Noncommutative Probability and Multiplicative Cascades

Abstract

Various aspects of standard model particle physics might be explained by a suitably rich algebra acting on itself, as suggested by Furey (2015). The present paper develops the asymptotics of large causal tree diagrams that combine freely independent elements in such an algebra. The Marčenko–Pastur law and Wigner’s semicircle law are shown to emerge as limits of normalized sum-over-paths of nonnegative elements assigned to the edges of causal trees. These results are established in the setting of noncommutative probability. Trees with classically independent positive edge weights (random multiplicative cascades) were originally proposed by Mandelbrot as a model displaying the fractal features of turbulence. The novelty of the present work is the use of noncommutative (free) probability to allow the edge weights to take values in an algebra. An application to theoretical neuroscience is also discussed.