From Parisi to Boltzmann: via GREM and TAP

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**This talk will be broadcast at 13:30 BT / 14:30 CET / 15:30 EET, December 17th, 2025 on MS Teams only. **

Meeting ID: 393 527 089 878
Passcode: dsm7py

Abstract

Introduced by Derrida in the 80s, the Generalized Random Energy Models (GREM) serve as simplified, abstract models that are completely solvable and have significantly contributed to our comprehension of specific facets within the Parisi theory for mean-field spin glasses. In this talk, after an overview of Parisi’s ultrametric principle for the free energy of the Sherrington-Kirkpatrick (SK) model, we will turn to the alternative approach due to Thouless, Anderson, and Palmer (TAP). The TAP approach provides a representation of the SK free energy as the maximum of a random functional over local magnetizations. A simple reformulation reveals that, when evaluated at its critical points, the TAP free energy becomes a nonlinear functional of empirical measures constructed from local fields. Motivated by this structure, we construct abstract GREM-like models whose Hamiltonians are nonlinear functionals of empirical measures associated with GREM-like variables. For the thermodynamic free energy of these abstract models, dual Parisi-like formulas hold and turn out to be finite-dimensional “collapsed” versions of the underlying infinite-dimensional Gibbs-Boltzmann principles. Crucially, the nonlinearity inherited from the TAP representation generates a correction term (absent in canonical GREM-like models) that substantially reduces the gap between the free energies of such abstract hierarchical models and Parisi’s solution for the SK. Based on joint work with Nicola Kistler and Marius Alexander Schmidt.

About Giulia

Giulia is currently a postdoc in the Probability Theory group at IAM, under the mentorship of Prof. Dr. Anton Bovier. She did her PhD at Goethe University Frankfurt. Her work centres around statistical mechanics with a focus on the Random Energy Model and its variations.