Matching lower bound for the number of solutions in random k-SAT (k ≥ 3)

Establish a matching lower bound on the logarithm of the number of satisfying assignments of random k-CNF formulas for k ≥ 3—i.e., for formulas with m clauses drawn uniformly from all k-clauses over n variables—at clause densities where the interpolation method provides an upper bound that aligns with cavity method predictions. Specifically, derive a lower bound that matches the known interpolation upper bound, thereby pinning down the typical exponential order of the number of solutions in the satisfiable regime for random k-SAT.

Background

The paper notes that for random k-CNFs with k ≥ 3, the interpolation method from mathematical physics yields an upper bound on the number of satisfying assignments, and this upper bound matches predictions from the cavity method. However, the corresponding lower bound that would confirm the tightness of these predictions across the satisfiable regime has not been established.

This contrasts with the present work on random 2-SAT, where a precise central limit theorem for the logarithm of the number of solutions is proved. For k ≥ 3, verifying the replica symmetric prediction has only been achieved for certain soft variants and far below the satisfiability threshold, underscoring the gap in knowledge for the standard random k-SAT model.