arXiv subjects

Tarek Hamdi

Publications and source records attributed to Tarek Hamdi.

At least 19 recordsLinked to original sources

arXiv2025

Finite free probability and $S$ transforms of Jacobi processes

We calculate the averaged characteristic polynomial and its finite $S-$ transform for the Hermitian Jacobi process at any fixed time $t$. We give a direct proof that this sequence of polynomials solves the backward heat equation linked to the one-dimensional Jacobi operator. We also expand the averaged characteristic polynomials in terms of Jacobi polynomials, using the dual Cauchy identity for multivariate Jacobi polynomials and their mutual orthogonality. The finite free $S-$transform is the finite free version of the free $S$ transform in that it behaves the same way with respect to the (finite) free multiplicative convolution. We present a finite difference and differential equation that the finite free $S$ transform of the averaged characteristic polynomials of the Hermitian Jacobi Process satisfies. In the high-dimensional limit, this yields a partial differential equation for the free $S$- transform of the free Jacobi process. We also prove a general technical lemma about the convergence of the finite differences of the finite free $ S$- transform.

math.PR↗

arXiv2025

Spectral distribution of the free Jacobi process with equal rank projections

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 $1/2$.

math.PR↗

arXiv2024

Quantum Mechanics of Arc-Sine and Semi-Circle Distributions: A Unified Approach

This paper continues the program of applying beyond physics the technique of \textbf{probabilistic quantization} and extending to the quantum mechanics associated with the arc--sine distributions our previous results on the semi--circle distribution. We derive analytical expressions for the momentum and kinetic energy operators using the arc--sine weighted Hilbert transform and express corresponding evolutions as Neumann series of Bessel functions. These series are applicable in various physical problems and in solving certain mixed difference equations and differential equations. Moreover, exploiting the similarity between the Jacobi sequences of the semi-circle and arc-sine measures, we establish a unified formulation of their quantum mechanics. We introduce the semicircle and arc--sine exponential vectors and the corresponding coherent states and prove that, for both measures, the vacuum distributions of the number operator in these states (arc--sine photon statistics) are a \textit{perturbation} of the geometric distribution (Gibbs states in Boson physics: see the Introduction below for a discussion of the physical meaning of this perturbation). The $*$--Lie algebra generated by canonical creation and annihilation operators of both probability measures is isomorphic to the $*$--Lie algebra generated by all rank-one operators in corresponding $L^2$-spaces. The paper concludes with appendices that discuss the integral and Neumann series representations of the $1$--parameter unitary groups generated by momentum in semi-circle and arc-sine cases.

math-ph↗

arXiv2024

Moments of the free Jacobi process: a matrix approach

We compute the large size limit of the moment formula derived in \cite{DHS} for the Hermitian Jacobi process at fixed time. Our computations rely on the polynomial division algorithm which allows to obtain cancellations similar to those obtained in Lemma 3 in \cite{Bia}. In particular, we identify the terms contributing to the limit and show they satisfy a double recurrence relation. We also determine explicitly some of them and revisit a special case relying on Carlitz summation identity for terminating $1$-balanced ${}_4F_3$ functions taken at unity.

math.PR↗

arXiv2022

Summing free unitary Brownian motions with applications to quantum information

Motivated by quantum information theory, we introduce a dynamical random state built out of the sum of $k \geq 2$ independent unitary Brownian motions. In the large size limit, its spectral distribution equals, up to a normalising factor, that of the free Jacobi process associated with a single self-adjoint projection with trace $1/k$. Using free stochastic calculus, we extend this equality to the radial part of the free average of $k$ free unitary Brownian motions and to the free Jacobi process associated with two self-adjoint projections with trace $1/k$, provided the initial distributions coincide. In the single projection case, we derive a binomial-type expansion of the moments of the free Jacobi process which extends to any $k \geq 3$ the one derived in \cite {DHH} in the special case $k=2$. Doing so give rise to a non normal (except for $k=2$) operator arising from the splitting of a self-adjoint projection into the convex sum of $k$ unitary operators. This binomial expansion is then used to derive a pde for the moment generating function of this non normal operator and for which we determine the corresponding characteristic curves.

