PDE's for the Dyson, Airy and Sine processes

Adler, Mark. "PDE's for the Dyson, Airy and Sine processes." Annales de l’institut Fourier 55.6 (2005): 1835-1846. <http://eudml.org/doc/116235>.

@article{Adler2005,
abstract = {In 1962, Dyson showed that the spectrum of a $n\times n$ random Hermitian matrix, whose entries (real and imaginary) diffuse according to $n^2$ independent Ornstein-Uhlenbeck processes, evolves as $n$ non-colliding Brownian particles held together by a drift term. When $n\rightarrow \infty $, the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend to the so-called Sine process. This lecture derives the distribution of the Airy Process at any given time and a PDE for the joint distribution at two different times. Similarly a PDE is found for the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process at different times $t_1$ and $t_2$, which itself is based on the joint probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE for the Dyson process is then subjected to an asymptotic analysis, consistent with the edge and bulk rescalings. The PDE’s obtained enable one to compute the asymptotic behavior of the joint distribution and the covariances for these processes at different times $t_1$ and $t_2$, when $t_2-t_1 \rightarrow \infty $.},
affiliation = {Brandeis University, department of mathematics, Waltham Mass 02454 (USA)},
author = {Adler, Mark},
journal = {Annales de l’institut Fourier},
keywords = {Dyson's Brownian motion; Airy process; coupled Gaussian hermitian matrices; coupled Gaussian Hermitian matrices},
language = {eng},
number = {6},
pages = {1835-1846},
publisher = {Association des Annales de l'Institut Fourier},
title = {PDE's for the Dyson, Airy and Sine processes},
url = {http://eudml.org/doc/116235},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Adler, Mark
TI - PDE's for the Dyson, Airy and Sine processes
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 6
SP - 1835
EP - 1846
AB - In 1962, Dyson showed that the spectrum of a $n\times n$ random Hermitian matrix, whose entries (real and imaginary) diffuse according to $n^2$ independent Ornstein-Uhlenbeck processes, evolves as $n$ non-colliding Brownian particles held together by a drift term. When $n\rightarrow \infty $, the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend to the so-called Sine process. This lecture derives the distribution of the Airy Process at any given time and a PDE for the joint distribution at two different times. Similarly a PDE is found for the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process at different times $t_1$ and $t_2$, which itself is based on the joint probability of the eigenvalues for coupled Gaussian Hermitian matrices. The PDE for the Dyson process is then subjected to an asymptotic analysis, consistent with the edge and bulk rescalings. The PDE’s obtained enable one to compute the asymptotic behavior of the joint distribution and the covariances for these processes at different times $t_1$ and $t_2$, when $t_2-t_1 \rightarrow \infty $.
LA - eng
KW - Dyson's Brownian motion; Airy process; coupled Gaussian hermitian matrices; coupled Gaussian Hermitian matrices
UR - http://eudml.org/doc/116235
ER -