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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
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References:
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