  We determine the asymptotics of the independence number of the random
$d$-regular graph for all $d \ge d_0$. It is highly concentrated, with
constant-order fluctuations around $n\alpha_* - c_*\log n$ for explicit
constants $\alpha_*(d)$ and $c_*(d)$. Our proof rigorously confirms the
one-step replica symmetry breaking heuristics for this problem, and we believe
the techniques will be more broadly applicable to the study of other
combinatorial properties of random graphs.
