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Title: On the TAP equations and the local magnetization of the Sherrington-Kirkpatrick (SK) mode.

Abstract: The TAP equations for the Sherrington-Kirkpatrick model are a set of high-dimensional, nonlinear, fixed-point equations of the local magnetization. In the seminal work [Comm. Math. Phys., 325(1):333-366, 2014], Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies below the Almeida-Thouless transition line (鈥渉igh temperature regime鈥�). However, it was unclear if this asymptotic solution coincides with the local magnetization. In this talk, I will introduce a new iterative scheme, motivated by the cavity equations of the SK model, and show that the new scheme is asymptotically the same as the so-called Approximate Message Passing (AMP) algorithm, a generalization of Bolthausen's iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration (and hence Bolthausen's iteration) converges to the local magnetization as long as the overlap is locally uniformly concentrated. If time permits, I will also briefly discuss the TAP equations in the low temperature regime. The talk is based on joint works with Wei-Kuo Chen (University of Minnesota).

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