Semicircular Law for Hadamard Products

Abstract

In nuclear physics, since the particles move with very high velocity in a small range, many excited states are seldom observed in very short time instances, and over long time periods there are no excitations. More generally, if a real physical system is not of full connectivity, the random matrix describing the interactions between the particles in the system will have a large proportion of zero elements. In this case, a sparse random matrix provides a more natural and relevant description of the system. Indeed, in neural network theory, the neurons in a person’s brain are large in number and are not of full connectivity with each other. Actually, the dendrites connected with one individual neuron are of much smaller number, probably several orders of magnitude, than the total number of neurons. Sparse random matrices are adopted in modeling these partially connected systems in neural network theory.