Spectral distribution of the free Jacobi process with equal rank projections

Nizar Demni; Tarek Hamdi

REVIEW

Spectral distribution of the free Jacobi process with equal rank projections

Nizar Demni, Tarek Hamdi

Probability · Submitted 13 Jan 2025 · 28 references

Pith's verdict: Solid extension of the free Jacobi spectral formula to equal rank alpha, with a real but fixable gap in the saddle point proof of the deformed chi-transform's analyticity.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 10, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2501.07552 (version 2)

Pith IDD7BRBB4Z, Pith's permanent record of this paper and its review

CitationNizar Demni, Tarek Hamdi. "Spectral distribution of the free Jacobi process with equal rank projections." arXiv:2501.07552 (2025). https://pith.science/paper/D7BRBB4Z

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

The main result is real and new: Theorem 1 gives an explicit local formula for the moment generating function of the equal-rank free Jacobi process for every alpha, extending your earlier alpha=1/2 work. The derivation via characteristic curves is clean, the recovery of the known alpha=1/2 case checks out, and the local inversion step for Theorem 1 is justified by the inverse function theorem with a positive derivative. That part deserves credit.

The genuinely new pieces are the alpha-deformed map V, its mapping properties on the positive half-line (Proposition 2), and the saddle point analysis leading to Proposition 3. The dynamical version of Kunisky's identity in Section 5 is also a nice contribution: the algebraic explanation via the Nica-Speicher semigroup and the moment identity from your earlier paper is elegant, and the PDE comparison in Proposition 5 is carefully done. The paper is not circular; it builds on established prior theorems.

Now the soft spots. The stress-test note is right about Section 4.2: the proof of Proposition 3 assumes, without proof, that a contour gamma exists inside |w|1/2 the punctured disc is multiply connected, so this is not automatic. This is load-bearing for the analyticity conclusion of Proposition 3, which the abstract advertises. It does not affect Theorem 1, whose local inversion is separate, but it needs to be fixed, either by an explicit construction or by a deformation argument showing the steepest-descent paths can be joined.

Two smaller issues: in Proposition 2, the interval sqrt(2alpha-1) <= u < 1 is meaningless for alpha<1/2; that is a typo (should be sqrt(1-2alpha), presumably), but it should be corrected. And in Section 2.4, the derivative of the RHS of (10) at u=1/(2alpha) appears miscomputed: the value should be e^t/(1-alpha), not (1+2alpha)^2/(2(1-alpha)e^t). Positivity still holds, so the conclusion survives, but the displayed value is wrong.

Overall this is a serious paper with one genuine gap and some minor blemishes. It deserves a proper referee, and I would send it back for revision rather than reject. If the contour issue is resolved, the paper is publishable essentially as is.

Referee report

Significance

If correct, the results substantially extend the previously known α=1/2 case to arbitrary equal-rank projections. The characteristic-curve derivation of Theorem 1 gives a unified proof that also recovers the known α=1/2 formula, and the saddle point analyticity result is advertised as a headline contribution. The dynamical Kunisky identity is an original and interesting contribution that connects a stationary distributional identity with a PDE statement for finite-time Cauchy-Stieltjes transforms. The paper's computations are detailed and mostly coherent, and the dependence on prior results is explicit and non-circular. However, two load-bearing issues—a false statement in Proposition 2 for α<1/2 and an unproved contour-existence assumption in Proposition 3—prevent the results from being fully acceptable in their present form.

Major objections (2)

[§3.1, Proposition 2] The statement for α<1/2 says that Ṽ_t([0,∞)) ⊊ (−1,1) for t∈[0,2], but the proof itself shows Ṽ_t(0)=1 and lim_{u→∞} Ṽ_t(u)=+∞, so the image is an interval of the form [m,∞) with m>−1, which is not a subset of (−1,1). The displayed symbol ⊊ appears to be a typo for a non-containment symbol, but as written the claimed phase-transition property for α<1/2 is false and the surrounding text in the introduction repeats the same claim. This needs correction: either the intended statement is different (e.g., that the image is not all of (−1,1)) or the proof must be revised to establish a meaningful property.
[§4.2, proof of Proposition 3] The saddle point proof assumes the existence of a simple closed contour γ inside the open disc |w|<α, encircling 0, avoiding the singularities at 0 and −(1−α), and passing through both critical points w_{+,+}(t) and w_{+,-}(t) with prescribed tangent directions. For α>1/2, the point −(1−α) lies strictly inside the disc on the negative real axis, and the two saddle points are also on that ray, so the domain is multiply connected and the simultaneous tangent conditions are not automatic. The sentence 'we assume that γ is chosen' does not provide an argument. Since the asymptotic expansion for a_n(t) and the conclusion that the local inverse of χ_t^{(α)} is analytic in the open unit disc rest on this contour, a proof of existence (or a different argument establishing the needed decay) is required.

