critical exponent. Should the scaling turn out to be [*isotropic*]{}, then $\zeta=1$, and for systems with a fixed aspect ratio $L_z=\gamma L$ the scaling law reduces to, $$L\Upsilon_\mu(T,L) =  \tilde u_\mu(tL^{1/\nu})\enspace.
\label{eUpsScale}$$ In this case, exactly at the transition temperature $T_{\rm c}$, one has $t=0$ and so $L\Upsilon_\mu$ is independent of system size $L$.

Henceforth, we will measure temperature and energies in units where $J_\perp=1$. Length will be measured in units where the grid spacing is unity.

Vortex Lattice Melting at Weak Pinning {#smelt}
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In this section we consider the first order vortex lattice melting transition at weak pinning strength. Our methods for identifying the melting transition and establishing that it is indeed a first order phase transition are the same as we have used in our earlier work[@OTmelt] on the more dilute $f=1/20$ system.

For a pure system with disorder strength $p=0$ the vortex line lattice, of vortex density $f=1/5$, will order into a ground state containing a square vortex lattice with lattice constant $\sqrt{5}$. There are two possible orientations of this square lattice with respect to the underlying grid, related to each other by a reflection through the $\hat x$ axis, as shown in Fig. \[f2\]a. In 