Approximately counting independent sets of a given size in bounded-degree graphs

We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $alpha_c(Delta)$ and provide (i) for $alpha < alpha_c(Delta)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $alpha n$ in $n$-vertex graphs of maximum degree $Delta$; and (ii) a proof that unless NP=RP, no such algorithms exist for $alpha>alpha_c(Delta)$. The critical density is the occupancy fraction of hard core model on the clique $K_{Delta+1}$ at the uniqueness threshold on the infinite $Delta$-regular tree, giving $alpha_c(Delta)simfrac{e}{1+e}frac{1}{Delta}$ as $Deltatoinfty$.