---
abstract: 'In this work we analyse the Parisi’s $\infty$-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 new pseudo-spectral code which allows for very accurate results. With this 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 $x_{\rm max}$ of the plateau of $q(x)$ and find that $\lim_{T\to 0} x_{\rm max}(T) > 0.5$.'
author:
- 'A. Crisanti'
- 'T. Rizzo'
date: 'V2.1, '
title: 'Analysis of the $\infty$-replica symmetry breaking solution of the Sherrington-Kirkpatrick model'
---

Introduction
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Since its proposal in the 80’s the behaviour of the Parisi $\infty$-replica symmetry breaking ($\infty$-RSB) solution of the Sherrington-Kirkpatrick model has been extensively investigated both qualitatively and quantitatively [@MPV; @FH]. Despite this enormous amount of work, which has revealed many of the properties of the solutions, a complete control of the solution is still missing. One of the reasons can be traced back to the fact that till now only low order expansions were used, 