The stochastic six-vertex model speed process

TL;DR

This work analyzes the stochastic six-vertex model on the quadrant with step initial data and a second-class particle at the origin, proving almost-sure convergence of the second-class particle’s speed to a random limit U supported on $[oldsymbol{ u}^{-1},oldsymbol{ u}]$ with density proportional to $x^{-3/2}$. The authors adapt the ASEP speed-propagation framework, introducing a geometric domination for third-class particles and sharp hydrodynamic bounds for the height function via integrable-probability techniques, including Meixner determinantal ensembles and Fredholm determinants. As a corollary, they construct and analyze the stochastic six-vertex speed process, showing ergodicity and stationarity for the multi-class dynamics and establishing a deterministic mapping to the ASEP speed process. The results advance understanding of multi-class invariant measures in KPZ universality, offering a robust approach to quantify speed fluctuations and to connect height-function fluctuations to particle-system speeds in integrable models.

Abstract

For the stochastic six-vertex model on the quadrant $\mathbb{Z}_{\geq0}\times\mathbb{Z}_{\geq0}$ 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 speed process, whose law we show to be ergodic and stationary for the dynamics of the multi-class stochastic six-vertex process. The proof follows the scheme developed in [ACG23] for ASEP and requires the development of 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.

Figures (8)

• Figure 1: The six allowed configurations for the stochastic six-vertex model
• Figure 2: A possible sampling of the stochastic six-vertex model on the quadrant with step initial data. The height function is denoted in blue.
• Figure 3: The allowed configurations for the multi-class stochastic six-vertex model, where red lines represent class $i$ and blue lines represent class $j$ for $i < j$.
• Figure 4: Left panel: step initial conditions with a second-class particle at the origin. Black arrows denote first-class particles, while the grey arrow denotes the second-class particle. Dashed lines denote holes. Right panel: a simulation of this process on a 200 by 200 square with $b_1=0.3$ and $b_2=0.6$ and with the second-class particle in red.
• Figure 5: A sketch of the densities of the processes $\bm{\mathcal{B}}^{(1)}$ in black at times $0,S$ and $S+T$ and $\bm{\mathcal{B}}^\text{step}$ in blue at times $S$ and $S+T$. At time $S$ the process $\bm{\mathcal{B}}^{(1,2,3)}$ is given exactly by the maximum of the two processes $\bm{\mathcal{B}}^{(1)}$ and $\bm{\mathcal{B}}^{\text{step}}$, while at time $S+T$ it is at least the maximum of $\bm{\mathcal{B}}^{(1)}$ and $\bm{\mathcal{B}}^{\text{step}}$.

Theorems & Definitions (126)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3: Height Function
• Definition 1.4
• Corollary 1.5: Existence of the Speed Process
• Theorem 1.6: Geometric Stochastic Domination
• Proposition 1.7: Lower Tail Bound
• Proposition 1.8: Upper Tail Bound
• Remark 1.9
• Remark 1.10