Benjamin Grant | Searcharxiv

arXiv2026

Topologizing infinite quivers and their mutations

We define several topological spaces whose points are quivers with a given infinite vertex set $X$. In the special case when $X$ is countably infinite, we show that two of the spaces of interest are homeomorphic to the Baire space $\mathbb{N}^\mathbb{N}$. We study properties of countably infinite quivers as subspaces of these topological spaces and prove a ``meta-theorem'' about hereditary properties of quivers. Furthermore, we approach the question of convergence for infinite mutation sequences in these spaces, providing a complete characterization of the (non-)density of the domains of convergence and divergence of infinite mutation sequences in one of these spaces and a partial characterization in the other. We then draw attention to a very special infinite quiver which we call the \emph{Fra\"iss\'e quiver} that draws a clear contrast between the behavior of finite and infinite mutation sequences. Finally, we reproduce (a very mild modification of) a previously-constructed topological space due to Ervin and Jackson as a subquotient of one of the spaces of interest.

math.CO↗

arXiv2026

Self-avoiding walks on cubic graphs and local transformations

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 $G_1=\phi(G)$, \[ \mu(G)^{-1}=g\bigl(\mu(G_1)^{-1}\bigr), \] equivalently $\mu(G_1)^{-1}$ is the unique solution $x\in(0,1)$ of $g(x)=\mu(G)^{-1}$, 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 $\gamma$ and $\eta$ (and $\nu$ 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 $\mu$ into infinite families of new quasi-transitive graphs whose connective constants are determined exactly as the unique roots of explicit algebraic equations.

math.CO↗