Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMatrix Dufresne Identity
221
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove a version of the classical Dufresne identity for matrix processes. In particular, we show that the inverse Wishart laws on the space of positive definite r x r matrices can be realized by the infinite time horizon integral of M_t times its transpose in which t -> M_t is a drifted Brownian motion on the general linear group. This solves a problem in the study of spiked random matrix ensembles which served as the original motivation for this result. Various known extensions of the Dufresne identity (and their applications) are also shown to have analogs in this setting. For example, we identify matrix valued diffusions built from M_t which generalize in a natural way the scalar processes figuring into the geometric Levy and Pitman theorems of Matsumoto and Yor.
rate research
Read More
We consider a multichannel wire with a disordered region of length $L$ and a reflecting boundary. The reflection of a wave of frequency $omega$ is described by the scattering matrix $mathcal{S}(omega)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $mathcal{Q}=-mathrm{i}, mathcal{S}^daggerpartial_omegamathcal{S}$ is another important matrix encoding temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, $mathcal{S}=mathrm{e}^{2mathrm{i}kL}mathcal{U}_Lmathcal{U}_R$ (with $mathcal{U}_L=mathcal{U}_R^mathrm{T}$ in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix: $widetilde{mathcal{Q}} =mathcal{U}_R,mathcal{Q},mathcal{U}_R^dagger = (2L/v),mathbf{1}_N -mathrm{i},mathcal{U}_L^daggerpartial_omegabig(mathcal{U}_Lmathcal{U}_Rbig),mathcal{U}_R^dagger$, where $k$ is the wave vector and $v$ the group velocity. We demonstrate that $widetilde{mathcal{Q}}$ can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, $Ltoinfty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for $mathcal{Q}$s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, the exponential functional representation is used to calculate the first moments $langlemathrm{tr}(mathcal{Q})rangle$, $langlemathrm{tr}(mathcal{Q}^2)rangle$ and $langlebig[mathrm{tr}(mathcal{Q})big]^2rangle$. Finally we derive a partial differential equation for the resolvent $g(z;L)=lim_{Ntoinfty}(1/N),mathrm{tr}big{big( z,mathbf{1}_N - N,mathcal{Q}big)^{-1}big}$ in the large $N$ limit.
We investigate conditions under which the identity matrix $I_n$ can be continuously factorized through a continuous $Ntimes N$ matrix function $A$ with domain in $mathbb{R}$. We study the relationship of the dimension $N$, the diagonal entries of $A$, and the norm of $A$ to the dimension $n$ and the norms of the matrices that witness the factorization of $I_n$ through $A$.
We present a list of equivalent expressions and extensions of Bougerols celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity in many directions, we give some new results and possible extensions and we try to point out open questions.
Using a divergent Bass-Burdzy flow we construct a self-repelling one-dimensional diffusion. Heuristically, it can be interpreted as a solution to an SDE with a singular drift involving a derivative of the local time. We show that this self-repelling diffusion inverts the second Ray-Knight identity on the line. The proof goes through an approximation by a self-repelling jump processes that has been previously shown by the authors to invert the Ray-Knight identity in the discrete.
The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_beta(N,r)$ consist of $N times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uniform measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N to infty$. In the unitary case ($beta=2$), we also find an explicit formula for the empirical spectral density for each $N$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
