Invariant random graphs with iid degrees in a general geography

Let D be a non-negative integer-valued random variable and let G = (V, E) be an infinite transitive finite-degree graph. Continuing the work of Deijfen and Meester (Adv Appl Probab 38:287–298) and Deijfen and Jonasson (Electron Comm Probab 11:336–346), we seek an Aut(G)-invariant random graph model with V as vertex set, iid degrees distributed as D and finite mean connections (i.e., the sum of the edge lengths in the graph metric of G of a given vertex has finite expectation). It is shown that if G has either polynomial growth or rapid growth, then such a random graph model exists if and only if ${\mathbb{E}D\,R(D)] < \infty}$ . Here R(n) is the smallest possible radius of a combinatorial ball containing more than n vertices. With rapid growth we mean that the number of vertices in a ball of radius n is of at least order exp(n ^c ) for some c > 0. All known transitive graphs have either polynomial or rapid growth. It is believed that no other growth rates are possible. When G has rapid growth, the result holds also when the degrees form an arbitrary invariant process. A counter-example shows that this is not the case when G grows polynomially. For this case, we provide other, quite sharp, conditions under which the stronger statement does and does not hold respectively. Our work simplifies and generalizes the results for ${G\,=\,\mathbb {Z}}$ in 4] and proves, e.g., that with ${G\,=\,\mathbb {Z}^d}$ , there exists an invariant model with finite mean connections if and only if ${\mathbb {E}D^{(d+1)/d}] < \infty}$ . When G has exponential growth, e.g., when G is a regular tree, the condition becomes ${\mathbb {E}D\,\log\,D] < \infty}$ .