Replica symmetry breaking in multi-species Sherrington-Kirkpatrick model

Abstract

In the Sherrington-Kirkpatrick (SK) and related mixed $p$-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally necessary) condition for symmetry breaking, has been a frequent object of study in spin glass theory. In this paper, we consider the analogous condition for the multi-species SK model, which concerns the eigenvectors of a Hessian matrix. The analysis is tractable in the two-species case with positive definite variance structure, for which we derive an explicit AT temperature threshold. To our knowledge, this is the first non-asymptotic symmetry breaking condition produced for a multi-species spin glass. As possible evidence that the condition is sharp, we draw further parallel with the classical SK model and show coincidence with a separate temperature inequality guaranteeing uniqueness of the replica symmetric critical point.

20 pages; updated Section 1.5 and proof of Theorem 1.1

References in corpus (20)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Dynamics of Majority Rule
• The Thermodynamic Limit in Mean Field Spin Glass Models
• Multitasking associative networks
• The Parisi formula has a unique minimizer
• The free energy in a multi-species Sherrington-Kirkpatrick model
• Equilibrium statistical mechanics of bipartite spin systems
• The Parisi formula for mixed $p$-spin models
• The replica symmetric behavior of the analogical neural network
• Quadratic replica coupling in the Sherrington-Kirkpatrick mean field spin glass model
• About the Almeida-Thouless transition line in the Sherrington-Kirkpatrick mean field spin glass mode
• A thermodynamical perspective of immune capabilities
• Free energy in the generalized Sherrington-Kirkpatrick mean field model
• Phase transitions in social sciences: two-populations mean field theory
• Some Properties of the Phase Diagram for Mixed $p$-Spin Glasses
• Scaling Limits for Multispecies Statistical Mechanics Mean-Field Models
• A statistical mechanics approach to autopoietic immune networks
• Mean field bipartite spin models treated with mechanical techniques
• Rigorous Results on the Bipartite Mean-Field Model
• Toward a quantitative approach to migrants integration

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