The Mezard-Parisi formula for diluted spin glasses: the Panchenko-Talagrand equality conjecture

Main theorem: for Poisson-diluted even-arity Ising models in the Panchenko-Talagrand class (factorized interactions $e^{\theta}=a(1+b\prod f_l)$ with the stated independence and positivity of moments of $-b$), with only first moments of the interaction and field, the limiting pressure exists and equals $\inf_r$ of the infimum of the Mezard-Parisi hierarchical functional of depth $r$. Covers Viana-Bray, symmetric diluted even-spin models and soft random even-$K$ SAT at positive temperature. Not covered: odd arity (for example K-SAT with odd K), models outside the positivity class, fixed-degree random regular graphs, identification of the optimal hierarchy depth, or zero temperature beyond the stated ground-state limits.

Main theorem: for Poisson-diluted even-arity Ising models in the Panchenko-Talagrand class (factorized interactions $e^{\theta}=a(1+b\prod f_l)$ with the stated independence and positivity of moments of $-b$), with only first moments of the interaction and field, the limiting pressure exists and equals $\inf_r$ of the infimum of the Mezard-Parisi hierarchical functional of depth $r$. Covers Viana-Bray, symmetric diluted even-spin models and soft random even-$K$ SAT at positive temperature. Not covered: odd arity (for example K-SAT with odd K), models outside the positivity class, fixed-degree random regular graphs, identification of the optimal hierarchy depth, or zero temperature beyond the stated ground-state limits.