[Paper Review] Tail estimates for the stationary stochastic six vertex model and ASEP

Benjamin Landon, Philippe Sosoe|arXiv (Cornell University)|Aug 31, 2023

Random Matrices and ApplicationsMathematics3 citations

TL;DR

This paper establishes sharp tail estimates for the height function in the stationary stochastic six vertex model and its degeneration to ASEP, using coupling techniques to derive matching upper and lower bounds with a tail exponent of $\frac{3}{2}$ for the upper tail, consistent with KPZ universality. The results extend to current and second class particle fluctuations in ASEP with stationary initial data.

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.}

Motivation & Objective

• To derive precise tail estimates for the height function in the stationary stochastic six vertex model in the moderate deviations regime.
• To establish that the upper tail exponent matches the KPZ universality class value of $\frac{3}{2}$, confirming predictions from random matrix theory.
• To extend these results to the asymmetric simple exclusion process (ASEP) via degeneration of the stochastic six vertex model.
• To analyze fluctuations of the current and location of a second class particle in ASEP with stationary (Bernoulli) initial data.
• To complement existing variance bounds by Balázs and Seppäläinen with exponential-scale tail estimates.

Proposed method

• Utilizes a coupling framework between different initial data in the stochastic six vertex model to control fluctuations without relying on contour integral formulas.
• Imposes a microscopic concavity condition on model parameters to ensure the validity of the coupling and tail estimates.
• Applies truncation techniques to time graphs of particle jump instructions in both ASEP and S6V models to enable convergence arguments.
• Employs a modified time graph for the truncated S6V model that converges to the ASEP time graph, ensuring convergence of second class particle dynamics.
• Uses deterministic estimates (Lemma C.1) to control parameter sensitivity in the coupling, particularly relating differences in particle positions to changes in jump rates.
• Relies on convergence in distribution of truncated processes to the untruncated ones, with uniformity in the error parameter $\varepsilon$.

Experimental results

Research questions

• RQ1 What is the correct tail exponent for the upper tail of the height function in the stationary stochastic six vertex model under moderate deviations?
• RQ2 Can upper and lower tail bounds of matching order be established for the height function in the S6V model under the microscopic concavity condition?
• RQ3 How do the tail estimates for the S6V model degenerate to yield results for ASEP with stationary initial data?
• RQ4 What are the tail behaviors of the current and second class particle location in ASEP at equilibrium?
• RQ5 Can the tail exponent of $\frac{3}{2}$ for the lower tail of the height function in S6V be matched with optimal bounds, or is it suboptimal?

Key findings

• The upper tail of the height function in the stationary stochastic six vertex model has matching upper and lower bounds with tail exponent $\frac{3}{2}$, consistent with KPZ universality.
• An upper bound for the lower tail of the height function is obtained with the same exponent $\frac{3}{2}$, though the optimal exponent is conjectured to be $3$.
• The results for the S6V model hold under a microscopic concavity condition on model parameters, which ensures the validity of the coupling construction.
• The tail estimates for the S6V model survive degeneration to ASEP, allowing the derivation of analogous tail bounds for the current and second class particle location in ASEP.
• The second class particle location in ASEP with stationary initial data is shown to satisfy tail estimates of order $\frac{3}{2}$, complementing the cube-root fluctuation results of Balázs and Seppäläinen.
• The convergence of truncated and modified time graphs ensures that the second class particle dynamics in the S6V model converge to those in ASEP, with high probability as $\varepsilon \to 0$.