On Normalized Multiplicative Cascades under Strong Disorder

The purpose of this note is to determine the weak limit of a sequence of (normalized) multiplicative cascade measures under strong disorder in terms of the extremes of an associated branching random walk, assuming i.i.d positive, non-lattice bond weights and a second moment condition. The solution is expressed as an almost sure coupling of random probability measures in the disorder parameter β > βc, that may be viewed in relation to the Aizenmann-Wehr metastates in physics and Aldous’ weak convergence paradigm in probability theory.