The stochastic six-vertex model speed process

Abstract

For the stochastic six-vertex model on the quadrant ${\mathbb{Z}}_{\ge 0}\times {\mathbb{Z}}_{\ge 0}$ with step initial conditions and a single second-class particle at the origin, we show almost sure convergence of the speed of the second-class particle to a random limit. This allows us to define the stochastic six-vertex model speed process, whose law we show to be ergodic and stationary for the dynamics of the multi-class stochastic six-vertex process. To prove this, we develop precise bounds on the fluctuations of the height function of the stochastic six-vertex model around its limit shape using methods from integrable probability. As part of the proof, we also obtain a novel geometric stochastic domination result that states that a second-class particle to the right of any number of third-class particles will at any fixed time be overtaken by at most a geometric number of third-class particles, which is of independent interest.

58 pages, 9 figures. Updated to match published version

References in corpus (23)

• Stochastic six-vertex model
• Stochastic higher spin vertex models on the line
• KPZ equation limit of higher-spin exclusion processes
• Current Fluctuations of the Stationary ASEP and Six-Vertex Model
• Competition interfaces and second class particles
• The TASEP speed process
• The ASEP and determinantal point processes
• The motion of a second class particle for the tasep starting from a decreasing shock profile
• Lower tail of the KPZ equation
• A phase transition for competition interfaces
• Jeu de taquin dynamics on infinite Young tableaux and second class particles
• Anomalous shock fluctuations in TASEP and last passage percolation models
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
• KPZ equation limit of stochastic higher spin six vertex model
• Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model
• The lower tail of the half-space KPZ equation
• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• Upper tail bounds for stationary KPZ models
• On the asymmetric zero-range in the rarefaction fan
• On the collision between two PNG droplets
• The stochastic telegraph equation limit of the stochastic higher spin six vertex model
• Tail estimates for the stationary stochastic six vertex model and ASEP