On the Replica Symmetry of a Variant of the Sherrington-Kirkpatrick Spin Glass

We consider $N$ i.i.d. Ising spins with mean $m\in (-1,1)$ whose interactions are described by a Sherrington-Kirkpatrick Hamiltonian with a quartic correction. This model was recently introduced by Bolthausen in \cite{Bolt2} as a toy model to understand whether a second moment argument can be used to derive the replica symmetric formula in the full high temperature regime if $m\neq 0$. In \cite{Bolt2}, Bolthausen suggested that a natural analogue of the de Almeida-Thouless condition for the toy model is \begin{equation}\label{eq:conj} \beta^2(1-m^2)^2\leq 1. \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, (1)\end{equation} Here, $\beta \geq 0$ corresponds to the inverse temperature. While the second moment method implies replica symmetry for $\beta $ sufficiently small, Bolthausen showed that the method fails to prove replica symmetry in the full region described by (1). A natural question that was left open in \cite{Bolt2} is whether (1) correctly characterizes the high temperature phase of the toy model. In this note, we show that this is indeed not the case. We prove that if $|m| \geq m_*$, for some $m_* \in (0,1)$, the limiting free energy of the toy model is negative for suitable $\beta $ that satisfy (1).