Intriguing Patterns in the Roots of the Derivatives of some Random Polynomials

Our observations show that the sets of real (respectively complex) roots of the derivatives of some classical families of random polynomials admit a rich variety of patterns looking like discretized curves. To bring out the shapes of the suggested curves, we introduce an original use of fractional derivatives. Then we set several conjectures and outline a strategy to explain the presented phenomena. This strategy is based on asymptotic geometric properties of the corresponding complex critical points sets.