On the Replica Symmetry of a Variant of the Sherrington-Kirkpatrick Spin Glass
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916514128461824 |
|---|---|
| author | Brennecke, Christian Schertzer, Adrien |
| author_facet | Brennecke, Christian Schertzer, Adrien |
| contents | 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} β^2(1-m^2)^2\leq 1. \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, (1)\end{equation} Here, $β\geq 0$ 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 \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 $β$ that satisfy (1). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04336 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Replica Symmetry of a Variant of the Sherrington-Kirkpatrick Spin Glass Brennecke, Christian Schertzer, Adrien Probability Mathematical Physics 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} β^2(1-m^2)^2\leq 1. \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, (1)\end{equation} Here, $β\geq 0$ 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 \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 $β$ that satisfy (1). |
| title | On the Replica Symmetry of a Variant of the Sherrington-Kirkpatrick Spin Glass |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2412.04336 |