On the replica symmetry of a variant of the Sherrington-Kirkpatrick spin glass

We consider N i.i.d. Ising spins with mean m∈(−1,1) whose interactions are described by a Sherrington-Kirkpatrick Hamiltonian with a quartic correction. This model was recently introduced by Bolthausen in [3] 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≠0. In [3], Bolthausen suggested that a natural analogue of the de Almeida-Thouless condition for the toy model is β2(1−m2)2≤1.(0.1) Here, β≥0 corresponds to the inverse temperature. While the second moment method implies replica symmetry for β sufficiently small, Bolthausen showed that the method fails to prove replica symmetry in the full region described by (0.1). A natural question that was left open in [3] is whether (0.1) 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|≥m∗, for some m∗∈(0,1), the limiting free energy of the toy model is negative for suitable β that satisfy (0.1).