Analysis of the $\infty$-replica symmetry breaking solution of the Sherrington-Kirkpatrick model

In this work we analyze the Parisi $\infty$-replica symmetry breaking solution of the Sherrington-Kirkpatrick model without external field using high order perturbative expansions. The predictions are compared with those obtained from the numerical solution of the $\infty$-replica symmetry breaking equations, which are solved using a pseudospectral code that allows for very accurate results. With these methods we are able to get more insight into the analytical properties of the solutions. We are also able to determine numerically the end point $x_{\max }$ of the plateau of $q\left(x\right)\mathrm{}$ and find that $\underset{\overrightarrow{T}0}{limop}{x}_{\max }\mathrm{}\left(T\right)>\mathrm{0.5.}\mathrm{}$