The Dyson Brownian Minor Process
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 971-1009.

Consider an n×n Hermitean matrix valued stochastic process {H t } t0 where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect.

In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the k×k minors in the upper left corner of H t . Projecting this process to a space-like path leads to a determinantal process for which we compute the kernel. This kernel contains the well known GUE minor kernel, and the Dyson Brownian motion kernel as special cases.

In the bulk scaling limit of this kernel it is possible to recover a time-dependent generalisation of Boutillier’s bead kernel.

We also compute the kernel for a process of intertwined Brownian motions introduced by Warren. That too is a determinantal process along space-like paths.

Nous considérons une processus stochastique fourni par une matrice {H t } t0 de taille n, dont les éléments évoluent selon un processus d’Ornstein-Uhlenbeck. Les valeurs propres de H t évoluent selon un mouvement Brownien de Dyson, c’est-à-dire qu’elles décrivent n processus d’Ornstein-Uhlenbeck répulsifs.

Dans cet article, nous considérons non seulement les valeurs propres de la matrice elle-même, mais aussi les valeurs propres combinées avec celles des mineurs principaux  ; c’est-à-dire les valeurs propres des sous-matrices dans le coin supérieur gauche de la matrice H t . Ce processus, projeté sur des chemins “spatiaux” appropriés, est un processus déterminantal dont nous fournissons le noyau  ; en outre, le noyau GUE-mineur et le noyau du processus de Dyson apparaissent tous deux comme des cas particuliers.

La limite dans le “bulk” de ce noyau fournit une généralisation, dépendante du temps, du noyau “bead” de Boutillier.

Nous calculons également le noyau pour un processus de mouvements browniens entrelacés introduit par Warren  ; celui-ci est également un processus déterminantal le long de chemins spatiaux.

DOI: 10.5802/aif.2871
Classification: 60B20, 60G55, 60J65, 60J10
Keywords: Dyson’s Brownian motion, bead kernel, extended kernels, Gaussian Unitary Ensemble
Mot clés : Mouvement Brownien de Dyson, le noyau “bead”, noyaux étendus, l’ensemble unitaire gaussien (GUE)
Adler, Mark 1; Nordenstam, Eric 2; Van Moerbeke, Pierre 3, 4

1 Brandeis University Department of Mathematics Waltham, Mass 02454 (USA)
2 Universität Wien Fakultät für Mathematik Oscar-Morgenstern-Platz 1 1090 Wien (Austria)
3 Brandeis University Waltham, Mass 02454 (USA)
4 Université de Louvain Department of Mathematics 1348 Louvain-la-Neuve (Belgium)
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     journal = {Annales de l'Institut Fourier},
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Adler, Mark; Nordenstam, Eric; Van Moerbeke, Pierre. The Dyson Brownian Minor Process. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 971-1009. doi : 10.5802/aif.2871.

[1] Adler, Mark; van Moerbeke, Pierre PDEs for the joint distributions of the Dyson, Airy and sine processes, Ann. Probab., Volume 33 (2005) no. 4, pp. 1326-1361 | MR | Zbl

[2] Adler, Mark; Nordenstam, Eric; van Moerbeke, Pierre Dyson’s Brownian motions on the spectra of consecutive minors, Stoch. Processes and their Appl., Volume 124 (2014), pp. 2023-2051 | MR

[3] Baryshnikov, Yu. GUEs and queues, Probab. Theory Related Fields, Volume 119 (2001) no. 2, pp. 256-274 | MR | Zbl

[4] Bender, Martin Global fluctuations in general β Dyson’s Brownian motion, Stochastic Process. Appl., Volume 118 (2008) no. 6, pp. 1022-1042 | MR | Zbl

[5] Borodin, Alexei Determinantal point processes, The Oxford handbook of random matrix theory, Oxford University Press, Oxford (2011), pp. 231-249 | MR | Zbl

[6] Borodin, Alexei; Ferrari, Patrik L. Large time asymptotics of growth models on space-like paths. I. PushASEP, Electron. J. Probab., Volume 13 (2008), pp. no. 50, 1380-1418 | MR | Zbl

[7] Borodin, Alexei; Ferrari, Patrik L.; Prähofer, Michael; Sasamoto, Tomohiro Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys., Volume 129 (2007) no. 5-6, pp. 1055-1080 | MR | Zbl

[8] Borodin, Alexei; Ferrari, Patrik L.; Prähofer, Michael; Sasamoto, Tomohiro; Warren, Jon Maximum of Dyson Brownian motion and non-colliding systems with a boundary, Electron. Commun. Probab., Volume 14 (2009), pp. 486-494 | MR | Zbl

[9] Borodin, Alexei; Ferrari, Patrik L.; Sasamoto, Tomohiro Two speed TASEP, J. Stat. Phys., Volume 137 (2009) no. 5-6, pp. 936-977 | MR | Zbl

[10] Borodin, Alexei; Rains, Eric M. Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys., Volume 121 (2005) no. 3-4, pp. 291-317 | MR | Zbl

[11] Boutillier, Cédric The bead model and limit behaviors of dimer models, Ann. Probab., Volume 37 (2009) no. 1, pp. 107-142 | MR | Zbl

[12] Defosseux, Manon Orbit measures and interlaced determinantal point processes, C. R. Acad. Sci. Paris, Series I, Volume 346 (2008) no. 13-14, pp. 783-788 | MR | Zbl

