Probability of k-connectedness for a random graph with given degree sequence
The answer probably mildly depends on how wild the $d_i$ are. A random $k$-regular graph is $k$-connected whp for $3 \leq k \leq n-4$ according to slides by Wormald (page 12), so I would expect that if, say, $k \leq d_i \leq \ell$ for constants $k,\ell$ (or even $\ell$ growing mildly with $n$) then the graph is still $k$-connected whp. The proof shouldn't be difficult (for constant $\ell$) using the configuration model.

A 1-regular graph is never connected (for $n &gt; 2$), and a random 2-regular graph is, if I remember correctly, disconnected whp. If you only have a few degree 2 vertices then perhaps the graph is 2-connected whp – that would be my guess.