Colored stochastic vertex models with U-turn boundary

TL;DR

The paper addresses the problem of understanding colored stochastic vertex models with a U-turn boundary by constructing an integrable framework in which vertex weights satisfy the Yang–Baxter and reflection equations. It develops a comprehensive set of recursive relations for partition functions and provides an explicit evaluation in a special boundary case, connecting these recursions to the Noumi representation of the affine Hecke algebra of type $ ilde{C}_n$. The key contributions include the introduction of three rotated $R$-vertices, proof of their integrability equations, and two main recursive formulas for $f_{oldsymbol{ u}}^{oldsymbol{\sigma}}(m{x})$, which together yield a structured approach to colored path ensembles with U-turn boundaries. This work has potential implications for algebraic combinatorics and integrable probability through connections to affine Hecke algebras and colored path models.

Abstract

In this paper, we introduce a class of colored stochastic vertex models with U-turn right boundary. The vertex weights in the models satisfy the Yang-Baxter equations and the reflection equation. Based on these equations, we derive recursive relations for partition functions of the models.

Figures (20)

• Figure 1: $\Gamma$ vertex with spectral parameter $x$
• Figure 2: $\Delta$ vertex with spectral parameter $x$
• Figure 3: Boltzmann weights for a cap vertex with spectral parameter $x$, where $i \in[n]$
• Figure 4: Model configuration when $n=2$, $L=4$, $\sigma=(1,2)$, $\mu=(4,-2)$
• Figure 5: Boltzmann weights for a $\Gamma-\Gamma$ vertex with spectral parameters $x,y$, where $i,j\in \{0\}\cup [\pm n]$ and $i<j$

Theorems & Definitions (8)

• Proposition 2.1
• Proposition 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• Theorem 3.3
• proof : Proof of Theorem \ref{['Thm3.2']}
• proof : Proof of Theorem \ref{['Thm3.3']}