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SMALL-SCALE EQUIDISTRIBUTION OF RANDOM WAVES GENERATED BY AN UNFAIR COIN FLIP

Part of: Partial differential equations on manifolds; differential operators Spectral theory and eigenvalue problems Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  29 November 2021

MIRIAM J. LEONHARDT Open the ORCID record for MIRIAM J. LEONHARDT [Opens in a new window]  and
MELISSA TACY
MIRIAM J. LEONHARDT*
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
MELISSA TACY
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

In this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$ . Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ( $p=0.5$ ) coin to retain equidistribution.

Keywords

eigenfunctionplane waverandom waverandom eigenfunctionequidistribution at small scalesunfair coin

MSC classification

Primary: 58J50: Spectral problems; spectral geometry; scattering theory 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Secondary: 60B10: Convergence of probability measures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 114 , Issue 2 , April 2023 , pp. 222 - 252
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Nathan Ross

The first author was supported by the University of Auckland’s Summer Scholar Scheme.

References

Berry, M. V., ‘Regular and irregular semiclassical wavefunctions’, J. Phys. A 10(12) (1977), 2083–2091.CrossRefGoogle Scholar
Burq, N. and Lebeau, G., ‘Injections de Sobolev probabilistes et applications’, Ann. Sci. Éc. Norm. Supér. (4) 46(6) (2013), 917–962.CrossRefGoogle Scholar
de Courcy-Ireland, M., ‘Shrinking scale equidistribution for monochromatic random waves on compact manifolds’, Int. Math. Res. Not. IMRN 2021(4) (2021), 3021–3055.CrossRefGoogle Scholar
Han, X., ‘Small scale equidistribution of random eigenbases’, Comm. Math. Phys. 349(1) (2017), 425–440.CrossRefGoogle Scholar
Han, X. and Tacy, M., ‘Equidistribution of random waves on small balls’, Comm. Math. Phys. 376(3) (2020), 2351–2377.CrossRefGoogle Scholar
Maples, K., ‘Quantum unique ergodicity for random bases of spectral projections’, Math. Res. Lett. 20(6) (2013), 1115–1124.CrossRefGoogle Scholar
SEGwiki, ‘Method of stationary phase’, available at https://wiki.seg.org/wiki/Method/_of/_ stationary/_phase (accessed 12 February 2021).Google Scholar
Sogge, C. D., ‘Localized ${L}^p$ -estimates of eigenfunctions: a note on an article of Hezari and Rivière’, Adv. Math. 289 (2016), 384–396.CrossRefGoogle Scholar
Tao, T., ‘Some recent progress on the restriction conjecture’, in: Fourier Analysis and Convexity , Applied and Numerical Harmonic Analysis (eds. Brandolini, L., Colzani, L., Travaglini, G. and Iosevich, A.) (Birkhäuser, Boston, MA, 2004), 217–243.CrossRefGoogle Scholar
Zelditch, S., ‘A random matrix model for quantum mixing’, Int. Math. Res. Not. IMRN 1996(3) (1996), 115–137.CrossRefGoogle Scholar
Zelditch, S., ‘Real and complex zeros of Riemannian random waves’, in: Spectral Analysis in Geometry and Number Theory, Contemporary Mathematics, 484 (eds. Kotani, M., Naito, H. and Tate, T.) (American Mathematical Society, Providence, RI, 2009), 321–342.CrossRefGoogle Scholar
Zelditch, S., ‘Quantum ergodicity of random orthonormal bases of spaces of high dimension’, Philos. Trans. Roy. Soc. A 372(2007) (2014), 20120511.CrossRefGoogle ScholarPubMed