Minor objections (4)

[§2.4, eq. (10)] The displayed value of the derivative of the RHS of (10) at u=1/(2α) seems inconsistent with the preceding computation of ∂_u V^{(α)}_t(1/(2α)): the paper gives (1+2α)^2/(2(1−α)e^t), while a direct computation appears to yield e^t/(1−α). Since the positivity of the derivative is what is used for local invertibility, the conclusion of Theorem 1 is unaffected, but the displayed formula should be corrected or checked.
[§3.1, proof of Proposition 2] In the α<1/2 case, the inequality is stated as 'valid for any √(2α−1) ≤ u < 1', but √(2α−1) is not real for α<1/2. The intended lower bound is √(1−2α), as used earlier in the same proof. This is a typo but should be fixed.
[§4.2, Proposition 3] The asymptotic formula contains redundant factors: both sides involve e^{-nt}, so the statement could be simplified to an asymptotic for a_n(t) alone. This is only a presentation issue.
[Abstract and Introduction] The phrase 'a α-deformation' should be 'an α-deformation' in the abstract and in the introduction, and the same construction appears with inconsistent notation V^{(α)}_{4αt} versus V^{(α)}_t in different sections.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

This paper derives an explicit local formula for the moment generating function of the equal-rank free Jacobi process for all α in (0,1), extending the α=1/2 case, and proves mapping properties plus a dynamical identity.

The reading

This paper studies the spectral distribution at finite times of the free Jacobi process when the two compressing projections have the same normalized rank α in (0,1). Its main result is an explicit local expression for the moment generating function $M_t^{{(α)}}$(z) around z=0 for arbitrary α, generalizing the previously solved case α=1/2. The expression is built from a map $J^{{(α)}}$_{4αt} that is the local inverse of an α-deformed χ-transform, and the paper proves mapping and analyticity properties of this map for large times and α≥1/2. It also gives a dynamical version of a known stationary identity relating the equal-rank process to the rank-(1/2,α) process. A sympathetic reader would care because this makes the finite-time spectrum of a natural operator-valued process accessible through one explicit analytic object.

What carries the argument

The engine is the characteristic-curve analysis of the transport PDE ∂$_tM_t^{{(α)}}$(z)=-z∂_z[(1-2α)$M_t^{{(α)}}$(z)+α(1-z)($M_t^{{(α)}}$(z))^2], which reduces along curves to a Riccati equation in one variable. Solving that equation produces the Möbius-exponential map $V^{{(α)}}$_{4αt}(u) and its local inverse $J^{{(α)}}$_{4αt}; local inversion of the characteristic-curve relation gives the formula in Theorem 1. For analyticity of the inverse, the paper applies the saddle point method to the deformed χ-transform $χ_t^{{(α)}}$(u), using Cauchy coefficient integrals over contours inside the disc of radius α; four critical points replace the two appearing at α=1/2, and real analysis of the second derivatives and of t+Re $φ_t^{{(α)}}$(w_{+,±}) yields exponential decay of the coefficients.

