Self-avoiding walks on cubic graphs and local transformations

Abstract

Despite its elementary definition, the self-avoiding walk (SAW) poses notoriously hard enumerative problems: exact connective constants are known for only a handful of infinite graphs, notably the honeycomb lattice \cite{ds}. We establish a general substitution principle for SAWs on infinite connected quasi-transitive cubic graphs under port-transitive vertex replacements, where each degree-$3$ vertex is replaced by a fixed finite three-port gadget. Writing $g(x)$ for the associated two-port SAW series, we prove that for $G_1=ϕ(G)$, \[ μ(G)^{-1}=g\bigl(μ(G_1)^{-1}\bigr), \] equivalently $μ(G_1)^{-1}$ is the unique solution $x\in(0,1)$ of $g(x)=μ(G)^{-1}$, thereby extending the Fisher-triangle relation of Grimmett--Li to arbitrary symmetric three-port gadgets. We also obtain the corresponding identity for bipartite graphs when one or both colour classes are transformed, and show that the critical exponents $γ$ and $η$ (and $ν$ under a standard regularity hypothesis) are invariant. For explicit gadget families, including complete-graph gadgets $K_N$ and Fisher-type constructions, these identities turn base graphs with known $μ$ into infinite families of new quasi-transitive graphs whose connective constants are determined exactly as the unique roots of explicit algebraic equations.