Pinning {#svg}
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We now consider the strong pinning limit, $p>p_c\simeq 0.22$, and the possible existence of a vortex glass phase. In our earlier work[@OTmelt] on the more dilute $f=1/20$ system, we presented preliminary evidence for the absence of a vortex glass phase within the XY model. However later work by Olsson[@OlssonVG] established that the model does indeed have a second order vortex glass transition at strong disorder strength. Moreover he found that critical scaling is isotropic. Here we follow the analysis of Olsson, looking at the disorder averaged helicity modulus in the $xy$ plane, $[\Upsilon_\perp]$, defined in Eq. (\[eUps\]).

In Fig. \[f6\]a we plot $[L\Upsilon_\perp]$ vs $T$ for three different disorder strengths, $p=0.3, 0.4$ and $0.55$, using for each case three different system sizes from $L=10$ to $25$, $L_z=(3/5)L$, as indicated in the figure. Results are averaged over $200-600$ independent realizations of the random disorder, depending on system size. As discussed following Eq. (\[eUpsScale\]), for each value of $p$, the common intersection of the curves for different $L$ locates the vortex glass transition, $T_{\rm g}(p)$, and thus allows us to map out the vortex glass transition line, plotted as the black squares and black solid line in Fig. \[f1\]. 