$q(x)$ directly point by point by computing $q(mx_{\rm max})$ where $m=i/n$, $(i=0,1,\ldots \, n)$ for fixed $n$. For any $m$ and $T$ the quantity $q(mx_{\rm max})$ is itself a power series in $\tau$ which can be summed up using Padè approximants. With this procedure the function $q(x)$ can be determined for different $x$-resolution just changing the value of $n$, e.g., $n=50,100,1000$, and using the value of $x_{\rm max}$ previously found, see Fig. \[fig:xmaxpad\]. In Figure \[fig:qxpad\] the function $q(x)$ is shown for various temperatures $T$.

This method can be extended to any function of $x$ or $q$, for example, we computed the overlap probability function $P(mq_{\rm max})$ in a wide range of temperature $T>0.3$, see Figure \[fig:pdqpad\]. We found that the best Padè approximant is given by the $(12,7)$. By using the relation (\[eq:pdq\]) we can have an independent estimation of $q(x)$ with which to test the precision of our results. By using a norm $d_\infty(q,q') = \max_{0\leq x \leq 1} |q(x) - q'(x)|$ and expansions up to order $20$ we find, for example, that $d_\infty(q,q') = O(10^{-5})$ for $T=0.6$ and $d_\infty(q,q') = O(10^{-4})$ for $T=0.4$.

The form of the function $q(x)$ confirms the prediction of Ref. [@VTP81] obtained from 