Extended reading notes

Core claim

Under the initial condition $μ_0^{{(α,α)}}$=δ_1, the paper proves that for every α∈(0,1) and t>0, the moment generating function is locally given by $M_t^{{(α)}}$(z)=\frac{-A(α)+\sqrt{A(α)^2z+(1-z)[$J^{{(α)}}$_{4αt}(z)]^2}}{1-z}, where A(α)=(1-2α)/(2α) and $J^{{(α)}}$_{4αt} is a local inverse at 1/(2α) of the map u↦4ξ_{2t}(u)/(1+ξ_{2t}(u))^2 deformed by α; equivalently, J is the local inverse of $V^{{(α)}}$_{4αt}(u)=\frac{(u-A(α)-1)(u-A(α))}{(u+A(α)+1)(u+A(α))}$e^{{2αut}}$. For α≥1/2 and t≥t_1(α), a saddle point analysis shows that the inverse of the deformed χ-transform u↦$χ_t^{{(α)}}$(u)=\frac{u(u+α)}{(u+1)(u+1-α)}$e^{{(1+2u)t}}$ is analytic in the open unit disc with exponentially decaying Taylor coefficients, with four critical points in place of the two that appear at α=1/2. The paper also proves a finite-time analogue of a stationary identity: after the map x↦(2x-1)^2 and the time change t/2, the normalized Cauchy-Stieltjes transforms of $μ_t^{{(α,α)}}$ and μ_{t/2}^{(1/2,α)} satisfy the same transport PDE.

Load-bearing premise

The saddle point proof in Section 4 assumes a contour γ inside the open disc of radius α that passes through the two critical points w_{+,+} and w_{+,-} with prescribed tangent directions and avoids all singularities; the paper states this assumption without proving that such a contour exists.

Editorial extensions

If this is right

For every α∈(0,1), the finite-time moments of the equal-rank free Jacobi process can in principle be read off from the Taylor coefficients of a single locally defined map J^{(α)}_{4αt}, extending the explicit α=1/2 formula.
For α≥1/2 and t≥t_1(α), the inverse of the α-deformed χ-transform is analytic in the unit disc with exponentially decaying coefficients; the phase transition seen at t=2 for α=1/2 splits into two times t_0(α)≤2≤t_1(α) when α>1/2.
The stationary identity relating μ_∞^{(α,α)} and the pushforward of μ_∞^{(1/2,α)} under x↦(2x-1)^2 follows from the rank-1/2 compression property of the Nica-Speicher semigroup, not only from the explicit density formulas.
Dynamically, the normalized Cauchy-Stieltjes transforms of μ_t^{(α,α)} and of the pushforward of μ_{t/2}^{(1/2,α)} under x↦(2x-1)^2 satisfy the same PDE; if their initial data agree, the finite-time distributions coincide for all t.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the local inverse formula is robust, it offers a computational route for the moments for all α and t: the coefficients of J^{(α)}_{4αt} could be computed by Lagrange inversion and compared with matrix-model simulations, bypassing the more intricate moment formulas.
The assumed saddle-point contour is the point most likely to need repair: a numerical search for a valid contour γ inside the disc of radius α through w_{+,+} and w_{+,-} for α just above 1/2 and t just above t_1(α) would either confirm the proof step or locate a gap.
The equality of PDEs without initial-data matching suggests a broader family of pairs (P,Q), not necessarily free, for which the two finite-time distributions remain related; the paper itself leaves open whether such pairs exist beyond freeness.
The quadratic transformation with Jacobi polynomials hints that other rank-parameter identities might exist for higher values of k in the Nica-Speicher semigroup, though no higher-order analogue is known.

Original 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 $\alpha \in (0,1)$. 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 $\alpha = 1/2$. Moreover, the obtained characteristic curves are encoded by an $\alpha$-deformation of the compositional inverse of the $\chi$-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 $\alpha$-deformation of the $\chi$-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 $(\alpha, \alpha)$ and $(1/2,\alpha)$ 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 $1/2$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,840 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.52 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

