Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Sine$_\beta$ operator   We show that Sine$_\beta$, the bulk limit of the Gaussian $\beta$-ensembles
is the spectrum of a self-adjoint random differential operator \[ f\to 2
{R_t^{-1}} \left[ \begin{array}{cc} 0 &-\tfrac{d}{dt} \tfrac{d}{dt} &0
\end{array} \right] f, \qquad f:[0,1)\to \mathbb R^2, \] where $R_t$ is the
positive definite matrix representation of hyperbolic Brownian motion with
variance $4/\beta$ in logarithmic time. The result connects the
Montgomery-Dyson conjecture about the Sine$_2$ process and the non-trivial
zeros of the Riemann zeta function, the Hilbert-P\'olya conjecture and de
Brange's attempt to prove the Riemann hypothesis. We identify the Brownian
carousel as the Sturm-Liouville phase function of this operator.
  We provide similar operator representations for several other finite
dimensional random ensembles and their limits: finite unitary or orthogonal
ensembles, Hua-Pickrell ensembles and their limits, hard-edge
$\beta$-ensembles, as well as the Schr\"odinger point process. In this more
general setting, hyperbolic Brownian motion is replaced by a random walk or
Brownian motion on the affine group.
  Our approach provides a unified framework to study $\beta$-ensembles that has
so far been missing in the literature. In particular, we connect It\^o's
classification of affine Brownian motions with the classification of limits of
random matrix ensembles.

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The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
1
The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics