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Spectral distribution of the free Jacobi process with equal rank projections

Nizar Demni, Tarek Hamdi

TL;DR

The paper addresses the finite-time spectral distribution of the free Jacobi process with equal rank projections by deriving characteristic curves for the transport PDE of the moment generating function and obtaining a local closed form in terms of an α‑deformed χ-transform inverse. The authors develop a Riccati-type framework along characteristics, specialize to α=1/2 to recover known unitary Brownian data, and extend the construction to general α through a detailed saddle-point/analytic inversion analysis of a deformed χ-transform. They also introduce mapping properties of the auxiliary V map, reveal phase-transition-like behavior, and perform a saddle-point analysis to establish analyticity of the local inverse and asymptotic decay of coefficients. A dynamical Kunisky identity is proved and extended via a Nica–Speicher semi-group perspective, linking finite-time Jacobi laws to balanced free copies, with discussions on initial-data matching and open questions for non-freeness. Overall, the work provides a comprehensive PDE/transform-based description of the finite-time spectral distribution for equal-rank free Jacobi processes and connects static and dynamic manifestations of Kunisky-type identities within free probability.

Abstract

The free Jacobi process is the radial part of the compression of the free unitary Brownian motion by two free orthogonal projections in a non commutative probability space. In this paper, we derive spectral properties of the free Jacobi process associated with projections having the same rank $α\in (0,1)$. To start with, we determine the characteristic curves of the partial differential equation satisfied by the moment generating function of its spectral distribution. Doing so leads for any fixed time $t >0$ to an expression of this function in a neighborhood of the origin, therefore extends our previous results valid for $α= 1/2$. Moreover, the obtained characteristic curves are encoded by an $α$-deformation of the compositional inverse of the $χ$-transform of the spectral distribution of the free unitary Brownian motion. In this respect, we study mapping properties of this deformation and use the saddle point method to prove that the compositional inverse of a $α$-deformation of the $χ$-transform of the free unitary Brownian motion is analytic in the open unit disc (for large enough time $t$). The last part of the paper is devoted to a dynamical version of a recent identity pointed out by T. Kunisky in \cite{Kun}. Actually, this identity relates the stationary distributions of the free Jacobi processes corresponding to the sets of parameters $(α, α)$ and $(1/2,α)$ respectively and we explain how it follows from the Nica-Speicher semi-group. Our dynamical version then relates the partial differential equations of the Cauchy-Stieltjes transforms of the densities of the finite-time spectral distributions. It also raises the problem of whether a dynamical analogue of the Nica-Speicher semi-group exists when the compressing projection has rank $1/2$.

Spectral distribution of the free Jacobi process with equal rank projections

TL;DR

The paper addresses the finite-time spectral distribution of the free Jacobi process with equal rank projections by deriving characteristic curves for the transport PDE of the moment generating function and obtaining a local closed form in terms of an α‑deformed χ-transform inverse. The authors develop a Riccati-type framework along characteristics, specialize to α=1/2 to recover known unitary Brownian data, and extend the construction to general α through a detailed saddle-point/analytic inversion analysis of a deformed χ-transform. They also introduce mapping properties of the auxiliary V map, reveal phase-transition-like behavior, and perform a saddle-point analysis to establish analyticity of the local inverse and asymptotic decay of coefficients. A dynamical Kunisky identity is proved and extended via a Nica–Speicher semi-group perspective, linking finite-time Jacobi laws to balanced free copies, with discussions on initial-data matching and open questions for non-freeness. Overall, the work provides a comprehensive PDE/transform-based description of the finite-time spectral distribution for equal-rank free Jacobi processes and connects static and dynamic manifestations of Kunisky-type identities within free probability.

Abstract

The free Jacobi process is the radial part of the compression of the free unitary Brownian motion by two free orthogonal projections in a non commutative probability space. In this paper, we derive spectral properties of the free Jacobi process associated with projections having the same rank . To start with, we determine the characteristic curves of the partial differential equation satisfied by the moment generating function of its spectral distribution. Doing so leads for any fixed time to an expression of this function in a neighborhood of the origin, therefore extends our previous results valid for . Moreover, the obtained characteristic curves are encoded by an -deformation of the compositional inverse of the -transform of the spectral distribution of the free unitary Brownian motion. In this respect, we study mapping properties of this deformation and use the saddle point method to prove that the compositional inverse of a -deformation of the -transform of the free unitary Brownian motion is analytic in the open unit disc (for large enough time ). The last part of the paper is devoted to a dynamical version of a recent identity pointed out by T. Kunisky in \cite{Kun}. Actually, this identity relates the stationary distributions of the free Jacobi processes corresponding to the sets of parameters and respectively and we explain how it follows from the Nica-Speicher semi-group. Our dynamical version then relates the partial differential equations of the Cauchy-Stieltjes transforms of the densities of the finite-time spectral distributions. It also raises the problem of whether a dynamical analogue of the Nica-Speicher semi-group exists when the compressing projection has rank .
Paper Structure (18 sections, 12 theorems, 172 equations)

This paper contains 18 sections, 12 theorems, 172 equations.

Table of Contents

  1. Introduction and main results
  2. Random matrices, matrix-valued stochastic processes and free probability theory
  3. Contribution and main results
  4. Analysis of the Characteristic curves: proof of Theorem \ref{['Th1']}
  5. A Ricatti equation
  6. Back to the case $\alpha =1/2$
  7. General ranks
  8. Local invertibility: end of the proof of Theorem \ref{['Th1']}
  9. Properties of the map $V_{4\alpha t}^{(\alpha)}$
  10. Variations of $V_{4\alpha t}^{(\alpha)}$ on the real line
  11. A deformed $\chi$-transform: a saddle point analysis
  12. Critical points of $\phi_t$
  13. Asymptotic behavior of $a_n^{(\alpha)}(t)$ as $n \rightarrow +\infty$
  14. Kunisky's identity: proof and dynamical extension
  15. Reminder
  16. ...and 3 more sections

Key Result

Theorem 1

For any $\alpha \in (0,1)$, set and assume $\mu_0^{(\alpha,\alpha)} = \delta_1$. Then, for any time $t > 0$, there exists a map $J_{4\alpha t}^{(\alpha)}$ locally around the origin such that the moment generating function $M_t^{(\alpha)}$ is given by: Equivalently, where

Theorems & Definitions (25)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • proof
  • Remark
  • Remark
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • ...and 15 more