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 $\alpha \in (0,1)$. 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 $\alpha = 1/2$. Moreover, the obtained characteristic curves are encoded by an $\alpha$-deformation of the compositional inverse of the $\chi$-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 $\alpha$-deformation of the $\chi$-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 $(\alpha, \alpha)$ and $(1/2,\alpha)$ 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 $1/2$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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.	background 1.00	0.06	0.03	asserted 1.00
s1	In this paper, we derive spectral properties of the free Jacobi process associated with projections having the same rank $\alpha \in (0,1)$.	contribution 0.99	0.95	0.23	asserted 1.00
s2	To start with, we determine the characteristic curves of the partial differential equation satisfied by the moment generating function of its spectral distribution.	method 0.94	0.93	0.04	asserted 1.00
s3	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 $\alpha = 1/2$.	result 0.70	0.91	0.14	asserted 1.00
s4	Moreover, the obtained characteristic curves are encoded by an $\alpha$-deformation of the compositional inverse of the $\chi$-transform of the spectral distribution of the free unitary Brownian motion.	result 0.86	0.86	0.12	asserted 1.00
s5	In this respect, we study mapping properties of this deformation and use the saddle point method to prove that the compositional inverse of a $\alpha$-deformation of the $\chi$-transform of the free unitary Brownian motion is analytic in the open unit disc (for large enough time $t$).	result 0.48	0.94	0.16	asserted 1.00
s6	The last part of the paper is devoted to a dynamical version of a recent identity pointed out by T. Kunisky in \cite{Kun}.	contribution 0.96	0.74	0.05	asserted 0.96
s7	Actually, this identity relates the stationary distributions of the free Jacobi processes corresponding to the sets of parameters $(\alpha, \alpha)$ and $(1/2,\alpha)$ respectively and we explain how it follows from the Nica-Speicher semi-group.	contribution 0.38	0.56	0.12	asserted 1.00
s8	Our dynamical version then relates the partial differential equations of the Cauchy-Stieltjes transforms of the densities of the finite-time spectral distributions.	result 0.53	0.88	0.11	asserted 1.00
s9	It also raises the problem of whether a dynamical analogue of the Nica-Speicher semi-group exists when the compressing projection has rank $1/2$.	future work 0.91	0.51	0.07	hedged possibility 0.38

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (28)

[12]

N. Demni . Free Jacobi processes. J. Theor. Proba . 21 (2008), 118-143

View on Pith 2008
[15]

N. Demni, T. Hamdi, T. Hmidi . Spectral distribution of the free Jacobi process. Indiana Univ. Math. J . 61 (2012), 1351-1368

View on Pith 2012
[20]

T. Hamdi . Spectral distribution of the free Jacobi process, revisit ed. Anal. PDE . 11, (8) 2137-2148, 2018

View on Pith 2018
[1]

G. Aubrun. Principal angles between random subspaces and polynomial s in two free projections. Conﬂuentes Mathematici , V olume 13, (2021) no. 2, pp. 3-10

View on Pith 2021
[2]

S. Belinschi, H. Bercovici . Partially deﬁned semigroups relative to multiplicative f ree convolution. Int. Math. Res. Not . (2005), no. 2, 65-101

View on Pith 2005
[3]

H. Bercovici, D. V. Voiculescu . L´ evy-Hincin type theorems for multiplicative and additi ve free convolution. Paciﬁc J. Math. 153. (1992), no. 2, 217-248

View on Pith 1992
[4]

P. Biane. Free Brownian motion, free stochastic calculus and random m atrices. Fields Inst. Commun., 12. American Mathematical Society, Providence , RI, 1997, 1-19

View on Pith 1997
[5]

P. Biane. Segal–Bargmann Transform, Functional Calculus on Matrix S paces and the Theory of Semi-circular and Circular Systems. J. Funct. Analysis. V olume 144, Issue 1, (1997), 232-286

View on Pith 1997
[6]

P. Biane, M. Capitaine, A. Guionnet . Large deviation bounds for matrix Brownian motion. Invent. Math . 152, (2003), no.2, 433-459

View on Pith 2003
[7]

M. Capitaine, M. Casalis . Asymptotic freeness by generalized moments for Gaussian a nd Wishart matrices. Application to beta random matrices. Indiana Univ. Math. J. 53, (2004), no. 2, 397-431

View on Pith 2004
[8]

M. Capitaine, C. Donati-Martin . Free Wishart processes. J. Theoret. Probab. 18, (2005), no. 2, 413-438

View on Pith 2005
[9]

B. Collins. Product of random projections, Jacobi ensembles and univer sality problems arising from free probability. Probab. Theory Relat. Fields , 133, 315-344 (2005)

View on Pith 2005
[10]

L. Deleaval, N. Demni . Moments of the Hermitian matrix Jacobi process. J. Theoret. Probab. 31, (2018), no.3, 1759-1778

View on Pith 2018
[11]

N. Demni . The Laguerre process and generalized Hartman-W atson law. Bernoulli, 13, (2007), no. 2, 556-580

View on Pith 2007
[13]

N. Demni . Free Jacobi process associated with one projection: local inverse of the ﬂow. Complex Anal. Oper. Theory . 10, (2016), no. 3, 527-543

