to comment on this approximation in section \[svg\].

To model vortex pinning due to quenched point randomness we use couplings[@OTmelt] $$\begin{aligned}
  \label{eJ}
   J_{i\mu}&=&J_\perp(1+p\epsilon_{i\mu}),\quad \mu=x,y\\ \nonumber
  J_{iz}&=&J_z  \quad {\rm constant},\end{aligned}$$ where the $\epsilon_{i\mu}$ are uncorrelated, uniformly distributed, random variables with $$\langle\epsilon_{i\mu}\rangle=0, \qquad \langle\epsilon_{i\mu}^2\rangle=1\enspace.$$ The disorder strength is thus controlled by the parameter $p$ in Eq. (\[eJ\]). We will vary $p$ from small to large values in order to systematically investigate the differences between weak and strong pinning.

In the following we will use $\langle\dots\rangle$ to denote the equilibrium average for a particular realization of the random pinning $\{\epsilon_{i\mu}\}$. We will use $[\dots ]$ to denote the average over several independent realizations of the $\{\epsilon_{i\mu}\}$.

In this work we use parameters, $$\begin{aligned}
f&=&1/5\enspace,\\
J_z&=&J_\perp/40\enspace.
\label{eparms}\end{aligned}$$ We use $J_z\ll J_\perp$ to enhance vortex line fluctuations along the $\hat z$ direction, thus allowing us to use systems with smaller $L_z$. The relatively dense value of $f$ was similarly chosen so as to have many vortex lines contained within systems of modest size, so as to permit finite size scaling analyses (particularly for the vortex glass) and to average over many independent realizations of the random disorder $\{\epsilon_{i\mu}\}$. While this high density leads to artificial 