2023/12/22 by Hall, Brian C., Ho, Ching-Wei, Jalowy, Jonas +1 · 4 citations
#30C10 #30C15 (Primary) 35F21 #30D20 #31A05 #46L54 (Secondary) #60B10 #60B20 #60G57 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2312.14883
We start with a random polynomial PN(z) of degree N with independent coefficients. We then consider a new polynomial PtN obtained by \lceil Nt\rceil applications of a fractional differential operator of the form za (d/dz)b, where a and b are real numbers. When b>0, we compute the limiting root distribution μt of PtN as N→∞. We show that μt is the push-forward of the limiting root distribution of PN under a transport map Tt. The map Tt is defined by flowing along the characteristic curves of a PDE satisfied by the log potential of μt. In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially with constant speed until they hit the origin, at which point, they cease to exist. For general a and b, the transport map Tt has a free probability interpretation as multiplication of an R-diagonal operator by an R-diagonal \textquotedblleft transport operator.\textquotedblright As an application, we obtain a push-forward characterization of the free self-convolution semigroup ⊕ of radial measures on ℂ. We also consider the case b<0, which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.