Free multiplicative cascades
Various aspects of standard model particle physics might be explained by a suitably rich algebra acting on itself, as suggested recently by Furey (2015). This talk discusses the statistical behavior of large causal tree diagrams that combine freely independent elements in such an algebra. It is shown that some of the familiar limiting distributions in random matrix theory (namely the Marchenko-Pastur law and Wigner's semicircle law) emerge in this setting as limits of normalized sums-over-paths of non-negative elements of the algebra assigned to the edges of the tree. These results are established in the setting of non-commutative 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 approach is the use of non-commutative (free) probability to allow the edge weights to take values in an algebra. Potential applications in theoretical neuroscience related to Alan Turing's famous "Imitation Game" paper are also discussed.