On convergence of Bolthausen's TAP iteration to the local magnetization

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Abstract
The Thouless, Anderson, and Palmer (TAP) equations state that the local magnetization in the Sherrington-Kirkpatrick mean-field spin glass model satisfies a system of nonlinear equations. In the seminal work [Comm. Math. Phys., 325(1):333-366, 2014], Bolthausen introduced a recursive scheme and showed that it converges and gives an asymptotic solution to the TAP equations assuming that the model lies inside the Almeida-Thouless transition line, but it was not understood if his scheme converges to the local magnetization. In this work, we present a positive answer to this question by showing that Bolthausen's scheme indeed approximates the local magnetization when the overlap is locally uniformly concentrated. Our approach introduces a new iterative scheme motivated by the cavity equations of the local magnetization, appearing in physics literature and rigorously established by Talagrand. This scheme makes it possible to quantify the distance to the local magnetization and is shown to be the same as that in Bolthausen's iteration asymptotically.

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