Self-avoiding walks and the Fisher transformation

TL;DR

This paper investigates how the Fisher transformation affects self-avoiding walks on cubic graphs, showing that iterative application leads to converging connective constants and enabling exact calculations for certain lattices.

Contribution

It provides a detailed analysis of the Fisher transformation's impact on self-avoiding walks and derives explicit formulas for connective constants in transformed graphs.

Findings

Connective constants converge geometrically to the golden mean.

Fisher transformation preserves critical exponents across iterations.

Explicit connective constants are computed for transformed hexagonal lattices.

Abstract

The Fisher transformation acts on cubic graphs by replacing each vertex by a triangle. We explore the action of the Fisher transformation on the set of self-avoiding walks of a cubic graph. Iteration of the transformation yields a sequence of graphs with common critical exponents, and with connective constants converging geometrically to the golden mean. We consider the application of the Fisher transformation to one of the two classes of vertices of a bipartite cubic graph. The connective constant of the ensuing graph may be expressed in terms of that of the initial graph. When applied to the hexagonal lattice, this identifies a further lattice whose connective constant may be computed rigorously.