the value $1.60\pm 0.01$ obtained with the expansions of previous sections.

Conclusions {#SecCon}
===========

In this paper we have studied the properties of the $\infty$-replica symmetry breaking solution of the Sherrington - Kirkpatrick model without external fields. Using high order expansions in $\tau = T_c - T$ we are able to compute the order parameter $q(x)$ and other relevant quantities for a large range of temperatures with high precision. In particular we found that $q(x)$ [*is not*]{} an odd function of $x$, confirming the prediction of Ref. [@Temes]. Direct consequence of this is that the overlap probability distribution function $P(q)$ has discontinuous derivatives at $q=0$. Another consequence of our findings is that the Parisi-Toulouse scaling becomes exact asymptotically for $T\to 0$ [*and*]{} $\beta x\to 0$, while for $T\to 0$ is a fairly good approximation. This is also consistent with the $T=0$ limit of the breaking point which we found to be $x_{max}(0)=.548\pm .005$.

Having reached very high orders we can reasonably speculate on the analytical properties of the function $ q(x) $. In particular we believe that

-   All the expansions in power of $ \tau  $ are asymptotic expansions.

-   At any temperature, the function $ q(x) $ is 