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Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions. (English. French summary) Zbl 1447.60024

Summary: We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases \(\beta =1,2,4\), this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of V. Gorin and M. Shkolnikov [Ann. Probab. 46, No. 4, 2287–2344 (2018; Zbl 1430.60011)] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy operators, introduced by A. Bloemendal and B. Virág [Probab. Theory Relat. Fields 156, No. 3–4, 795–825 (2013; Zbl 1356.60014)]. The Feynman-Kac formulas involve functionals of a reflected Brownian motion and its local times, thus, allowing to study the limiting operators by tools of stochastic analysis. We derive a first result in this direction by obtaining a new distributional identity for a reflected Brownian bridge conditioned on its local time at zero. A key feature of our proof consists of a novel strong invariance result for certain non-negative random walks and their occupation times that is based on the Skorokhod reflection map.

MSC:

60B20 Random matrices (probabilistic aspects)
60H25 Random operators and equations (aspects of stochastic analysis)
47D08 Schrödinger and Feynman-Kac semigroups
60J55 Local time and additive functionals
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[1] G. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics118. Cambridge University Press, Cambridge, 2010. · Zbl 1184.15023
[2] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab.33 (2005) 1643-1697. · Zbl 1086.15022
[3] R. Bass and D. Khoshnevisan. Strong approximations to Brownian local time. Progr. Probab.33 (1992) 43-65. · Zbl 0789.60062
[4] R. F. Bass and D. Khoshnevisan. Laws of the iterated logarithm for local times of the empirical process. Ann. Probab.23 (1995) 388-399. · Zbl 0845.60079 · doi:10.1214/aop/1176988391
[5] K. E. Bassler, P. J. Forrester and N. E. Frankel. Edge effects in some perturbations of the Gaussian unitary ensemble. J. Math. Phys.51 (2010), Article ID 123305. · Zbl 1314.82028 · doi:10.1063/1.3521288
[6] R. N. Bhattacharya and R. R. Rao. Normal Approximation and Asymptotic Expansions. John Wiley & Sons, New York, 1976. · Zbl 0331.41023
[7] P. Billingsley. Convergence of Probability Measures, 2nd edition. John Wiley & Sons, New York, 1999. · Zbl 0944.60003
[8] A. Bloemendal and B. Virág. Limits of spiked random matrices I. Probab. Theory Related Fields156 (2013) 795-825. · Zbl 1356.60014
[9] A. Bloemendal and B. Virág. Limits of spiked random matrices II. Ann. Probab.44 (2016) 2726-2769. · Zbl 1396.60004 · doi:10.1214/15-AOP1033
[10] X. Chen. Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs157. American Mathematical Society, Providence, 2010.
[11] R. Dudley. Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics63. Cambridge University Press, Cambridge, 1999. · Zbl 0951.60033
[12] I. Dumitriu and A. Edelman. Matrix models for beta ensembles. J. Math. Phys.43 (2002) 5830-5847. · Zbl 1060.82020 · doi:10.1063/1.1507823
[13] A. Edelman. Stochastic differential equations and random matrices. In SIAM Conference on Applied Linear Algebra. The College of William and Mary, Williamsburg, 2003. Available at http://math.mit.edu/ edelman/homepage/talks/siam2003.ppt.
[14] A. Edelman and B. Sutton. From random matrices to stochastic operators. J. Stat. Phys.127 (2007) 1121-1165. · Zbl 1131.15025 · doi:10.1007/s10955-006-9226-4
[15] P. Forrester. Log-Gases and Random Matrices. London Mathematical Society Monographs Series34. Princeton University Press, Princeton, 2010. · Zbl 1217.82003
