February 2024 Complexity of bipartite spherical spin glasses
Benjamin McKenna
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 636-657 (February 2024). DOI: 10.1214/22-AIHP1327


This paper characterizes the annealed complexity of bipartite spherical spin glasses, both pure and mixed. This means we give exact variational formulas for the asymptotics of the expected numbers of critical points and of local minima. This problem was initially considered by Auffinger and Chen (J. Stat. Phys. 157 (2014) 40–59), who gave upper and lower bounds on this complexity. We find two surprising connections between pure bipartite and pure single-species spin glasses, which were studied by Auffinger, Ben Arous, and Černý (Comm. Pure Appl. Math. 66 (2013) 165–201). First, the local minima of any pure bipartite model lie primarily in a low-energy band, similar to the single-species case. Second, for a more restricted set of pure (p,q) bipartite models, the complexity matches exactly that of a pure p+q single-species model.

Cet article caractérise la complexité annealed des verres de spin sphériques bipartis, à la fois purs et mixtes. Nous donnons des formules exactes et variationnelles pour la limite des nombres de points critiques et de minima locaux. Ce problème a d’abord été considéré par Auffinger et Chen (J. Stat. Phys. 157 (2014) 40–59), qui ont donné des bornes supérieures et inférieures sur cette complexité. Nous trouvons deux connexions surprenantes entre les verres de spin purs et bipartis, et les purs et monospécifiques, qui ont été étudiés par Auffinger, Ben Arous, et Černý (Comm. Pure Appl. Math. 66 (2013) 165–201). D’abord, les minima locaux de n’importe quel modèle pur et biparti se situent principalement dans une bande de basse énergie, similaire au cas monospécifique. De plus, pour un ensemble restreint des modèles purs et bipartis de type (p,q), la complexité correspond exactement à celle d’un modèle pur et monospécifique de type p+q.

Funding Statement

This work was supported in part by NSF grant DMS-1812114.


We wish to thank Tuca Auffinger for bringing the bipartite spin glass model to our attention, and Gérard Ben Arous, Paul Bourgade, Wei-Kuo Chen, Krishnan Mody, Jean-Christophe Mourrat, and Ofer Zeitouni for helpful discussions. We are also grateful to the referees for greatly improving the readability of the paper, to Brice Huang and Mark Sellke for help correcting some formulas in a previous version of this paper, and to Yan Fyodorov for pointing us towards additional references on bipartite spin glasses in the physics literature.


Download Citation

Benjamin McKenna. "Complexity of bipartite spherical spin glasses." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 636 - 657, February 2024. https://doi.org/10.1214/22-AIHP1327


Received: 21 June 2021; Revised: 18 May 2022; Accepted: 25 September 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718393
Digital Object Identifier: 10.1214/22-AIHP1327

Primary: 82B44
Secondary: 60B20 , 60G15

Keywords: Bipartite spin glasses , Dyson equation , Kac–Rice formula , Landscape complexity , spherical spin glasses

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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