  The cavity and TAP equations are high-dimensional systems of nonlinear
equations of the local magnetization in the Sherrington-Kirkpatrick model. 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 inside the Almeida-Thouless transition line.
However, it was unclear if this asymptotic solution coincides with the local
magnetization. In this work, motivated by the cavity equations, we introduce a
new iterative scheme and establish a weak law of large numbers. We show that
our new scheme is asymptotically the same as the so-called Approximate Message
Passing 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 Bolthausen's scheme both
converge to the local magnetization as long as the overlap is locally uniformly
concentrated.
