Analytical Solution for Multivariate Statistics in Random Multiplicative Cascades

This contribution is intended to serve as an introduction-by-example for prospective students and the casual observer. For more details and greater rigour, the interested reader should consult the published papers. What are multiplicative random cascade models (MRCM’s), and why are they relevant? Turbulence in fluids must necessarily obey the Navier-Stokes equation, while the QCD lagrangian in principle describes all high-energy interactions and final states. Nevertheless, many experimental results cannot be reproduced from the underlying theories, even though they are known. We hence look to models to reproduce essential features of physical data and hence point the way. This approach has a long and fruitful history. Multiplicative variables do not obey classical central limit theorems but rather exhibit large fluctuations even in the limit of large numbers. Large fluctuations, unexplainable in terms of additive variables, are seen in the energy dissipation ǫ in fully developed turbulence and in the number and density of pions in the final states of high-energy hadronic collisions. The models contain random variables because distributions of relevant quantities differ greatly from event to event in high energy physics and correspondingly in different spatial “snapshots” or pieces of time series of ǫ. Hierarchical structures or cascades are used because they can easily be made scale invariant and are hence well placed to mimick the scale invariance seen in the Richardson cascades, the original Kolmogorov theory, as well as some deviations from it. In high-energy ee, lepton-hadron and hadronhadron collisions, particle production can be described at least in part as cascades of partons and hadrons.