# PDE's for the Dyson, Airy and Sine processes

Mark Adler^{[1]}

- [1] Brandeis University, department of mathematics, Waltham Mass 02454 (USA)

Annales de l’institut Fourier (2005)

- Volume: 55, Issue: 6, page 1835-1846
- ISSN: 0373-0956

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topAdler, 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 -

## References

top- M. Adler, P. van Moerbeke, PDE's for the joint distributions of the Dyson. Airy and Sine Processes, (2005) Zbl1093.60021MR2150191
- M. Adler, P. van Moerbeke, The spectrum of coupled random matrices, Annals of Math. 149 (1999), 921-976 Zbl0936.15018MR1709307
- M. Adler, P. van Moerbeke, A PDE for the joint distribution of the Airy process, (2003) Zbl1093.60021
- F.J. Dyson, A Brownian-Motion Model for the Eigenvalues of a Random Matrix, Journal of Math. Phys. 3 (1962), 1191-1198 Zbl0111.32703MR148397
- P.J. Forrester, T. Nagao, G. Honner, Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nucl. Phys. B 553 (1999), 601-643 Zbl0944.82012MR1707162
- K. Johansson, Discrete Polynuclear Growth and Determinantal Processes, (2002) Zbl1031.60084
- K. Johansson, The Arctic circle boundary and the Airy process, (2003) Zbl1096.60039
- M. Prähofer, H. Spohn, Scale Invariance of the PNG Droplet and the Airy Process, J. Stat. Phys. 108 (2002), 1071-1106 Zbl1025.82010MR1933446
- C.A. Tracy, H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174 Zbl0789.35152MR1257246
- C.A. Tracy, H. Widom, A system of differential equations for the Airy process, Elect. Comm. in Prob. 8 (2003), 93-98 Zbl1067.82031MR1987098
- C.A. Tracy, H. Widom, Differential equations for Dyson processes, (2003) Zbl1124.82007MR2103903
- H. Widom, On asymptotics for the Airy process, (2003) Zbl1073.82033MR2054175

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