# zbMATH — the first resource for mathematics

Operator limit of the circular beta ensemble. (English) Zbl 1452.60009
In [the authors, Invent. Math. 177, No. 3, 463–508 (2009; Zbl 1204.60012)] and [R. Killip and M. Stoiciu, Duke Math. J. 146, No. 3, 361–399 (2009; Zbl 1155.81020)], the bulk scaling limit of the Gaussian and circular beta ensembles were derived, and the counting functions of the limit processes were characterized via coupled systems of stochastic differential equations. In [F. Nakano, J. Stat. Phys. 156, No. 1, 66–93 (2014; Zbl 1303.82019)] and [the authors, Invent. Math. 209, No. 1, 275–327 (2017; Zbl 1443.60008)], it was shown that the scaling limit of the circular beta ensemble is the same as Sine$$_\beta$$, the bulk limit of the Gaussian beta ensemble. In the same paper, the authors also constructed a stochastic differential operator with a spectrum given by Sine$$_\beta$$ and it is shown that several random matrix limits can be described via differential operators parameterized by certain random walks or diffusions. The main results of the paper is the following statement:
Theorem 1. There is a probability space with a standard hyperbolic Brownian motion $$\mathcal B$$ and an array of stopping times $$0=\tau_{n,n} < \tau_{n,n-1}< \dots < \tau_{n,0} = \infty$$ so that $${\mathcal B}(\tau_{n,\lceil(1-t) n\rceil})$$, $$t\in [0, 1)$$ has the law of the random walk on the hyperbolic plane $$\mathbf H$$ used to generate Circ$$_{\beta,n}$$. This provides a coupling of Sine$$_\beta$$ and the sequence of operators Circ$$_{\beta,n}$$. There exists an a.s. finite positive random variable $$N$$ so that in this coupling $$\|r\,\mathrm{Sine}_\beta - r\, \mathrm{Circ}_{\beta, n}\|^2_{HS} \le\frac{\log^6n}{n}$$ a.s. in the Hilbert-Schmidt norm for all $$n\ge N$$.
The paper is organized as follows. Section 2 reviews the framework introduced in [Zbl 1443.60008] to study random matrix ensembles via differential operators. A modulus of the continuity estimate for the hyperbolic Brownian motion implies that the random walk is close to the Brownian path (Section 3). Section 4 shows that, if two paths are close and escape to the boundary of $$\mathbf H$$ similarly as a geodesic, then the corresponding operators are also close. Finally, Section 5 uses the linear rate of escape for the hyperbolic Brownian motion to complete the proof of Theorem 1. Some of the technical facts needed are collected in the appendix.

##### MSC:
 60B20 Random matrices (probabilistic aspects) 47B80 Random linear operators 47E05 General theory of ordinary differential operators 51M10 Hyperbolic and elliptic geometries (general) and generalizations
Full Text:
##### References:
 [1] Allez, R. and Dumaz, L. (2014). From sine kernel to Poisson statistics. Electron. J. Probab. 19 no. 114, 25. · Zbl 1334.60078 [2] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge. · Zbl 1184.15023 [3] Bhatia, R. and Elsner, L. (1994). The Hoffman-Wielandt inequality in infinite dimensions. Proc. Indian Acad. Sci. Math. Sci. 104 483-494. · Zbl 0805.47017 [4] Bourgade, P., Najnudel, J. and Nikeghbali, A. (2013). A unitary extension of virtual permutations. Int. Math. Res. Not. IMRN 18 4101-4134. · Zbl 1314.60023 [5] Davies, E. B. and Mandouvalos, N. (1988). Heat kernel bounds on hyperbolic space and Kleinian groups. Proc. Lond. Math. Soc. (3) 57 182-208. · Zbl 0643.30035 [6] Dumitriu, I. and Edelman, A. (2002). Matrix models for beta ensembles. J. Math. Phys. 43 5830-5847. · Zbl 1060.82020 [7] Edelman, A. and Sutton, B. D. (2007). From random matrices to stochastic operators. J. Stat. Phys. 127 1121-1165. · Zbl 1131.15025 [8] Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34. Princeton Univ. Press, Princeton, NJ. · Zbl 1217.82003 [9] Holcomb, D. and Valkó, B. (2015). Large deviations for the $$\text{Sine}_{\beta}$$ and $$\text{Sch}_{\tau}$$ processes. Probab. Theory Related Fields 163 339-378. · Zbl 1330.60047 [10] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam. · Zbl 0684.60040 [11] Karpelevic, F. I., Tutubalin, V. N. and Šur, M. G. (1959). Limit theorems for compositions of distributions in the Lobacevskii plane and space. Theory Probab. Appl. 4 399-402. [12] Killip, R. and Nenciu, I. (2004). Matrix models for circular ensembles. Int. Math. Res. Not. 50 2665-2701. · Zbl 1255.82004 [13] Killip, R. and Stoiciu, M. (2009). Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles. Duke Math. J. 146 361-399. · Zbl 1155.81020 [14] Maples, K., Najnudel, J. and Nikeghbali, A. (2019). Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble. Ann. Probab. 47 2417-2458. · Zbl 07114720 [15] Meckes, E. S. and Meckes, M. W. (2016). Self-similarity in the circular unitary ensemble. Discrete Anal. Paper No. 9, 14. · Zbl 1376.15028 [16] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam. [17] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics 30. Cambridge Univ. Press, Cambridge. [18] Nakano, F. (2014). Level statistics for one-dimensional Schrödinger operators and Gaussian beta ensemble. J. Stat. Phys. 156 66-93. · Zbl 1303.82019 [19] Ramírez, J. A. and Rider, B. (2009). Diffusion at the random matrix hard edge. Comm. Math. Phys. 288 887-906. · Zbl 1183.47035 [20] Ramírez, J. A., Rider, B. and Virág, B. (2011). Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Amer. Math. Soc. 24 919-944. · Zbl 1239.60005 [21] Valkó, B. and Virág, B. (2009). Continuum limits of random matrices and the Brownian carousel. Invent. Math. 177 463-508. · Zbl 1204.60012 [22] Valkó, B. and Virág, B. (2017). The $$\text{Sine}_{\beta}$$ operator. Invent. Math. 209 275-327. · Zbl 1443.60008 [23] Wigner, E.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.