Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model

Abstract

It is shown that the power expansion of the Gibbs potential of the SK model up to second order in the exchange couplings leads to the TAP equation. This result remains valid for the general (including a ferromagnetic exchange) SK model. Theorems of power expansions and resolvent techniques are employed to solve the convergence problem. The convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations.