Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures

Let $Z_N$ be the partition function of the Sherrington-Kirkpatrick model at inverse temperature $\beta >1$. Replica theory (Kondor near the critical temperature, Parisi and Rizzo at all temperatures) predicts that the probability that $N^{-1}\log Z_N$ exceeds its typical value by $\delta$ decays like $\exp (-Na{\delta }^{6\mathrm{/}5})$ with an explicit $a$; Aronow and Lopatto recently proved bounds of this order. The exponent is the Legendre transform of the Parisi problem constrained to carry an atom $\theta$ at zero. We show that the constrained minimum exceeds the equilibrium free energy by $c_{\beta }{\theta }^5(1+o(1))$, where $c_{\beta }=\frac{9}{640}{\beta }^3{\rho }_{\beta }(0{)}^{-3}$ and ${\rho }_{\beta }(0)>0$ is the density at zero of the Parisi measure. With the fractional-moment formula this gives the exponent $\frac{5}{6}(6c_{\beta }{)}^{-1\mathrm{/}5}{\delta }^{6\mathrm{/}5}(1+o(1))$ as $N\to \mathrm{\infty }$ and then $\delta \to 0$. The constrained minimizer has a marginal first positive contact, to leading order at $3\theta \mathrm{/}(2{\rho }_{\beta }(0))$ with an atom $\theta \mathrm{/}2$. This plateau calculus applies to every mixture with ${\xi }^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)=0$ whose Parisi measure accumulates at zero; a small field $h$ opens a gap of order $\mathrm{\mid }h{\mathrm{\mid }}^{2\mathrm{/}3}$, and an atom of the Parisi measure at zero, present in every pure $p$-spin model with $p\ge 3$, makes the rate leave zero linearly. Near criticality the constant tends to Kondor’s value $9\mathrm{/}5120$ and the marginal regime ends at a de Almeida-Thouless threshold. For the Sherrington-Kirkpatrick model at every $\beta >1$, a finite-step curvature inequality from an unrefereed preprint shows that every constrained minimizer either has an interval of positive contacts, all marginal, or has exactly two atoms, and that the marginal regime ends at a unique de Almeida-Thouless threshold ${\theta }_c(\beta )$. The laws at every $\beta >1$ use that the Parisi measure accumulates at zero, a theorem of Lopatto also proved in earlier work of the author.