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, Bolthausen (Commun Math Phys 325(1):333–366, 2014) 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.

空腔和TAP方程是Sherrington-Kirkpatrick模型中局部磁化强度非线性方程的高维系统。在开创性工作中，Bolthausen（Commun Math Phys 325（1）：333–366，2014）引入了一种迭代方案，如果该模型位于Almeida-Thouless转换线内，则该迭代方案将为TAP方程生成渐近解。但是，尚不清楚该渐近解是否与局部磁化一致。在这项工作中，受腔方程的启发，我们引入了一个新的迭代方案并建立了一个大数的弱定律。我们证明了我们的新方案与所谓的近似消息传递算法（Bolthausen迭代的一般化）渐近相同，该算法已广泛应用于压缩感知，贝叶斯推理等中。基于此，