View on Pith 2016
[14]

N. Demni, T. Hamdi . Relating moments of self-adjoint polynomials in two ortho gonal projections. Adv. Oper. Theory . 8, (2023), no. 1, Paper No. 7, 19 pp

View on Pith 2023
[16]

N. Demni, T. Hamdi, A. Souaissi . The Hermitian Jacobi process: a simpliﬁed formula for the m oments and applications to optical ﬁber MIMO channels. Funct. Anal. Appl . 54 (2020), no. 4, 257-271

View on Pith 2020
[17]

N. Demni, T. Hamdi . Moments of the free Jacobi process: a matrix approach. Canadian Mathematical Bulletin (2025): 1–26

View on Pith 2025
[18]

F.J. Dyson . A Brownian-motion model for the eigenvalues of a random mat rix. J. Math. Phys. , 3, (1962), 1191-1198

View on Pith 1962
[19]

D. J. Gross, A. Matytsin . Some properties of large-N two-dimensional Yang-Mills th eory. Nuclear Physics B . Volume 437, Issue 3, 20 March 1995, Pages 541-584

View on Pith 1995
[21]

T. Hamdi . Liberation, free mutual information and orbital free entr opy. Nagoya Math. J . 239 (2020), 205-231

View on Pith 2020
[22]

T. Koornwinder . Quadratic transformations for orthogonal polynomials in one and two variables. Adv. Stud. Pure Math. , 419-447, (2018)

View on Pith 2018
[23]

D. Kunisky . Generic MANOV A limit theorems for products of projections . Available on arXiv

View on Pith
[24]

J. Mingo, R. Speicher . Free Probability and Random Matrices. Fields Institute Monographs, Springer , 2017

View on Pith 2017
[25]

A. Nica, R. Speicher . Lectures on the combinatorics of free probability. London M athematical Society Lecture Note Series , Vol. 335, Cambridge University Press, 2006

View on Pith 2006
[26]

D. Voiculescu. The analogue of entropy and of Fisher’s information measur e in free probability theory. VI. Liberation and mutual free information. Adv. Math . 146, (1999), 101-166

View on Pith 1999
[27]

K. W. Wachter. The limiting empirical measure of multiple discriminant ra tios. The Annals of Statistics , pages 937-957, 1980

View on Pith 1980
[28]

J. Warren, M. Yor . The Brownian burglar: conditioning Brownian motion by its local time process. Lecture Notes in Math., 1686, Springer-Verlag, Berlin, (1998), 328-342. Department of Mathematics, New York University in Abu Dhabi, Saa diyat Island, P.O. Box 129188, Abu Dhabi, UAE Email address : nd2889@nyu.edu Department of Management Information Systems,...

View on Pith 1998

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [12]

N. Demni . Free Jacobi processes. J. Theor. Proba . 21 (2008), 118-143

View on Pith 2008

[2] [15]

N. Demni, T. Hamdi, T. Hmidi . Spectral distribution of the free Jacobi process. Indiana Univ. Math. J . 61 (2012), 1351-1368

View on Pith 2012

[3] [20]

T. Hamdi . Spectral distribution of the free Jacobi process, revisit ed. Anal. PDE . 11, (8) 2137-2148, 2018

View on Pith 2018

[4] [1]

G. Aubrun. Principal angles between random subspaces and polynomial s in two free projections. Conﬂuentes Mathematici , V olume 13, (2021) no. 2, pp. 3-10

View on Pith 2021

[5] [2]

S. Belinschi, H. Bercovici . Partially deﬁned semigroups relative to multiplicative f ree convolution. Int. Math. Res. Not . (2005), no. 2, 65-101

View on Pith 2005

[6] [3]

H. Bercovici, D. V. Voiculescu . L´ evy-Hincin type theorems for multiplicative and additi ve free convolution. Paciﬁc J. Math. 153. (1992), no. 2, 217-248

View on Pith 1992

[7] [4]

P. Biane. Free Brownian motion, free stochastic calculus and random m atrices. Fields Inst. Commun., 12. American Mathematical Society, Providence , RI, 1997, 1-19

View on Pith 1997

[8] [5]

P. Biane. Segal–Bargmann Transform, Functional Calculus on Matrix S paces and the Theory of Semi-circular and Circular Systems. J. Funct. Analysis. V olume 144, Issue 1, (1997), 232-286

