  We propose a mean field theory for the localization of damage in a
quasistatic fuse model on a cylinder. Depending on the quenched disorder
distribution of the fuse thresholds, we show analytically that the system can
either stay in a percolation regime up to breakdown, or start at some current
level to localize starting from the smallest scale (lattice spacing), or
instead go to a diffuse localization regime where damage starts to concentrate
in bands of width scaling as the width of the system, but remains diffuse at
smaller scales. Depending on the nature of the quenched disorder on the fuse
thresholds, we derive analytically the phase diagram of the system separating
these regimes and the current levels for the onset of these possible
localizations. We compare these predictions to numerical results.
