On the Martin boundary for discrete TASEP

TL;DR

The paper classifies Gibbs measures for a five-vertex/quadrant system, coherent systems on a $p$-deformed Gelfand–Tsetlin graph, and the Martin boundary of discrete-time TASEP with geometric jumps, revealing a large family of measures parameterized by analytic functions in a space $\mathcal{F}$. It builds coherent systems $\{M_n^{\Phi}\}$ via Grothendieck polynomials and shows independence of these systems; it identifies a finite GT-type subfamily $\{M_n^{\mathcal{A},\mathcal{B}}\}$ for which LLN/CLT-type results hold with fluctuations tied to independent GUE eigenvalues. The analysis leverages integral representations and steepest-descent asymptotics of Grothendieck polynomials to connect to TASEP dynamics and to derive extremality for certain coherent systems. The work advances the understanding of the Martin boundary in this deformed setting and provides explicit asymptotic descriptions, including LLN/CLT and Gaussian fluctuations, while leaving open whether the constructed list is exhaustive. Overall, the study blends representation-theoretic deformation, integrable probability, and local limit predictions for 2D lattice models, with implications for local limits and universality in the five-vertex/tasep framework.

Abstract

We study a problem with three equivalent formulations: describing Gibbs measures for five-vertex model in quadrant; classifying coherent systems on a p-deformation of the Gelfand-Tsetlin graph related to Grothendieck polynomials; finding the Martin boundary for discrete time TASEP with p-geometric jumps. We find a wide family of the Gibbs measures, parameterized by certain analytic functions. A subset of our measures have probabilistic interpretation as interacting particle systems with fixed particles speeds. In contrast to previous related boundary problems, we find that admissible speeds are not arbitrary, but must be larger than $\frac{p}{1-p}$. For this subset we further establish Law of Large Numbers and Central Limit Theorem, connecting the fluctuations to families of independent GUE eigenvalues. As a consequence, the measures from the subset are extreme points of the Martin boundary. It remains open whether our list of measures is exhaustive.

Figures (6)

• Figure 1: Left panel: six vertex weights. Right panel: stochastic five-vertex weights.
• Figure 2: Left: boundary conditions in quadrant. Right: a configuration and a subdomain $\Omega'$. With given boundary conditions $\Omega'$ has two configurations of conditional probabilities $\frac{c_1^2 c_2^2}{c_1^2 c_2^2+a_1 a_2 b_1 b_2}$ and $\frac{a_1 a_2 b_1 b_2}{c_1^2 c_2^2+a_1 a_2 b_1 b_2}$ --- the second one vanishes for the five-vertex weights.
• Figure 4: Top half: A possible step of TASEP with geometric jumps from $Y(2)$ to $Y(3)$, where $\mu=(4,2)$ and $\lambda=(4,3,2)$. The probability of this step is $(1-p)^2p^3$. Bottom half: the partition function $Z_{3, \lambda/\mu}$ of the five-vertex model, with the same $\lambda,\mu$ as in the top half. The value of this partition function is $(1-p)^2p^3$.
• Figure 5: Weights of possible local vertex configurations in the five-vertex model.
• Figure 6: The five-vertex model configuration corresponding to a path starting from $\varnothing\to(2)\to (3,1)\to (5,1,0)\to (5,1,0,0)\to(5,3,0,0,0)\to\dots$.

Theorems & Definitions (79)

• Theorem A: Theorem \ref{['coherent-thm']} in the text
• Theorem B: Theorem \ref{['limittheo']} in the text
• Theorem C: Theorem \ref{['Theorem_extreme']} in the text
• Proposition 2.1: HJKSS21
• Proposition 2.2: Yel16
• Proposition 2.3: Yel16
• Proposition 2.4
• proof
• Proposition 2.5
• proof