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 the hard core model on the clique $K_{\Delta+1}$ at the uniqueness threshold on the infinite $\Delta$-regular tree, giving $\alpha_c(\Delta)\sim\frac{e}{1+e}\frac{1}{\Delta}$ as $\Delta\to\infty$. Our methods apply more generally to anti-ferromagnetic 2-spin systems and motivate new questions in extremal combinatorics.