Are there any results on whether a large random graph with a given degree distribution is likely to be connected?

In Erdős-Rényi graphs with $n$ vertices and $m$ edges, we have two sudden transitions (for large $n$):

- A giant component appears above the threshold $m/n = 1/2$.
- The graph becomes connected above the threshold $m/n = (\ln n)/2$.

There is a result analogous to (1) above by Molloy and Reed for random graphs with a given degree distribution. If $d$ denotes the vertex degree and $\langle \cdot \rangle$ denotes the average, then the quantity of interest is $Q = \langle d^2 \rangle - 2\langle d \rangle$. A giant component suddenly appears above the threshold $Q > 0$.

**Question:** Is there a result analogous to (2) for random graphs with a fixed degree sequence, in the large graph limit? Is there a quantity that can be computed from the degree distribution, and when it crosses a threshold, the graph suddenly becomes connected (in the $n\rightarrow\infty$ limit)? Let us assume that there are no isolated vertices ($d\ne 0$).

**Clarification update:** Let me try to give a more precisely specified version of the problem. Suppose we have $n$ vertices. Of these, precisely $n_d = f_d n$ have degree $d$: thus we have a degree sequence
$$(
\overbrace{0,\dots,0}^{\text{$n_0$ times}},\;
\overbrace{1,\dots,1}^{\text{$n_1$ times}},\;
\overbrace{2,\dots,2}^{\text{$n_2$ times}}, \dots).
$$
Choose one simple (labelled) graph with this degree sequence uniformly at random.

What conditions do we need to have on the $f_d$ (the degree distribution), or on $n_d$, so that in the $n \rightarrow \infty$ limit the graph is connected with probability 1?

Clearly, if the $f_0 \ne 0$, then there are isolated vertices and the graph is not connected. Therefore, one condition is that $f_0 = 0$.