  A replica-symmetry-breaking phase transition is predicted in a host of
disordered media. The criticality of the transition has, however, long been
questioned below its upper critical dimension, six, due to the absence of a
critical fixed point in the renormalization-group flows at one-loop order. A
recent two-loop analysis revealed a possible strong-coupling fixed point but,
given the uncontrolled nature of perturbative analysis in the strong-coupling
regime, debate persists. Here we examine the nature of the transition as a
function of spatial dimension and show that the strong-coupling fixed point can
go through a Hopf bifurcation, resulting in a critical limit cycle and a
concomitant discrete scale invariance. We further investigate a different
renormalization scheme and argue that the basin of attraction of the
strong-coupling fixed point/limit cycle may thus stay finite for all
dimensions.