math.PR↗

arXiv2022

Open Quantum Random Walks and Quantum Markov chains on Trees II: The recurrence

In the present paper, we construct QMC (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution of OQRW. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in Ref. [9]. Using such a construction, we are able to construct QMCs on tress associated with OQRW. Our investigation leads to the detection of the phase transition phenomena within the proposed scheme. This kind of phenomena appears first time in this direction. Moreover, mean entropies of QMCs are calculated.

math-ph↗

arXiv2022

Open Quantum Random Walks and Quantum Markov chains on Trees I: Phase transitions

In the present paper, we construct QMC (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution $P_ρ$ of OQRW. However, we are going to look at the probability distribution as a Markov field over the Cayley tree. Such kind of consideration allows us to investigated phase transition phenomena associated for OQRW within QMC scheme. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in Ref. [10]. Using such a construction, we are able to construct QMCs on tress associated with OQRW. Our investigation leads to the detection of the phase transition phenomena within the proposed scheme. This kind of phenomena appears first time in this direction. Moreover, mean entropies of QMCs are calculated.

math-ph↗

arXiv2022

Relating moments of self-adjoint polynomials in two orthogonal projections

Given two orthogonal projections $\{P,Q\}$ in a non commutative tracial probability space, we prove relations between the moments of $P+Q$, of $\sqrt{-1}(PQ-QP)$ and of $P+QPQ$ and those of the angle operator $PQP$. Our proofs are purely algebraic and enumerative and does not assume $P,Q$ satisfying Voiculescu's freeness property or being in general position. As far as the sum and the commutator are concerned, the obtained relations follow from binomial-type formulas satisfied by the orthogonal symmetries associated to $P$ and $Q$ together with the trace property. In this respect, they extend those corresponding to the cases where one of the two projections is rotated by a free Haar unitary operator or more generally by a free unitary Brownian motion. As to the operator $P+QPQ$, we derive autonomous recurrence relations for the coefficients (double sequence) of the expansion of its moments as linear combinations of those of $PQP$ and determine explicitly few of them. These relations are obtained after a careful analysis of the structure of words in the alphabet $\{P, QPQ\}$. We close the paper by exploring the connection of our previous results to the so-called Kato's dual pair. Doing so leads to new identities satisfied by their moments.

math.PR↗

arXiv2021

The quantum mechanics canonically associated to free probability Part I: Free momentum and associated kinetic energy

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the \textbf{standard semi--circle random variable} $X$, characterized by the fact that its probability distribution is the semi--circle law $μ$ on $[-2,2]$. We prove that, in the identification of $L^2([-2,2],μ)$ with the $1$--mode interacting Fock space $Γ_μ$, defined by the orthogonal polynomial gradation of $μ$, $X$ is mapped into position operator and its canonically associated momentum operator $P$ into $i$ times the $μ$--Hilbert transform $H_μ$ on $L^2([-2,2],μ)$. In the first part of the present paper, after briefly describing the simpler case of the $μ$--harmonic oscillator, we find an explicit expression for the action, on the $μ$--orthogonal polynomials, of the semi--circle analogue of the translation group $e^{itP}$ and of the semi--circle analogue of the free evolution $e^{itP^2/2}$ respectively in terms of Bessel functions of the first kind and of confluent hyper--geometric series. These results require the solution of the \textit{inverse normal order problem} on the quantum algebra canonically associated to the classical semi--circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of $e^{-tH_μ}$ and $e^{-itH_μ^2/2}$ on the $μ$--orthogonal polynomials is difficult, % aaa ask T if it is solved the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.

math.OA↗

arXiv2021

On star-Moments of the compression of the free unitary Brownian motion by a free projection

In this paper, we derive explicit expressions for the moments and for the mixed moments of the compression of a free unitary Brownian motion by a free projection. While the moments of this non-normal operator are readily derived using analytical or combinatorial methods, we only succeeded to derive its mixed ones after solving a non-linear partial differential equation for their two-variables generating function. Nonetheless, the combinatorics of non crossing partitions lead to another expression of the lowest-order mixed moment. We shall also give some interest in odd alternating moments. In particular, we derive a linear partial differential equation for their generating function and discuss the combinatorial approach to these moments when the rank of the projection equals $1/2$.

