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

TL;DR

The paper develops a universal framework for self-avoiding walks on infinite quasi-transitive cubic graphs under local vertex-replacements by finite gadgets. It proves a substitution identity for connective constants, $bc(G)^{-1}=gigl(bc(G_1)^{-1}igr)$, and extends this to bipartite graphs with $bc(G)^{-2}=higl(bc(G_{ ext{e}})^{-1}igr)$, where $h$ is determined by gadget data; it further shows the critical exponents $gamma$ and $eta$ are invariant and $nu$ is invariant under mild hypotheses. The results cover both single- and dual-color transformations, and include explicit gadget families such as replacing degree-3 vertices with $K_N$ graphs as well as generalized Fisher transformations. Overall, the work provides exact connective-constant relations and exponent-invariance results that extend the Fisher-transformation framework beyond triangles to general finite three-port gadgets, with broad implications for SAW universality on quasi-transitive graphs.

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 $h(x)=g_{φ_1}(x)\,g_{φ_2}(x)$ when both are transformed. We further present explicit families of examples, including replacing each degree-3 vertex by a complete-graph gadget $K_N$.

Figures (4)

• Figure 1.1: Local replacement at a degree-$3$ vertex: (a) the original star, (b) Fisher transformation (triangle gadget), and (c) replacement by a $K_4$ gadget. See also Definition 3.1 and Figure 3.1 in the draft.
• Figure 1.2: Local regular-polygon patches for the five Euclidean type-vectors of transitive TLF-planar cubic graphs.
• Figure 6.1: A local transformation consisting of a $K_7$ gadget. In this case, $g_7(x)=x^2+5x^3+20x^4+60x^5+120x^6+120x^7$.
• Figure 6.2: A generalized Fisher transformation: (a) a cycle graph $H$ on $N=7$ vertices with two distinguished vertices $v_1,v_2$ spaced $k=2$ edges apart and (b) the gadget $\widetilde{H}$ constructed according to Example \ref{['ex:GFT']}.

Theorems & Definitions (12)

• Definition 3.1: Local transformation
• Definition 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• proof : Proof of Theorem \ref{['thm:main2']}
• proof : Proof of Theorem \ref{["thm:main2'"]}
• proof : Proof of Theorem 3.5
• Example 6.1: Complete-graph gadget $K_N$
• Example 6.2: Generalized Fisher transformations