Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization

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We consider the Sherrington-Kirkpatrick model of spin glasses at high-temperature and no external field, and study the problem of sampling from the Gibbs distribution \mu in polynomial time. We prove that, for any inverse temperature \beta<1/2, there exists an algorithm with complexity O(n^2) that samples from a distribution \mu^{alg} which is close in normalized Wasserstein distance to \mu. Namely, there exists a coupling of \mu and \mu^{alg} such that if (x,x^{alg})\in\{-1,+1\}^n\times \{-1,+1\}^n is a pair drawn from this coupling, then n^{-1}\mathbb E\{||x-x^{alg}||_2^2\}=o_n(1). The best previous results, by Bauerschmidt and Bodineau and by Eldan, Koehler, and Zeitouni, implied efficient algorithms to approximately sample (under a stronger metric) for \beta<1/4.
We complement this result with a negative one, by introducing a suitable "stability" property for sampling algorithms, which is verified by many standard techniques. We prove that no stable algorithm can approximately sample for \beta>1, even under the normalized Wasserstein metric.
Our sampling method is based on an algorithmic implementation of stochastic localization, which progressively tilts the measure \mu towards a single configuration, together with an approximate message passing algorithm that is used to approximate the mean of the tilted measure.