On the New Dimer lambda_d x-Expansion, Triangular and Hexagonal Lattices too

TL;DR

This paper explores a new formal series expansion for the dimer problem's lambda_d parameter, proposing it may have better asymptotic properties than previous inverse d expansions, with applications to triangular and hexagonal lattices.

Contribution

Introduces and analyzes an x-expansion method for lambda_d, applied to triangular and hexagonal lattices, offering potentially improved asymptotic behavior over traditional expansions.

Findings

x-expansion shows promising asymptotic properties

Application to triangular and hexagonal lattices yields satisfactory results

Series may outperform inverse d expansions in asymptotic analysis

Abstract

In recent work the author presented a formal expansion for lambda_d associated to the dimer problem on a d-dimensional rectangular lattice. Expressed in terms of d it yielded a presumed asymptotic expansion for lambda_d in inverse powers of d. We also considered an expansion in powers of x, a formal variable ultimately set equal to 1. We believe this series has better asymptotic properties than the expansion in inverse powers of d. We discuss this, and apply the same method to the two-dimensional triangular and hexagonal lattices. Viewed as a test of the x-expansion the results on those two lattices are satisfactory, if not thoroughly convincing.