2024/12/05 by Christian Brennecke, Brennecke, Christian, Adrien Schertzer +1
Physics and Astronomy · #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2412.04336
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 \citeBolt2 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 \citeBolt2, Bolthausen suggested that a natural analogue of the de Almeida-Thouless condition for the toy model is β2(1-m2)2≤ 1. (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 (1). A natural question that was left open in \citeBolt2 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| ≥ m_*, for some m_* ∈ (0,1), the limiting free energy of the toy model is negative for suitable β that satisfy (1).