View on Pith 1997

[9] [6]

P. Biane, M. Capitaine, A. Guionnet . Large deviation bounds for matrix Brownian motion. Invent. Math . 152, (2003), no.2, 433-459

View on Pith 2003

[10] [7]

M. Capitaine, M. Casalis . Asymptotic freeness by generalized moments for Gaussian a nd Wishart matrices. Application to beta random matrices. Indiana Univ. Math. J. 53, (2004), no. 2, 397-431

View on Pith 2004

[11] [8]

M. Capitaine, C. Donati-Martin . Free Wishart processes. J. Theoret. Probab. 18, (2005), no. 2, 413-438

View on Pith 2005

[12] [9]

B. Collins. Product of random projections, Jacobi ensembles and univer sality problems arising from free probability. Probab. Theory Relat. Fields , 133, 315-344 (2005)

View on Pith 2005

[13] [10]

L. Deleaval, N. Demni . Moments of the Hermitian matrix Jacobi process. J. Theoret. Probab. 31, (2018), no.3, 1759-1778

View on Pith 2018

[14] [11]

N. Demni . The Laguerre process and generalized Hartman-W atson law. Bernoulli, 13, (2007), no. 2, 556-580

View on Pith 2007

[15] [13]

N. Demni . Free Jacobi process associated with one projection: local inverse of the ﬂow. Complex Anal. Oper. Theory . 10, (2016), no. 3, 527-543

View on Pith 2016

[16] [14]

N. Demni, T. Hamdi . Relating moments of self-adjoint polynomials in two ortho gonal projections. Adv. Oper. Theory . 8, (2023), no. 1, Paper No. 7, 19 pp

View on Pith 2023

[17] [16]

N. Demni, T. Hamdi, A. Souaissi . The Hermitian Jacobi process: a simpliﬁed formula for the m oments and applications to optical ﬁber MIMO channels. Funct. Anal. Appl . 54 (2020), no. 4, 257-271

View on Pith 2020

[18] [17]

N. Demni, T. Hamdi . Moments of the free Jacobi process: a matrix approach. Canadian Mathematical Bulletin (2025): 1–26

View on Pith 2025

[19] [18]

F.J. Dyson . A Brownian-motion model for the eigenvalues of a random mat rix. J. Math. Phys. , 3, (1962), 1191-1198

View on Pith 1962

[20] [19]

D. J. Gross, A. Matytsin . Some properties of large-N two-dimensional Yang-Mills th eory. Nuclear Physics B . Volume 437, Issue 3, 20 March 1995, Pages 541-584

View on Pith 1995

[21] [21]

T. Hamdi . Liberation, free mutual information and orbital free entr opy. Nagoya Math. J . 239 (2020), 205-231

View on Pith 2020

[22] [22]

T. Koornwinder . Quadratic transformations for orthogonal polynomials in one and two variables. Adv. Stud. Pure Math. , 419-447, (2018)

View on Pith 2018

[23] [23]

D. Kunisky . Generic MANOV A limit theorems for products of projections . Available on arXiv

View on Pith

[24] [24]

J. Mingo, R. Speicher . Free Probability and Random Matrices. Fields Institute Monographs, Springer , 2017

View on Pith 2017

[25] [25]

A. Nica, R. Speicher . Lectures on the combinatorics of free probability. London M athematical Society Lecture Note Series , Vol. 335, Cambridge University Press, 2006

View on Pith 2006

[26] [26]

D. Voiculescu. The analogue of entropy and of Fisher’s information measur e in free probability theory. VI. Liberation and mutual free information. Adv. Math . 146, (1999), 101-166

View on Pith 1999

[27] [27]

K. W. Wachter. The limiting empirical measure of multiple discriminant ra tios. The Annals of Statistics , pages 937-957, 1980

View on Pith 1980

[28] [28]

J. Warren, M. Yor . The Brownian burglar: conditioning Brownian motion by its local time process. Lecture Notes in Math., 1686, Springer-Verlag, Berlin, (1998), 328-342. Department of Mathematics, New York University in Abu Dhabi, Saa diyat Island, P.O. Box 129188, Abu Dhabi, UAE Email address : nd2889@nyu.edu Department of Management Information Systems,...

View on Pith 1998