[16] V. Gorin and M. Shkolnikov. Stochastic Airy semigroup through tridiagonal matrices. Ann. Probab.46 (2018) 2287-2344. Available at arXiv:1601.06800v1. · Zbl 1430.60011 · doi:10.1214/17-AOP1229
[17] Y. Hariya. A pathwise interpretation of the Gorin-Shkolnikov identity. Electron. Commun. Probab.21 (2016) 1-6. · Zbl 1346.60130 · doi:10.1214/16-ECP10
[18] D. Holcomb and G. R. M. Flores. Edge scaling of the \(\beta \)-Jacobi ensemble. J. Stat. Phys.149 (2012) 1136-1160. · Zbl 1257.82006 · doi:10.1007/s10955-012-0634-3
[19] S. Jacquot and B. Valko. Bulk scaling limit of the Laguerre ensemble. Electron. J. Probab.11 (2011) 314-346. · Zbl 1225.60015 · doi:10.1214/EJP.v16-854
[20] A. Javanmarda, A. Montanari and F. Ricci-Tersenghi. Phase transitions in semidefinite relaxations. Proc. Natl. Acad. Sci. USA113 (2016) E2218-E2223. · Zbl 1359.62188 · doi:10.1073/pnas.1523097113
[21] R. L. Karandikar. On pathwise stochastic integration. Stochastic Process. Appl.57 (1995) 11-18. · Zbl 0816.60047 · doi:10.1016/0304-4149(95)00002-O
[22] R. Kozhan. Rank one non-Hermitian perturbations of Hermitian \(\beta \)-ensembles of random matrices. J. Stat. Phys.168 (2017) 92-108. · Zbl 1372.15030 · doi:10.1007/s10955-017-1792-0
[23] M. Krishnapur, B. Rider and B. Virág. Universality of the stochastic Airy operator. Comm. Pure Appl. Math.69 (2016) 145-199. · Zbl 1364.60015
[24] G. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics123. Cambridge University Press, Cambridge, 2010. · Zbl 1210.60002
[25] P. Lax. Functional Analysis. John Wiley & Sons, New York, 2002. · Zbl 1009.47001
[26] I. G. Macdonald. Symmetric Functions and Hall Polynomials, 2nd edition. The Clarendon Press, Oxford University Press, New York, 2015. · Zbl 1332.05002
[27] S. Péché. The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields134 (2006) 127-173. · Zbl 1088.15025
[28] J. Pitman. The distribution of local times of a Brownian bridge. In Séminaire de Probabilités XXXIII 388-394. Lecture Notes in Math.1709. Springer-Verlag, Berlin, 1999. · Zbl 0945.60081
[29] J. Pitman. The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest. Ann. Probab.27 (1999) 261-283. · Zbl 0954.60060 · doi:10.1214/aop/1022874819
[30] J. Pitman. Combinatorial Stochastic Processes. Lecture Notes in Math.1875. Springer-Verlag, Berlin, 2006. · Zbl 1103.60004
[31] J. Ramírez and B. Rider. Diffusion at the random matrix hard edge. Comm. Math. Phys.288 (2009) 887-906. · Zbl 1183.47035
[32] J. Ramírez, B. Rider and B. Virág. Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Amer. Math. Soc.24 (2011) 919-944. · Zbl 1239.60005
[33] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften293. Springer-Verlag, Berlin, 1999. · Zbl 0917.60006
[34] B. Rider and P. Waters. Universality of the stochastic Bessel operator. Preprint, 2016. Available at arXiv:1610.01637. · Zbl 1422.60037 · doi:10.1007/s00440-018-0888-z
[35] H. F. Trotter. A property of Brownian motion paths. Illinois J. Math.2 (1958) 425-433. · Zbl 0117.35502 · doi:10.1215/ijm/1255454547
[36] B. Valkó and B. Virág. Continuum limits of random matrices and the Brownian carousel. Invent. Math.177 (2009) 463-508. · Zbl 1204.60012
[37] B. Valkó and B. Virág. Operator limit of the circular beta ensemble, 2017. Available at arXiv:1710.06988v1. · Zbl 1452.60009
[38] B. Valkó and B. Virág. The \(sine_{\beta }\) operator. Invent. Math.209 (2017) 275-327. · Zbl 1443.60008
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