direct reason for thinking thus.

t.asym If $\gh_n$ are finite connected graphs with bounded average degree whose random weak limit is a probability measure $\rtd$ on infinite rooted graphs, then $$\lim_{n \to\infty} {1 \over |\verts(\gh_n)|} \log \cp(\gh_n)
=
\asym(\rtd)
\,.$$ The same limit holds in probability when $\gh_n$ are random with bounded expected average degree.

r.quotient If there is a homomorphism that is not an isomorphism from one transitive graph $\gh_1$ onto another $\gh_2$ of the same degree, then it is clear that $p_k(o;\gh_1) \le p_k(o; \gh_2)$ for every $k$, with strict inequality for some $k$. Therefore, $\asym(\gh_1) > \asym(\gh_2)$. In particular, among all transitive graphs $\gh$ of degree $d$, the maximum of $\asym(\gh)$ is achieved uniquely for $\gh$ the regular tree of degree $d$. This maximum value is calculated explicitly in \[x\].wired-free/. The uniqueness of the maximum, \[t\].asym/, and tightness imply that if a sequence of finite regular graphs $\gh_n$ of degree $d$ does not tend to the $d$-regular tree, $T$, then $\liminf_n |\verts(\gh_n)|^{-1} \log\cp(\gh_n) < \asym(T)$; this is Theorem 4.5 of \[b\].McKay/.

The only difficulty in deducing \[t\].asym/ from \[p\].finite/ is the interchange of limit and summation. In order to accomplish this, we shall use the following lemmas.

