The enumeration of independent sets on some lattices

TL;DR

This paper investigates the entropy constants of independent sets on various lattices, establishing equivalences for some and deriving bounds for more complex structures using a new transfer multiplicity concept.

Contribution

It introduces the concept of transfer multiplicity to analyze complex lattices and provides bounds for their entropy constants, extending previous methods.

Findings

Entropy constants are the same for certain plane, cylindrical, and toroidal lattices.

Bounds are obtained for crossed quadratic, generalized aztec diamond, and 8-8-4 lattices.

Transfer multiplicity enables analysis of complex lattice structures.

Abstract

In this paper, firstly we show that the entropy constants of the number of independent sets on certain plane lattices are the same as the entropy constants of the corresponding cylindrical and toroidal lattices. Secondly, we consider three more complex lattices which can not be handled by a single transfer matrix as in the plane quadratic lattice case. By introducing the concept of transfer multiplicity, we obtain the lower and upper bounds of the entropy constants of crossed quadratic lattice, generalized aztec diamond lattice and 8-8-4 lattice.