Origin of the term "connective constant"

Question

Let $G$ be a vertex-transitive locally finite graph and $c_n$ the number of self-avoiding walks in $G$ starting from some fixed vertex $v_0$. One can easily see that $c_{m+n} \leq c_m c_n$ and hence Fekete's subadditive lemma gives that there exists the limit $\mu := \lim_{n\to\infty} c_n^{1/n}$. This quantity is known in the literature as the connective constant of $G$.

My question is basically what is the source of this term? I.e. it seems to be a strange choice of words for this quantity, so is there some application where it plays a role in some kind of "connectedness" which would excuse the name?

Answer 1

Q: Is there some application where $\mu$ plays a role in some kind of "connectedness" which would excuse the name?

A: The application is to crystalline structure. The name originates from Hammersley, who introduced [1,2] the "connective constant" to characterize the bonds in a crystal.

In a certain sense [the connective constant] measures the richness of the connexions in a crystal.

For example, if each atom in a branching process has $M$ direct descendant atoms to each of which there is a one-way bond, then $\mu=\log M$.

[1] J. M. Hammersley, Percolation processes II. The connective constant, Proc. Camb. Phil. Soc. 53 (1957), 642–645.
[2] S. R. Broadbent and J. M. Hammersley, Percolation processes (1957).