Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Colored Line Ensembles for Stochastic Vertex Models

Amol Aggarwal, Alexei Borodin

Abstract

In this paper we assign a family of $n$ coupled line ensembles to any $U_q (\widehat{\mathfrak{sl}}_{n+1})$ colored stochastic fused vertex model, which satisfies two properties. First, the joint law of their top curves coincides with that of the colored height functions for the vertex model. Second, the $n$ line ensembles satisfy an explicit Gibbs property prescribing their laws if all but a few of their curves are conditioned upon. We further describe several examples of such famlies of line ensembles, including the ones for the colored stochastic six-vertex and $q$-boson models. The appendices (which may be of independent interest) include an explanation of how the $U_q (\widehat{\mathfrak{sl}}_{n+1})$ colored stochastic fused vertex model degenerates to the log-gamma polymer, and an effective rate of convergence of the colored stochastic six-vertex model to the colored ASEP.

Colored Line Ensembles for Stochastic Vertex Models

Abstract

In this paper we assign a family of coupled line ensembles to any colored stochastic fused vertex model, which satisfies two properties. First, the joint law of their top curves coincides with that of the colored height functions for the vertex model. Second, the line ensembles satisfy an explicit Gibbs property prescribing their laws if all but a few of their curves are conditioned upon. We further describe several examples of such famlies of line ensembles, including the ones for the colored stochastic six-vertex and -boson models. The appendices (which may be of independent interest) include an explanation of how the colored stochastic fused vertex model degenerates to the log-gamma polymer, and an effective rate of convergence of the colored stochastic six-vertex model to the colored ASEP.
Paper Structure (50 sections, 54 theorems, 258 equations, 23 figures)

This paper contains 50 sections, 54 theorems, 258 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Preface
  3. Colored Stochastic Six-Vertex Model
  4. Colored Line Ensembles
  5. Colored Line Ensembles for the $q=0$ Stochastic Six-Vertex Model
  6. Outline
  7. Notation
  8. Yang--Baxter Equation and Partition Functions
  9. Yang--Baxter Equation
  10. Height Functions and Partition Functions
  11. Nonsymmetric Functions
  12. Properties of $f$ and $G$
  13. Probability Measures and Matchings
  14. Ascending $fG$ Measures
  15. Matching Between Colored Stochastic Six-Vertex Models and $\mathbb{P}_{\mathop{\mathrm{fG}}\nolimits}^{\sigma}$
  16. ...and 35 more sections

Key Result

Theorem 1.6

Sample a colored six-vertex ensemble $\mathcal{E}$ on $\mathcal{D}_{M;N}$ according to the stochastic six-vertex model with $q = 0$; all parameters of $\bm{x}$ equal to $x$ and of $\bm{y}$ equal to $y$; and $\sigma$-entrance data. For each $c \in \llbracket 1, n \rrbracket$ define $H_c : \llbracket where $\mathfrak{h}_{\ge c}^{\leftarrow}$ is the colored height function with respect to $\mathcal{

Figures (23)

  • Figure 1: Depicted above are various degenerations of the $U_q (\widehat{\mathfrak{sl}}_{n+1})$ colored fused stochastic vertex model.
  • Figure 2: Shown to the left is a vertex with arrow configuration $(a, b; c, d) = (2, 1; 1, 2)$, where red and blue are colors $1$ and $2$, respectively. Shown to the right is a colored model on the quadrant.
  • Figure 3: The $R_z$ weights are depicted above.
  • Figure 4: Shown above are two colored line ensembles. To the left, $\bm{\mathsf{L}}$ is simple; to the right, $\widehat{\bm{\mathsf{L}}}$ is not simple and is $\llbracket 1, 2 \rrbracket \times \llbracket 5, 6 \rrbracket$-compatible with $\bm{\mathsf{L}}$.
  • Figure 5: Depicted above are the $L_{x;s}$ weights.
  • ...and 18 more figures

Theorems & Definitions (154)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2: CSVMST
  • Definition 2.3
  • Definition 2.4
  • ...and 144 more