The Dyson Brownian Minor Process

[Le Processus Brownien des mineurs de Dyson]

Adler, Mark ; Nordenstam, Eric ; Van Moerbeke, Pierre

Annales de l'Institut Fourier, Tome 64 (2014), p. 971-1009 / Harvested from Numdam

Résumé

Nous considérons une processus stochastique fourni par une matrice $\lbrace H_t\rbrace _{t\ge 0}$ 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.

Abstract

Consider an $n\times n$ Hermitean matrix valued stochastic process $\lbrace H_t\rbrace _{t\ge 0}$ 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 $k\times k$ 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.

Détail

Publié le : 2014-01-01

MR 3330161 | Zbl 06387298

DOI : https://doi.org/10.5802/aif.2871
Classification:  60B20,  60G55,  60J65,  60J10
Mots clés: Mouvement Brownien de Dyson, le noyau “bead”, noyaux étendus, l’ensemble unitaire gaussien (GUE)

Bibliographie

[1] Adler, Mark; Van Moerbeke, Pierre PDEs for the joint distributions of the Dyson, Airy and sine processes, Ann. Probab., Tome 33 (2005) no. 4, pp. 1326-1361 | MR 2150191 | Zbl 1093.60021

[2] Adler, Mark; Nordenstam, Eric; Van Moerbeke, Pierre Dyson’s Brownian motions on the spectra of consecutive minors, Stoch. Processes and their Appl., Tome 124 (2014), pp. 2023-2051 | MR 3188348

[3] Baryshnikov, Yu. GUEs and queues, Probab. Theory Related Fields, Tome 119 (2001) no. 2, pp. 256-274 | MR 1818248 | Zbl 0980.60042

[4] Bender, Martin Global fluctuations in general $\beta $ Dyson’s Brownian motion, Stochastic Process. Appl., Tome 118 (2008) no. 6, pp. 1022-1042 | MR 2418256 | Zbl 1139.60342

[5] Borodin, Alexei Determinantal point processes, The Oxford handbook of random matrix theory, Oxford University Press, Oxford (2011), pp. 231-249 | MR 2932631 | Zbl 1238.60055

[6] Borodin, Alexei; Ferrari, Patrik L. Large time asymptotics of growth models on space-like paths. I. PushASEP, Electron. J. Probab., Tome 13 (2008), pp. no. 50, 1380-1418 | MR 2438811 | Zbl 1187.82084

[7] Borodin, Alexei; Ferrari, Patrik L.; Prähofer, Michael; Sasamoto, Tomohiro Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys., Tome 129 (2007) no. 5-6, pp. 1055-1080 | MR 2363389 | Zbl 1136.82028

[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., Tome 14 (2009), pp. 486-494 | MR 2559098 | Zbl 1189.60169

[9] Borodin, Alexei; Ferrari, Patrik L.; Sasamoto, Tomohiro Two speed TASEP, J. Stat. Phys., Tome 137 (2009) no. 5-6, pp. 936-977 | MR 2570757 | Zbl 1183.82062

[10] Borodin, Alexei; Rains, Eric M. Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys., Tome 121 (2005) no. 3-4, pp. 291-317 | MR 2185331 | Zbl 1127.82017

[11] Boutillier, Cédric The bead model and limit behaviors of dimer models, Ann. Probab., Tome 37 (2009) no. 1, pp. 107-142 | MR 2489161 | Zbl 1171.82006

[12] Defosseux, Manon Orbit measures and interlaced determinantal point processes, C. R. Acad. Sci. Paris, Series I, Tome 346 (2008) no. 13-14, pp. 783-788 | MR 2427082 | Zbl 1157.60027

[13] Defosseux, Manon Orbit measures, random matrix theory and interlaced determinantal processes, C. R. Acad. Sci. Paris, Tome 46 (2010) no. 1, pp. 209-249 | Numdam | MR 2641777 | Zbl 1216.15024

[14] Doob, Joseph L. Classical potential theory and its probabilistic counterpart, Springer-Verlag, Berlin, Classics in Mathematics (2001), pp. xxvi+846 (Reprint of the 1984 edition) | MR 1814344 | Zbl 0990.31001

