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

Nizar Demni (1) , Tarek Hamdi , Abdessatar Souissi

1 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: Unitary Brownian motion; Orthogonal projection; Jacobi Unitary Ensemble; Schur polynomials; Symmetric Jacobi polynomials; MIMO channels; Shannon capacity

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]

Nizar Demni, Tarek Hamdi, Abdessatar Souissi. The Hermitian Jacobi process: simplified formula for the moments and application to optical fibers MIMO channels. 2024. ⟨hal-03170472⟩