Connective constant ($\kappa$)

If $b_n$ counts the $n$-step self-avoiding walks from a fixed vertex of a transitive lattice, the connective constant is the exponential growth rateSubmultiplicativity of $b_n$ and the Fekete lemma applied to $\log b_n$ prove existence of this limit.

For an infinite connected locally finite graph with a uniform degree bound, let ${\sigma }_n(v)$ count its length-$n$ self-avoiding walks from $v$. The suprema satisfy ${\sigma }_{m+n}\le {\sigma }_m{\sigma }_n$, so the Fekete lemma applied to their logarithms gives a finite connective constant. Unlike the usual rooted definition on a vertex-transitive graph, this definition explicitly takes the supremum over roots.