[Paper Review] Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$

Alexander Glazman|arXiv (Cornell University)|Feb 21, 2014

Theoretical and Computational Physics23 references3 citations

TL;DR

This paper introduces a weighted self-avoiding walk model on a skewed $θ$-rhombus lattice ($\mathbb{Z}^2$) with plaquet weights derived from the Yang-Baxter equation and parafermionic observables. It proves that for a critical family of weights parametrized by $\theta \in [\frac{\pi}{3}, \frac{2\pi}{3}]$, the connective constant is exactly $\frac{1}{u_1}$, with explicit formulas for $u_1, u_2, v, w_1, w_2$ in terms of $\theta$, generalizing the honeycomb lattice result of Duminil-Copin and Smirnov.

ABSTRACT

We consider a self-avoiding walk on the dual $\mathbb{Z}^2$ lattice. This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. We consider a family of critical weights parametrized by angle $θ\in[\fracπ{3},\frac{2π}{3}]$. For $θ=\fracπ{3}$, this can be mapped to the self-avoiding walk on the honeycomb lattice. The connective constant in this case was proved to be equal to $\sqrt{2+\sqrt{2}}$ by Duminil-Copin and Smirnov in \cite{DS10}. We generalize their result.

Motivation & Objective

• To extend the exact solution of the connective constant for self-avoiding walks from the honeycomb lattice to a broader class of weighted walks on a skewed $\mathbb{Z}^2$ lattice.
• To identify a critical family of weights satisfying the Yang-Baxter equation and parafermionic observables' consistency condition.
• To prove that the exponential growth rate of weighted walks (the connective constant) is exactly $1/u_1$ for this family of weights.
• To provide explicit, closed-form expressions for the critical weights $u_1, u_2, v, w_1, w_2$ in terms of the angle $\theta$.

Proposed method

• The model is defined on a rhombic lattice with plaquet configurations (empty, single arc, straight line, two arcs) each assigned a weight depending on the angle $\theta$.
• The partition function is defined as the sum of weights over all self-avoiding walks starting from a fixed edge midpoint.
• The connective constant is derived as the limit $\lim_{n\to\infty} \sqrt[n]{\tilde{c}_n} = 1/u_1$, where $\tilde{c}_n$ is the weighted sum of walks with $n$ arcs.
• Critical weights are derived from the Yang-Baxter equation and parafermionic observable consistency, ensuring the observable satisfies a discrete holomorphicity condition.
• The weights are expressed explicitly using trigonometric functions of $\theta$, with $u_1$ as the inverse of the connective constant.
• The solution is validated by showing that for $\theta = \pi/3$, the model reduces to the honeycomb lattice case with known connective constant $\sqrt{2 + \sqrt{2}}$.

Experimental results

Research questions

• RQ1 Can the exact solution for the connective constant of self-avoiding walks be extended beyond the honeycomb lattice to a broader class of weighted models on $\mathbb{Z}^2$?
• RQ2 What family of weights on a rhombic lattice ensures the parafermionic observable is discrete holomorphic and satisfies the Yang-Baxter equation?
• RQ3 Is there a critical weight regime for which the connective constant is exactly $1/u_1$ for a continuous family of angles $\theta \in [\pi/3, 2\pi/3]$?
• RQ4 How do the weights derived from the Yang-Baxter equation and parafermionic observables relate to the known critical weights for the $O(n)$ model?
• RQ5 Can the model be generalized to allow different lengths for $\theta$-arcs and $(\pi-\theta)$-arcs while preserving the exact connective constant?

Key findings

• For $\theta = \pi/3$, the model reduces to the honeycomb lattice, and the connective constant is $\sqrt{2 + \sqrt{2}}$, consistent with Duminil-Copin and Smirnov's result.
• The critical weights $u_1, u_2, v, w_1, w_2$ are explicitly given in terms of $\theta$ via trigonometric expressions involving $\sin$ functions.
• The connective constant is exactly $1/u_1$, with $u_1$ given by $\frac{\sin(5\pi/4)\sin(5\pi/8 + 3\theta/8)}{\sin(5\pi/4 + 3\theta/8)\sin(5\pi/8 - 3\theta/8)}$.
• The weights satisfy the Yang-Baxter equation and ensure the parafermionic observable is discrete holomorphic, enabling exact solvability.
• The solution remains valid even when $\theta$-arcs and $(\pi - \theta)$-arcs are assigned arbitrary positive integer lengths, preserving the connective constant as $1/u_1$.
• For $\theta = \pi/2$, the weights are symmetric: $u_1 = u_2$ and $w_1 = w_2$, reflecting lattice rotational invariance.