arXiv subjects

Nizar Demni

Publications and source records attributed to Nizar Demni.

At least 19 recordsLinked to original sources

arXiv2026

Brownian motion on the complex general group and the non-compact Jacobi process

In this paper, we introduce and study a new matrix-valued process built out of truncations of the Brownian motion on complex positive definite matrices and of its inverse. In particular, we prove that its eigenvalue process is a non-compact Jacobi particle system on $(1,\infty)$. Surprisingly, the same particle system occurred in \cite{BDW} in relation with the Brownian motion on the non-compact complex Grassmann manifold. In this respect, we provide some evidence supporting this occurrence and based on the realization of the Grassmann manifold as a generalized Poincaré disc. We proceed further to the spectral study of the large size limit of our matrix model and prove results that parallel those obtained by the second author for the free Jacobi process.

math.PR↗

arXiv2026

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

Brownian motion and stochastic areas on complex full flag manifolds

We show that the Brownian motion on the complex full flag manifold can be represented by a matrix-valued diffusion obtained from the unitary Brownian motion. This representation actually leads to an explicit formula for the characteristic function of the joint distribution of the stochastic areas on the full flag manifold. The limit law for those stochastic areas is shown to be a multivariate Cauchy distribution with independent and identically distributed entries. Using a deep connection between area functionals on the flag manifold and winding functionals on complex spheres, we establish new results about simultaneous Brownian windings on the complex sphere and their asymptotics. As a byproduct, our work also unveils a new probabilistic interpretation of the Jacobi operators and polynomials on simplices.

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↗

arXiv2025

Polyanalytic Hermite polynomials associated with the elliptic Ginibre model

Motivated by the connection between the eigenvalues of the complex Ginibre matrix model and the magnetic Laplacian in the complex plane, we derive analogues of the complex Hermite polynomials for the elliptic Ginibre model. To this end, we appeal to squeezed creation and annihilation operators arising from the Bogoliubov transformation of creation and annihilation operators on the Bargmann-Fock space. The obtained polynomials are then expressed as linear combinations of products of Hermite polynomials and share the same orthogonality relation with holomorphic Hermite polynomials. Moreover, this expression allows to identify them with the 2D-Hermite polynomials associated to a unimodular complex symmetric 2x2 matrix. Afterwards, we derive, for any Landau level, a closed formula for the kernel of the isometry mapping the basis of (rescaled) holomorphic Hermite polynomials to the corresponding complex Hermite polynomials. This kernel is also interpreted in terms of the two-photon coherent states and the metaplectic representation of the SU(1,1) group.

math-ph↗

arXiv2024

Orthogonal Random Features: Explicit Forms and Sharp Inequalities

Random features have been introduced to scale up kernel methods via randomization techniques. In particular, random Fourier features and orthogonal random features were used to approximate the popular Gaussian kernel. Random Fourier features are built in this case using a random Gaussian matrix. In this work, we analyze the bias and the variance of the kernel approximation based on orthogonal random features which makes use of Haar orthogonal matrices. We provide explicit expressions for these quantities using normalized Bessel functions, showing that orthogonal random features does not approximate the Gaussian kernel but a Bessel kernel. We also derive sharp exponential bounds supporting the view that orthogonal random features are less dispersed than random Fourier features.

cs.LG↗

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↗

arXiv2023

Stochastic areas, Horizontal Brownian Motions, and Hypoelliptic Heat Kernels

The monograph is devoted to the study of stochastic area functionals of Brownian motions and of the associated heat kernels on Lie groups and Riemannian manifolds. It is essentially self-contained and as such can serve as a textbook on the theory of Brownian motions and horizontal Brownian motions on manifolds. Emphasis is put on concrete examples which allows us to concretely illustrate the rich and deep interactions between stochastic calculus, Riemannian and sub-Riemannian geometry, the theory of complex and quaternionic symmetric spaces and random matrices.

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

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

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

Generalized stochastic areas, Winding numbers, and hyperbolic Stiefel fibrations

We study the Brownian motion on the non-compact Grassmann manifold $\frac{\mathbf{U}(n-k,k)} {\mathbf{U}(n-k)\mathbf{U}(k)}$ and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group $\mathbf{U}(n-k,k)$ are then given.

math.PR↗

arXiv2021

Hartman-Watson distribution and hyperbolic-like Heat kernels

We relate Gruet formula for the heat kernel on real hyperbolic spaces to the commonly used one derived from Millson induction. The bridge between both formulas is settled by Yor result on the joint distribution of a Brownian motion and of its exponential functional at fixed time. This result allows further to relate Gruet formula with real parameter to the heat kernel of the hyperbolic Jacobi operator and to derive a new integral representation for the heat kernel of the Maass Laplacian. When applied to harmonic AN groups, Yor result yields also new a integral representation of their corresponding heat kernels which does not distinguish the parity of the dimension of the center of the Lie group N.

math.PR↗

arXiv2020

Explicit expressions of the Hua-Pickrell semi-group

In this paper, we study the one-dimensional Hua-Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semi-group density through the associated Legendre function in the cut. Next, we focus on the general (not necessarily stationary) case for which we prove an intertwining relation between Hua-Pickrell diffusions corresponding to different sets of parameters. Using Cauchy Beta integral on the one hand and Girsanov's Theorem on the other hand, we discuss the connection between the stationary and general cases. Afterwards, we prove our main result providing novel integral representations of the Hua-Pickrell semi-group density, answering a question raised by Alili, Matsumoto and Shiraishi (Séminaire de Probabilités, 35, 2001). To this end, we appeal to the semi-group density of the Maass Laplacian and extend it to purely-imaginary values of the magnetic field. In the last section, we use the Karlin-McGregor formula to derive an expression of the semi-group density of the multi-dimensional Hua-Pickrell particle system introduced by T. Assiotis.

math.PR↗

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

Polyanalytic Reproducing Kernels on the Quantized Annulus

While dealing with the constant-strength magnetic Laplacian on the annulus, we complete J. Peetre's work. In particular, the eigenspaces associated with its discrete spectrum are true-polyanalytic spaces with respect to the invariant Cauchy-Riemann operator, and we write down explicit formulas for their reproducing kernels. The latter are expressed by means of the fourth Jacobi theta function and of its logarithmic derivatives when the magnetic field strength is an integer. Under this quantization condition, we also derive the transformation rule satisfied by the reproducing kernel under the automorphism group of the annulus.

math-ph↗

arXiv2020

Generalized Bessel functions of dihedral-type: expression as a series of confluent Horn functions and Laplace-type integral representation

In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as a infinite series of confluent Horn functions. The key ingredient leading to this expression is an extension of an identity involving Gegenbauer polynomials proved in a previous paper by the authors, together with the use of the Poisson kernel for these polynomials. In particular, we derive an integral representation of this generalized Bessel function over the standard simplex. The second part of this paper is concerned with even dihedral systems and boundary values of one of the variables. Still assuming that the multiplicity function is constant, we obtain a Laplace-type integral representation of the corresponding generalized Bessel function, which extends to all even dihedral systems a special instance of the Laplace-type integral representation proved in \cite{Amr-Dem}.

math.CA↗

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↗