Tail estimates for the stationary stochastic six vertex model and ASEP

TL;DR

The paper analyzes tail exponents for KPZ-class observables in the stationary stochastic six vertex model and, via degeneration, the ASEP. It develops a coupling-based probabilistic framework anchored by microscopic concavity and a Rains–EJS identity to obtain exponential tail bounds with the optimal $\frac{3}{2}$ exponent for the upper tail of the height and current, and related bounds for the lower and left tails. These results extend beyond one-point convergence, providing exponential tail information and enabling passage to ASEP to bound both current fluctuations and second-class particle positions at equilibrium. The work thus enhances the understanding of KPZ-tail universality in integrable and near-integrable systems, connecting height fluctuations, two-point functions, and particle-current statistics.

Abstract

This work studies the tail exponents for the height function of the stationary stochastic six vertex model in the moderate deviations regime. For the upper tail of the height function we find upper and lower bounds of matching order, with a tail exponent of $\frac{3}{2}$, characteristic of KPZ distributions. We also obtain an upper bound for the lower tail of the same order. Our results for the stochastic six vertex model hold under a restriction on the model parameters for which a certain "microscopic concavity" condition holds. Nevertheless, our estimates are sufficiently strong to pass through the degeneration of the stochastic six vertex model to the ASEP. We therefore obtain tail estimates for both the current as well as the location of a second class particle in the ASEP with stationary (Bernoulli) initial data. Our estimates complement the variance bounds obtained in the seminal work of Balázs and Seppäläinen.}

Figures (7)

• Figure 1: An example of a six vertex directed path ensemble.
• Figure 2: Weights of the possible vertex configurations of the stochastic six vertex model. The final configuration on the right is viewed as two paths bouncing off each other instead of crossing.
• Figure 3: An example of a six vertex directed path ensemble with second class particles. The boundary data $\eta_0$ with less particles has incoming arrows to the vertices $(2, 1)$ and $(4, 1)$. The boundary data $\xi_0$ includes also an incoming arrow at $(1, 2)$. The ensemble $\eta$ is given by only the black arrows while $\xi$ is the union of the black and grey arrows.
• Figure 4: Top row: weights associated to evolution of second class particle in absence of incoming black arrows at same vertex. Second row: deterministic evolution of second class arrow encountering vertex with an incoming black arrow. The grey arrow deterministically fills the empty outgoing edge. The cases where the grey arrow is incoming vertically from the south are similar.
• Figure 5: Antiparticle construction of second class particle. Top row indicates probabilities of exiting eastwards or upwards when the second class particle encounters a vertex with an additional incoming black arrow and, necessarily, two outgoing black arrows. Bottom row: deterministic evolution when second class particle encounters a vertex with no additional incoming arrow; it simply follows the outgoing arrow. Cases where second class particle enters from the south are similar.

Theorems & Definitions (31)

• Definition 2.1
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7
• Lemma 3.1
• Lemma 3.2
• Corollary 3.3
• Proposition 4.1