The Almeida–Thouless conjecture for the replica-symmetric regime of the planted Sherrington–Kirkpatrick model

Let $c$ denote the planted-clique proportion, let $\theta$ and $\theta_1$ be the model parameters, and let $q$, $\mu$, $h$, and $z$ be the quantities appearing in the planted Sherrington–Kirkpatrick model's replica-symmetric description. Almeida–Thouless conjecture. The replica-symmetric regime of the planted Sherrington–Kirkpatrick model is the region where

$$\mathbb{E}\theta^2\bigl((1-c)\operatorname{sech}^4(\theta\sqrt q z+h)+c\operatorname{sech}^4(\theta\sqrt q z+\theta_1\mu+h)\bigr)<1.$$

The paper presents this as the planted-model counterpart of a conjecture for the standard Sherrington–Kirkpatrick model. It is intended to characterize exactly the temperature regime in which replica symmetry holds, beyond the equality criterion involving the planted model's variational quantity and free energy.