The Integrable Snake Model

TL;DR

The paper introduces the integrable snake model, a Fibonacci-weighted generalisation of lozenge tilings defined via pure snake bijections on a discrete torus, and establishes exact solvability through Kasteleyn theory. It proves that the partition function and correlation functions admit explicit determinant representations, provided the weighting satisfies a positivity condition $\alpha^2-4\gamma\delta\ge 0$, and derives a rich determinantal structure for random snake configurations via pushforwards through the shape map. Through a sequence of scaling limits—$m_1\to\infty$, then $n\to\infty$, and a continuous-time limit on $\mathbb{R}\times\mathbb{Z}_n$—the model yields determinantal processes with kernels that generalise the extended discrete sine kernel and connect to non-colliding walkers and the exclusion process on a ring (ASEP), via a cyclic Karlin–McGregor formula and a traffic representation. These connections provide a new integrable-probability framework linking dimer-like models, bead/bead-like processes, and standard interacting particle systems, with explicit kernels and stationary laws. The results illuminate how combinatorial tiling structures underpin stochastic dynamics and offer tractable tools for analyzing multi-point correlations and scaling limits in related systems.

Abstract

A pure snake configuration is a bijection $σ:\mathbb{Z}^2 \to \mathbb{Z}^2$ containing no two-cycles and such that for each $x \in \mathbb{Z}^2$ we have $σ(x) \in \{ x , x+ \mathbf{e}^1, x+\mathbf{e}^2 , x- \mathbf{e}^2 \}.$ The non-trivial cycles of a pure snake configuration may be regarded as a collection of non-intersecting paths in $\mathbb{Z}^2$ that may travel right, up, or down (but not left) from a given vertex. Pure snake configurations are a generalisation of lozenge tilings, which are in natural correspondence with paths that only travel right or up. We introduce a partition function on a finite version of this model and study the probabilistic properties of random pure snake configurations chosen according to their contribution to this partition function. Under a suitable weighting, the model is integrable in the sense that we have access to explicit formulas for its partition function and correlation function. We utilise the integrable structure of this model in several applications through its various scaling limits, such as to prove a traffic representation of ASEP on the ring, generalising the analogous result for TASEP by the first author.

Figures (5)

• Figure 1: A generalised snake configuration. The snakelets are highlighted in red. If the two-cycles corresponding to snakelets were removed and replaced with fixed points, then we would obtain a pure snake configuration.
• Figure 2: The diagram illustrates one possible nest, $N$. The underlying snake configuration $\sigma$ is marked by the blue points and the snakelets are marked with red points.
• Figure 3: The red lines join the points in the same equivalence class, i.e. they represent the vertical gaps.
• Figure 4: The blue curve denotes the ellipse $E_{\gamma,\delta}$ and the dotted red curve is the circle $C_\beta$. The points on the blue curve of the form $1 + \gamma w + \delta w^{-1}$ where $w^{n} = (-1)^{\eta}$ are denoted in black. The points lying inside the red curve are $L_\eta$ and outside are $R_\eta$.
• Figure :

Theorems & Definitions (79)

• Theorem 1.1
• Theorem 2.1
• Remark 1
• Theorem 2.2
• Lemma 2.3
• Proposition 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8