2013/02/20 by Nizar Demni, Taoufik Hmidi, Demni, Nizar +1
Mathematics · #35F20 #60B20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1302.4844
openalex publication_date 2013/02/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Given an orthogonal projection P and a free unitary Brownian motion Y = (Yt)t ≥ 0 in a W⋆-non commutative probability space such that Y and P are ⋆-free in Voiculescu's sense, the main result of this paper states that P and YtPYt⋆ are in general position at any time t. To this end, we study the dynamics of the unitary operator SYtSYt⋆ where S = 2P-1. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say μt. Then, we provide a flow on the interval [-1,1] in such a way that the Herglotz transform of μt composed with this flow is governed by both the Herglotz transforms of the initial (t=0) and the stationary (t = ∞) distributions. This fact allows to compute the weight that μt assigns to z=1 leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to x=1 follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay e-nt in the expansion of the n-th moment of SYtSYt⋆.