[15] Dyson, Freeman J. A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys., Tome 3 (1962), pp. 1191-1198 | MR 148397 | Zbl 0111.32703

[16] Eichelsbacher, Peter; König, Wolfgang Ordered random walks, Electron. J. Probab., Tome 13 (2008), pp. no. 46, 1307-1336 | MR 2430709 | Zbl 1189.60092

[17] Ferrari, Patrik L.; Frings, René On the partial connection between random matrices and interacting particle systems, J. Stat. Phys., Tome 141 (2010), pp. 613-637 | MR 2733398 | Zbl 1205.82104

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

[19] Forrester, Peter J.; Nordenstam, Eric The anti-symmetric GUE minor process, Mosc. Math. J., Tome 9 (2009) no. 4, p. 749-774, 934 | MR 2663989 | Zbl 1191.15032

[20] Johansson, Kurt Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields, Tome 123 (2002) no. 2, pp. 225-280 | MR 1900323 | Zbl 1008.60019

[21] Johansson, Kurt Discrete polynuclear growth and determinantal processes, Comm. Math. Phys., Tome 242 (2003) no. 1-2, pp. 277-329 | MR 2018275 | Zbl 1031.60084

[22] Johansson, Kurt The arctic circle boundary and the Airy process, Ann. Probab., Tome 33 (2005) no. 1, pp. 1-30 | MR 2118857 | Zbl 1096.60039

[23] Johansson, Kurt Non-intersecting, simple, symmetric random walks and the extended Hahn kernel, Ann. Inst. Fourier (Grenoble), Tome 55 (2005) no. 6, pp. 2129-2145 | Numdam | MR 2187949 | Zbl 1083.60079

[24] Johansson, Kurt Random matrices and determinantal processes, Mathematical statistical physics, Elsevier B. V., Amsterdam (2006), pp. 1-55 | MR 2581882

[25] Johansson, Kurt; Nordenstam, Eric Eigenvalues of GUE minors, Electron. J. Probab., Tome 11 (2006), pp. no. 50, 1342-1371 | MR 2268547 | Zbl 1127.60047

[26] Johansson, Kurt; Nordenstam, Eric Erratum to: “Eigenvalues of GUE minors” [Electron. J. Probab. 11 (2006), no. 50, 1342–1371; MR2268547], Electron. J. Probab., Tome 12 (2007), p. 1048-1051 (electronic) | MR 2268547 | Zbl 1134.60344

[27] Karlin, Samuel; Mcgregor, James Coincidence probabilities, Pacific J. Math., Tome 9 (1959), pp. 1141-1164 | MR 114248 | Zbl 0092.34503

[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., Tome 75 (2003) no. 6, pp. 369-390 | MR 2029612 | Zbl 1206.82044

[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, Tome 7 (1975), pp. 83-122 | MR 380979 | Zbl 0366.60081

[32] Mehta, Madan Lal Random matrices, Elsevier/Academic Press, Amsterdam, Pure and Applied Mathematics (Amsterdam), Tome 142 (2004), pp. xviii+688 | MR 2129906 | Zbl 1107.15019

[33] Nagao, Taro; Forrester, Peter J. Multilevel dynamical correlation functions for dyson’s brownian motion model of random matrices, Physics Letters A, Tome 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., Tome 15 (2010), pp. no. 3, 75-95 | MR 2578383 | Zbl 1193.60015

[36] Okounkov, Andrei; Reshetikhin, Nicolai The birth of a random matrix, Mosc. Math. J., Tome 6 (2006) no. 3, p. 553-566, 588 | MR 2274865 | Zbl 1130.15014

[37] Spohn, Herbert Interacting Brownian particles: a study of Dyson’s model, Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986), Springer, New York (IMA Vol. Math. Appl.) Tome 9 (1987), pp. 151-179 | MR 914993 | Zbl 0674.60096

[38] Tracy, Craig A.; Widom, Harold Differential equations for Dyson processes, Comm. Math. Phys., Tome 252 (2004) no. 1-3, pp. 7-41 | MR 2103903 | Zbl 1124.82007

[39] Warren, Jon Dyson’s Brownian motions, intertwining and interlacing, Electron. J. Probab., Tome 12 (2007), pp. no. 19, 573-590 | MR 2299928 | Zbl 1127.60078