Hamiltonian Cycles on a Random Three-coordinate Lattice

TL;DR

This paper studies the enumeration of Hamiltonian cycles on a specific type of random lattice and expresses the related partition function as an elliptic integral, advancing understanding of combinatorial structures in statistical models.

Contribution

It provides a formula for counting Hamiltonian cycles on a three-coordinate lattice and expresses the partition function as an elliptic integral, linking combinatorics and mathematical physics.

Findings

Number of Hamiltonian cycles as a function of vertices v

Partition function expressed as an elliptic integral

Insights into combinatorial structures in statistical models

Abstract

Consider a random three-coordinate lattice of spherical topology having 2v vertices and being densely covered by a single closed, self-avoiding walk, i.e. being equipped with a Hamiltonian cycle. We determine the number of such objects as a function of v. Furthermore we express the partition function of the corresponding statistical model as an elliptic integral.