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A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^{d}$

Youssef Lazar

TL;DR

This paper recasts the problem of counting self-avoiding walks on $\\mathbb{Z}^d$ as a lattice-point counting task in multidimensional domains, and uses Fourier analysis together with the Poisson summation formula to derive an asymptotic for $c_n$ (and its dimensionally extended form $c_n(d)$). The leading term is governed by the volume term corresponding to the zero Fourier mode, and an explicit high-dimensional integral involving products of $\\mathrm{sinc}$ and Bessel functions $J_1$ encodes the main behavior. The authors establish a unified framework that not only yields asymptotics for $c_n$ but also provides a route to the mean squared displacement of SAWs, offering insights transferable to lattices of any dimension $d\\ge 2$. While the exact evaluation of the resulting multidimensional integrals remains challenging, the method connects SAW enumeration to concrete analytically tractable objects and opens a path toward rigorous bounds and further dimensional generalizations.

Abstract

We reduce the problem of counting self-avoiding walks in the square lattice to a problem of counting the number of integral points in multidimensional domains. We obtain an asymptotic estimate of the number of self-avoiding walks of length $n$ in the square lattice. This new formalism gives a natural and unified setting in order to study the properties of the number of self-avoiding walks in the lattice $\mathbb{Z}^{d}$ of any dimension $d\geq 2$.

A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^{d}$

TL;DR

This paper recasts the problem of counting self-avoiding walks on as a lattice-point counting task in multidimensional domains, and uses Fourier analysis together with the Poisson summation formula to derive an asymptotic for (and its dimensionally extended form ). The leading term is governed by the volume term corresponding to the zero Fourier mode, and an explicit high-dimensional integral involving products of and Bessel functions encodes the main behavior. The authors establish a unified framework that not only yields asymptotics for but also provides a route to the mean squared displacement of SAWs, offering insights transferable to lattices of any dimension . While the exact evaluation of the resulting multidimensional integrals remains challenging, the method connects SAW enumeration to concrete analytically tractable objects and opens a path toward rigorous bounds and further dimensional generalizations.

Abstract

We reduce the problem of counting self-avoiding walks in the square lattice to a problem of counting the number of integral points in multidimensional domains. We obtain an asymptotic estimate of the number of self-avoiding walks of length in the square lattice. This new formalism gives a natural and unified setting in order to study the properties of the number of self-avoiding walks in the lattice of any dimension .

Paper Structure

This paper contains 18 sections, 5 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. The origins of the problem
  3. Set-up and goal of the paper
  4. Main results
  5. Reduction to a lattice point counting in multidimensional domains
  6. Arithmetical interpretation of $c_{n}$
  7. The geometry of the problem
  8. Poisson Summation Formula
  9. Proof of Theorem \ref{['main']}
  10. An approximation of $c_{n}$
  11. The Fourier transform of $\psi_{jk}$.
  12. An integral asymptotic formula for $c_n$ and end of the proof of Theorem \ref{['main']}
  13. How to obtain an exact Asymptotic estimate of $c_{n}$?
  14. The main term of the asymptotic expansion for $c_{n}$
  15. Some notations
  16. ...and 3 more sections

Key Result

Theorem 1.1

As $n$ gets large, the number $c_n$ of self-avoiding walks of length $n$ behaves as follows, where for each $\xi \in \mathbb{R}^{2n}$, and $J_{1}$ is the first order Bessel function of the first kind.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3