On the ubiquity of the Cauchy distribution in spectral problems

Michael Aizenman; Simone Warzel

doi:10.1007/s00440-014-0587-3

Probability Theory and Related Fields

October 2015, Volume 163, Issue 1–2, pp 61–87 | Cite as

On the ubiquity of the Cauchy distribution in spectral problems

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Michael Aizenman
Simone WarzelEmail author

Article

First Online: 06 November 2014

Received: 31 December 2013

Revised: 14 October 2014

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3 Citations

Abstract

We consider the distribution of the values at real points of random functions which belong to the Herglotz–Pick (HP) class of analytic mappings of the upper half plane into itself. It is shown that under mild stationarity assumptions the individual values of HP functions with singular spectra have a Cauchy type distribution. The statement applies to the diagonal matrix elements of random operators, and holds regardless of the presence or not of level repulsion, i.e. applies to both random matrix and Poisson-type spectra.

Mathematics Subject Classification

Primary 60E99 Secondary 15B52

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Notes

Acknowledgments

We thank A. Knowles, L. Erdös, Y.V. Fyodorov and O. Zeitouni for relevant comments. M. Aizenman was supported in parts by the NSF grant PHY-1104596 and by the Weston Visiting Professorship at the Weizmann Institute of Science; S. Warzel was supported in part by the von Neumann Visiting Professorship at the Princeton Institute for Advanced Study.

Appendix A: Boole’s identity for the Stieltjes transform of singular measures

Following is the proof of Proposition 2.6, which we assume is known to experts. For convenience we restate the result, which extends an identity of Boole [8] from the case of pure-point measure $\mu $ to general singular measures.

Theorem 7.1

Let

$$\begin{aligned} F(z)\, = \, \int \frac{\mu _F(du)}{u-z} \end{aligned}$$

(7.1)

with the spectral measure $\mu _F$ which is finite and purely singular with respect to the Lebesgue measure $\mathcal L$ (or equivalently: ${{\mathrm{Im \,}}}F(x+i0) =0$ for a.e. $x\in \mathbb {R}$). Then for any $ t >0 $:

$$\begin{aligned} \mathcal L ( \{ x \in \mathbb {R}\, | \, F(x+i0) \ge t \} ) \, = \, \frac{\mu _F(\mathbb {R})}{t}. \end{aligned}$$

(7.2)

Proof

The monotone convergence theorem implies that

$$\begin{aligned} \mathcal L( \{ x \in \mathbb {R}\, | \, F(x+i0) \ge t \} ) \, = \, \lim _{\eta \rightarrow \infty } \int \frac{\eta ^2}{x^2 + \eta ^2} \, {{\mathrm{\mathrm 1}}}[ F(x+i0) \ge t] \, dx. \end{aligned}$$

(7.3)

The proof is based on the observation that the distribution of the random variable $ F(x+i0) $ with respect to the Cauchy probability measure $\frac{\eta }{ x^2 + \eta ^2} \, \frac{dx}{\pi } $ is uniquely characterized by its characteristic function, which by contour integration is:

$$\begin{aligned} \int e^{i \tau F(x+i0)} \frac{\eta }{ x^2 + \eta ^2} \, \frac{dx}{\pi } = e^{i\tau {{\mathrm{Re \,}}}F(i\eta )} \, e^{- |\tau | \, {{\mathrm{Im \,}}}F(i\eta ) }, \end{aligned}$$

(7.4)

where the integral was evaluated for $ \tau > 0 $ using a contour integration argument using the analyticity of $ F$ in the upper half plane $ \mathbb {C}^+ $. For $ \tau < 0 $ the characteristic function is obtained by complex conjugation from the one for $ \tau > 0 $ (since $F(x+i0)$ is real). Equation (7.4) shows that with respect to the Cauchy probability measure, the distribution of the variable $ F(x+i0) $ is itself Cauchy centered at $ {{\mathrm{Re \,}}}F(i\eta ) $ of width $ {{\mathrm{Im \,}}}F(i\eta ) $. As a consequence,

$$\begin{aligned} \lim _{\eta \rightarrow \infty } \int \frac{\eta }{x^2 + \eta ^2} \, {{\mathrm{\mathrm 1}}}[ F(x+i0) \ge t] \, dx \,&= \, \lim _{\eta \rightarrow \infty } \frac{\eta \, {{\mathrm{Im \,}}}F(i\eta ) }{ t - {{\mathrm{Re \,}}}F(i\eta )} \, = \, \frac{\mu _F(\mathbb {R})}{t}, \end{aligned}$$

(7.5)

since $ \lim _{\eta \rightarrow \infty } {{\mathrm{Re \,}}}F(i\eta ) = \lim _{\eta \rightarrow \infty } {{\mathrm{Im \,}}}F(i\eta )= 0 $ and $ \lim _{\eta \rightarrow \infty } \eta \, {{\mathrm{Im \,}}}F(i\eta ) = \mu _F(\mathbb {R}) $. $\square $

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Authors and Affiliations

Michael Aizenman
- 1
Simone Warzel
- 2
Email author

1.Departments of Physics and MathematicsPrinceton UniversityPrincetonUSA
2.Zentrum MathematikTU MünchenGarchingGermany

On the ubiquity of the Cauchy distribution in spectral problems

Abstract

Mathematics Subject Classification

Notes

Acknowledgments

References

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