Is there any result that relates the minimum degree of a geometric random graph to its k-vertex connectivity? I read papers where they pose the condition $$d_{\rm min} \geq k$$ to imply that the graph is k-connected (vertex connectivity). Is that an approximation or an asymptotic result? In my case, I need, if not an exact result, a lower bound

$$P({\rm graph~is~connected}) \geq f(d_{\rm min}).$$

I could not find references so far.