diagram, showing consistency between microscale soliton dynamics and macroscopic diffusive behavior. A key methodological advance is an extended SRBM invariance principle for overdetermined Skorokhod decompositions, enabling analysis when boundary behavior occurs on many boundary cells beyond the principal faces, with a weak $\mathcal{S}$-condition ensuring higher-order reflections vanish in the limit. The paper also proves that SBBS reduces to PushTASEP as $\varepsilon\to1$ and provides sharp soliton-time estimates (solitons occur for a $1/\sqrt{n}$ fraction of times) and precise $d=2$ asymptotics, revealing a persistent “solitonic bias” at the diffusive scale. Overall, the results unify SBBS and PushTASEP within a single diffusion-limit framework and introduce a versatile invariant-principle tool that may apply to other interacting-particle systems with complex boundary dynamics.

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 $ε$. This model interpolates between two known integrable systems: the Box-Ball System (as $ε\rightarrow 0$) and the PushTASEP (as $ε\rightarrow 1$). 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 show that under diffusive scaling, both SBBS (for any carrier capacity) and PushTASEP converge weakly to semimartingale reflecting Brownian Motions (SRBMs) on the Weyl chamber with explicit covariance and reflection matrices, which are consistent with the microscale relations between these systems. The reflection matrix for SBBS is determined by how 2-solitons behave and exhibit ``solitonic bias'' visible in the diffusive scale. 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, which is crucial for analyzing the complex solitonic interaction in SBBS. We believe this tool may be of independent interest.

Figures (4)

• Figure 1: Sample trajectories of SBBS with $\varepsilon=0, 0.01, 0.9$ with initial configuration $\eta_{0}$ starting with 10 consecutive 1's followed by 20 consecutive 0's followed by 5 consecutive 1's followed by 0's. At $\varepsilon=0$, SBBS coincides with BBS, and the 10-soliton and the 5-soliton swap their momentum with nonlinear scattering in position. At $\varepsilon=0.01$, some balls from these solitons are left behind, creating further interactions. At $\varepsilon=0.9$, initial soliton structures are forgotten rapidly, and the system behaves like a PushTASEP.
• Figure 2: A sample trajectory of SBBS with infinite capacity, $\varepsilon=0.1$, and initial configuration $\zeta_{0}$ starting with 200 consecutive 1's followed by 0's. The initial soliton falls apart due to the carrier's failure to pick up balls, and the long-term dynamics are governed by singleton balls performing independent random walks and the spontaneous emergence of short solitons.
• Figure 3: Left: Transition kernel for SBBS $\zeta^{c,\varepsilon}$ with $d=2$. Each nonzero transition is shown with the corresponding independent coin flips with failure probability $\varepsilon$ for the two balls as a binary string. Right: Sample paths of $\zeta^{c,\varepsilon}$ in diffusive scaling for capacities $c=1$ and $c=2$ with shared coin flips for all times. The mean reflection vector $\hat{R}^{c,\varepsilon}$ is $(0, \varepsilon)^{\top}$ for $c=1$ and $(1-\varepsilon, 1)^{\top}$ for $c= 2$, respectively. The vector for $c = 2$ is slanted more in the tangential direction, Demonstrating the stronger solitonic bias.
• Figure 4: Transition kernel for the gap process of the infinite-capacity 3-ball SBBS. The transition kernel at the origin is shown in the picture on the right. Four boundary cells are indicated by the regions shaded in yellow.

Theorems & Definitions (52)

• Theorem 2.1: PushTASEP is $\varepsilon\nearrow 1$ scaling limit of SBBS
• Remark 2.2: Unit capacity SBBS and PushTASEP
• Theorem 2.3: Solitons exist only for $1/\sqrt{n}$-fraction of times
• Corollary 2.4: Expected location of the balls
• Theorem 2.5
• Theorem 2.6: Diffusive scaling limit of SRBM
• Theorem 2.7: Diffusive scaling limit of PushTASEP
• Corollary 2.8: Unit capacity SRBM and PushTASEP in diffusive scaling
• Remark 2.9: Coordinate-wise recursive construction of SRBM on the Weyl chamber
• Definition 2.10: semimartingale reflecting Brownian Motion