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The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space

Part of: Equilibrium statistical mechanics Applications to specific types of physical systems

Published online by Cambridge University Press:  28 January 2021

Jean-Christophe Mourrat
Jean-Christophe Mourrat*
Affiliation:
DMA, Ecole normale supérieure, CNRS, PSL University, Paris, France; Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
*
e-mail: mourrat@dma.ens.fr
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Abstract

The Parisi formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We showthat this quantity can be recast as the solution of a Hamilton–Jacobi equation in the Wasserstein space of probability measures on the positive half-line.

Keywords

Spin glassHamilton–Jacobi equationWasserstein space

MSC classification

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 607 - 629
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This work was partially supported by the ANR grants LSD (ANR-15-CE40-0020-03) and Malin (ANR-16-CE93-0003).

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