Self-avoiding walk, connective constant, cubic graph, Fisher transformation, quasi-transitive graph

Abstract: We study self-avoiding walks (SAWs) on infinite quasi-transitive cubic graphs under \emph{local transformations} that replace each degree-$3$ vertex by a finite, symmetric three-port gadget. To each gadget we associate a two-port SAW generating function $g(x)$, defined by counting SAWs that enter and exit the gadget through prescribed ports. Our first main result shows that, if $G$ is cubic and $G_1=φ(G)$ is obtained by applying the local transformation at every vertex, then the connective constants $μ(G)$ and $μ(G_1)$ satisfy the functional relation [ μ(G)^{-1}=g\bigl(μ(G_1)^{-1}\bigr). ] We next consider critical exponents defined via susceptibility-type series that do not rely on an ambient Euclidean dimension, and prove that the exponents $γ$ and $η$ are invariant under local transformations; moreover $ν$ is invariant under a standard regularity hypothesis on SAW counts (a common slowly varying function). Our second set of results concerns bipartite graphs, where the local transformation is applied to one colour class (or to both classes, possibly with different gadgets). In this setting we obtain an analogous relation [ μ(G)^{-2}=h\bigl(μ(G_{\mathrm e})^{-1}\bigr), ] with $h(x)=xg(x)$ when only one class is transformed and $h(x)=g_{φ1}(x)\,g{φ_2}(x)$ when both are transformed. We further present explicit families of examples, including replacing each degree-3 vertex by a complete-graph gadget $K_N$.