Open Access
2024 Bounds on the covariance matrix of the Sherrington–Kirkpatrick model
Ahmed El Alaoui, Jason Gaitonde
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Electron. Commun. Probab. 29: 1-13 (2024). DOI: 10.1214/24-ECP582


We consider the Sherrington-Kirkpatrick model with no external field and inverse temperature β<1 and prove that the expected operator norm of the covariance matrix of the Gibbs measure is bounded by a constant depending only on β. This answers an open question raised by Talagrand, who proved a bound of C(β)(logn)8. Our result follows by establishing an approximate formula for the covariance matrix which we obtain by differentiating the TAP equations and then optimally controlling the associated error terms. We complement this result by showing diverging lower bounds on the operator norm, both at the critical and low temperatures.


We are grateful to Ronen Eldan for valuable conversations about bounding the covariance matrix of disordered spin systems which inspired this work. We also thank the anonymous referees for their feedback, and for the remark on the predicted O(β4) behavior in the bound Eq. (3.1).


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Ahmed El Alaoui. Jason Gaitonde. "Bounds on the covariance matrix of the Sherrington–Kirkpatrick model." Electron. Commun. Probab. 29 1 - 13, 2024.


Received: 13 August 2023; Accepted: 4 March 2024; Published: 2024
First available in Project Euclid: 28 March 2024

Digital Object Identifier: 10.1214/24-ECP582

Primary: 82D30

Keywords: Sherrington-Kirkpatrick model , Spin glasses , TAP free energy

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