Spectral distribution of the free Jacobi process, revisited

2017/11/20 by Hamdi, Tarek · 1 citation
#FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1711.07382

Abstract

We obtain a description for the spectral distribution of the free Jacobi process for any initial pair of projections. This result relies on a study of the unitary operator RU_tSU_t^* where R,S are two symmetries and U_t a free unitary Brownian motion, freely independent from \R,S\. In particular, for non-null traces of R and S, we prove that the spectral measure of RU_tSU_t^* possesses two atoms at ±1 and an L^∞-density on the unit circle \mathbbT, for every t>0. Next, via a Szegő type transform of this law, we obtain a full description of the spectral distribution of PU_tQU_t^* beyond the τ(P)=τ(Q)=1/2 case. Finally, we give some specializations for which these measures are explicitly computed.

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• Finite free probability and S transforms of Jacobi processes

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