Stationary measure for six-vertex model on a strip

Abstract

We study the stochastic six-vertex model on a strip $$\left\{(x,y)\in\mathbb{Z}^2: 0\leq y\leq x\leq y+N\right\}$$ with two open boundaries. We develop a `matrix product ansatz' method to solve for its stationary measure, based on the compatibility of three types of local moves of any down-right path. The stationary measure on a horizontal path turns out to be a tilting of the stationary measure of the asymmetric simple exclusion process (ASEP) with two open boundaries. Similar to open ASEP, the statistics of this tilted stationary measure as the number of sites $N\rightarrow\infty$ (with the bulk and boundary parameters fixed) also exhibit a phase diagram, which is a tilting of the phase diagram of open ASEP. We study the limit of mean particle density as an example.

Figures (6)

• Figure 1: Outgoing edges on down-right paths and set of vertices $\mathop{\mathrm{U}}\nolimits(\mathcal{P},\mathcal{Q})$. The lower thick path is $\mathcal{P}$ and upper thick path is $\mathcal{Q}$. The gray edges are outgoing edges of $\mathcal{P}$ and $\mathcal{Q}$. Outgoing edges of $\mathcal{P}$ are labelled from the up-left of the path to the down-right of the path: $p_1=\rightarrow$, $p_2=\uparrow$, $p_3=\rightarrow$, $p_4=\uparrow$, $p_5=\uparrow$. The thick nodes are vertices in $\mathop{\mathrm{U}}\nolimits(\mathcal{P},\mathcal{Q})$.
• Figure 2: Sample configuration of stochastic six-vertex model on a strip. Down-right paths $\mathcal{P}$ and $\mathcal{Q}$ are the same as in Figure \ref{['fig:outgoing arrows down-right path']} and are omitted.
• Figure 3: Jump rates in the open ASEP, where we often take $L=q$ and $R=1$.
• Figure 4: (a) Phase diagram for open ASEP. (b) Phase diagram for six-vertex model on a strip.
• Figure 5: Upper translation of a horizontal path via three local moves.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Theorem 2.3
• proof
• Theorem 2.4: DEHP93
• Remark 2.5
• Remark 2.6
• Remark 2.7