math.OA↗

arXiv2021

Quantum Markov Chains on the Comb graphs: Ising model

In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an application of this construction, it is proved the existence of the disordered phase for the Ising type models (within QMC scheme) over the Comb graphs. Moreover, it is also established that the associated QMC has clustering property with respect to translations of the graph. We stress that this paper is the first one where a nontrivial example of QMC over non-regular graphs is given.

math-ph↗

arXiv2020

Support of the Brown measure of the product of a free unitary Brownian motion by a free self-adjoint projection

The first part of this paper is devoted to the Brown measure of the product of the free unitary Brownian motion by an arbitrary free non negative operator. Our approach follows the one recently initiated by Driver-Hall-Kemp though there are substantial differences at the analytical side. In particular, the corresponding Hamiltonian system is completely solvable and the characteristic curve describing the support of the Brown measure has a non-constant (in time) argument. In the second part, we specialize our findings to the product of the free unitary Brownian motion by a free self-adjoint projection and obtain an explicit description of its support.

math.SP↗

arXiv2020

The Hermitian Jacobi process: simplified formula for the moments and application to optical fibers MIMO channels

Using a change of basis in the algebra of symmetric functions, we compute the moments of the Hermitian Jacobi process. After a careful arrangement of the terms and the evaluation of the determinant of an `almost upper-triangular' matrix, we end up with a moment formula which is considerably simpler than the one derived in \cite{Del-Dem}. As an application, we propose the Hermitian Jacobi process as a dynamical model for optical fibers MIMO channels and compute its Shannon capacity for small enough power at the transmitter. Moreover, when the size of the Hermitian Jacobi process is larger than the moment order, our moment formula may be written as a linear combination of balanced terminating ${}_4F_3$-series evaluated at unit argument.

math.PR↗

arXiv2020

Schur-Weyl duality and the Product of randomly-rotated symmetries by a unitary Brownian motion

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an autonomous ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression.

math.PR↗

arXiv2017

Spectral distribution of the free Jacobi process, revisited

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 $\pm1$ and an $L^\infty$-density on the unit circle $\mathbb{T}$, 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.

math.PR↗

arXiv2017

Free mutual information for two projections

The present paper provides a proof of $i^*( \mathbb{C}P+\mathbb{C}(I-P); \mathbb{C}Q+\mathbb{C}(I-Q) )=-χ_{orb}(P,Q)$ for two projections $P,Q$ without any extra assumptions. An analytic approach is adopted to the proof, based on a subordination result for the liberation process of symmetries associated with $P,Q$.

math.PR↗

arXiv2017

Liberation, free mutual information and orbital free entropy

We present here some connections between the liberation process for projections $(P,Q)\mapsto(P,U_tQU_t^*)$ and its counterpart $(R,S)\mapsto(R,U_tSU_t^*)$ for symmetries when the projections $\{P,Q\}$ and the symmetries $\{R,S\}$ are associated, where $U_t$ is a free unitary Brownian motion freely independent from $\{P,Q\}$ (and so $\{R,S\}$). We relate the moments of their actions on the operators $X_t:=PU_tQU_t^*$ and $Y_t:=U_tRU_t^*S$ and use this to prove a relationship between the corresponding spectral measures (hereafter $μ_t$ and $ν_t$). On the other hand, we focus in the process of unitary random variables $Y_t$ in the case of arbitrary trace values $τ(R),τ(S)$. More precisely, we use stochastic calculus to derive a partial differential equation (PDE for short) for its Herglotz transform and use it to develop subordination results in terms of Löwner equations. The paper is closed with an improved proof of $i^*\left( \mathbb{C}P+\mathbb{C}(I-P); \mathbb{C}Q+\mathbb{C}(I-Q) \right)=-χ_{orb}\left(P,Q\right)$ as an application.

math.PR↗

arXiv2016

Inverse of the flow and moments of the free Jacobi process associated with a single projection

This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we notice that the flow solves a radial Löwner equation and as such, the general theory of Löwner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov-Clark measure at $z=1$ of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that $z=1$ belongs only to the discrete spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow's inverse.

math.PR↗