[13] Defosseux, Manon Orbit measures, random matrix theory and interlaced determinantal processes, C. R. Acad. Sci. Paris, Volume 46 (2010) no. 1, pp. 209-249 | Numdam | MR | Zbl

[14] Doob, Joseph L. Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer-Verlag, Berlin, 2001, pp. xxvi+846 (Reprint of the 1984 edition) | MR | Zbl

[15] Dyson, Freeman J. A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys., Volume 3 (1962), pp. 1191-1198 | MR | Zbl

[16] Eichelsbacher, Peter; König, Wolfgang Ordered random walks, Electron. J. Probab., Volume 13 (2008), pp. no. 46, 1307-1336 | MR | Zbl

[17] Ferrari, Patrik L.; Frings, René On the partial connection between random matrices and interacting particle systems, J. Stat. Phys., Volume 141 (2010), pp. 613-637 | MR | Zbl

[18] Forrester, Peter J.; Nagao, Taro Determinantal correlations for classical projection processes, J. Stat. Mech.: Theory and Exp., Volume 8 (2011) no. 8 (P08011)

[19] Forrester, Peter J.; Nordenstam, Eric The anti-symmetric GUE minor process, Mosc. Math. J., Volume 9 (2009) no. 4, p. 749-774, 934 | MR | Zbl

[20] Johansson, Kurt Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields, Volume 123 (2002) no. 2, pp. 225-280 | MR | Zbl

[21] Johansson, Kurt Discrete polynuclear growth and determinantal processes, Comm. Math. Phys., Volume 242 (2003) no. 1-2, pp. 277-329 | MR | Zbl

[22] Johansson, Kurt The arctic circle boundary and the Airy process, Ann. Probab., Volume 33 (2005) no. 1, pp. 1-30 | MR | Zbl

[23] Johansson, Kurt Non-intersecting, simple, symmetric random walks and the extended Hahn kernel, Ann. Inst. Fourier (Grenoble), Volume 55 (2005) no. 6, pp. 2129-2145 | Numdam | MR | Zbl

[24] Johansson, Kurt Random matrices and determinantal processes, Mathematical statistical physics, Elsevier B. V., Amsterdam, 2006, pp. 1-55 | MR

[25] Johansson, Kurt; Nordenstam, Eric Eigenvalues of GUE minors, Electron. J. Probab., Volume 11 (2006), pp. no. 50, 1342-1371 | MR | Zbl

[26] Johansson, Kurt; Nordenstam, Eric Erratum to: “Eigenvalues of GUE minors” [Electron. J. Probab. 11 (2006), no. 50, 1342–1371; MR2268547], Electron. J. Probab., Volume 12 (2007), p. 1048-1051 (electronic) | MR | Zbl

[27] Karlin, Samuel; McGregor, James Coincidence probabilities, Pacific J. Math., Volume 9 (1959), pp. 1141-1164 | MR | Zbl

[28] Katori, Makoto; Tanemura, Hideki Scaling limit of vicious walks and two-matrix model, Phys Rev E Stat Nonlin Soft Matter Phys (2002), pp. 66(1 Pt 1):011105

[29] Katori, Makoto; Tanemura, Hideki Functional central limit theorems for vicious walkers, Stoch. Stoch. Rep., Volume 75 (2003) no. 6, pp. 369-390 | MR | Zbl

[30] König, Wolfgang; Schmid, Patrik Random walks conditioned to stay in Weyl chambers of type C and D (2009) (arXiv:0911.0631v1)

[31] Macchi, Odile The coincidence approach to stochastic point processes, Advances in Appl. Probability, Volume 7 (1975), pp. 83-122 | MR | Zbl

[32] Mehta, Madan Lal Random matrices, Pure and Applied Mathematics (Amsterdam), 142, Elsevier/Academic Press, Amsterdam, 2004, pp. xviii+688 | MR | Zbl

[33] Nagao, Taro; Forrester, Peter J. Multilevel dynamical correlation functions for dyson’s brownian motion model of random matrices, Physics Letters A, Volume 247 (1998) no. 1-2, pp. 42-46

[34] Nordenstam, Eric Interlaced particles in tilings and random matrices, Swedish Royal Institute of Technology (KTH) (2009) (Ph. D. Thesis)

[35] Nordenstam, Eric On the shuffling algorithm for domino tilings, Electron. J. Probab., Volume 15 (2010), pp. no. 3, 75-95 | MR | Zbl

[36] Okounkov, Andrei; Reshetikhin, Nicolai The birth of a random matrix, Mosc. Math. J., Volume 6 (2006) no. 3, p. 553-566, 588 | MR | Zbl

[37] Spohn, Herbert Interacting Brownian particles: a study of Dyson’s model, Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986) (IMA Vol. Math. Appl.), Volume 9, Springer, New York, 1987, pp. 151-179 | MR | Zbl

[38] Tracy, Craig A.; Widom, Harold Differential equations for Dyson processes, Comm. Math. Phys., Volume 252 (2004) no. 1-3, pp. 7-41 | MR | Zbl

[39] Warren, Jon Dyson’s Brownian motions, intertwining and interlacing, Electron. J. Probab., Volume 12 (2007), pp. no. 19, 573-590 | MR | Zbl

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