Density of bridges for critical random regular graph

Question

Let $G_{n,d}$ be the space of all $d$-regular graphs with $n$ vertices. Now choose a graph from $G_{n,d}$ uniform at random. Once obtained do independent bond-percolation on it, i.e. keep an edge with probability $p$ and discard it with prob. $1-p$. It is known that this model has a critical point at $p_c = 1/(d-1)$. An interesting question is what is the average number of bridges or better the average density of bridges for $n\rightarrow \infty$. My numerical analysis suggests it is exactly $p_c$ for $n\rightarrow \infty$. That is what remains is a forest. How would one show this rigorously? Is the configuration model of use?