Critical disorder and universality class of the MBL transition

Assuming $W_{c}$ exists for the random-field Heisenberg chain, determine $W_{c}/J$ and the scaling near the transition (Kosterlitz-Thouless-like versus a finite correlation-length exponent $\nu$, subject to the Harris-Chayes bound $\nu \ge 2$ in $d = 1$). Answer: $W_{c}$ with uncertainty and the scaling form of the characteristic length near $W_{c}$.

Avalanche-stability numerics (chain coupled to an ideal bath at one end) extrapolated with a controlled scaling theory and consistent with renormalization-group predictions.