Proof of the satisfiability conjecture for large k

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_*(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha<\alpha_*$, and unsatisfiable for $\alpha>\alpha_*$. We show that the threshold $\alpha_*(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the 1-RSB universality class

On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs

Let Omega_q=Omega_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Gamma_q be the graph with vertex set Omega_q where two vertices are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_{n,m;k}, k >= 2, the random k-uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Gamma_q is connected if d is sufficiently large and q >~ (d/log d)^{1/(k-1)}. This is optimal to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma_q is O(n) w.h.p, where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber Dynamics Markov Chain on Omega_q is ergodic w.h.p