# Average Length of Cycles in Rectangular Lattice

## TL;DR

This paper analyzes the cycle structure in rectangular lattices using transfer matrix methods, deriving generating functions and asymptotic formulas for average cycle length for lattices up to size 7.

## Contribution

It introduces a bivariate generating function approach to count cycles and determine their average length in rectangular lattices, extending to larger widths.

## Key findings

- Average cycle length in 3×N lattice is approximately 3.166N + 0.961.
- Derived generating functions for lattices up to size 7.
- Method generalizes to larger lattice widths.

## Abstract

We study the number of cycles and their average length in $L\times N$ lattice by using classical method of transfer matrix. In this work, we derive a bivariate generating function $G_3(y, z)$ in which a coefficient of $y^i z^j$ is the number of cycles of length $i$ in $3\times j$ lattice. By using the bivariate generating function, we show that the average length of cycles in $3\times N$ lattice is $\alpha N + \beta + o(1)$ where $\alpha$ and $\beta$ are some algebraic numbers approximately equal to 3.166 and 0.961, respectively. We argue generalizations of this method for $L\ge 4$, and obtain a generating function of the number of cycles in $L\times N$ lattice for $L$ up to 7.