  We examine the many-body localization (MBL) phase transition in
one-dimensional quantum systems with quenched randomness and short-range
interactions. Following recent works, we use a strong-randomness
renormalization group (RG) approach where the phase transition is due to the
so-called avalanche instability of the MBL phase. We show that the critical
behavior can be determined analytically within this RG. On a rough
$\textit{qualitative}$ level the RG flow near the critical fixed point is
similar to the Kosterlitz-Thouless (KT) flow as previously shown, but there are
important differences in the critical behavior. Thus we show that this MBL
transition is in a new universality class that is different from KT. The
divergence of the correlation length corresponds to critical exponent $\nu
\rightarrow \infty$, but the divergence is weaker than for the KT transition.
