Duke Mathematical Journal

Quantum ergodicity on large regular graphs

Nalini Anantharaman and Etienne Le Masson

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We propose a version of the quantum ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of “most” eigenfunctions. We consider expander graphs with few short cycles (for instance random large regular graphs). Our method mimics the proof of quantum ergodicity on manifolds: it uses microlocal analysis on regular trees, as introduced by the second author in an earlier paper.

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Duke Math. J., Volume 164, Number 4 (2015), 723-765.

First available in Project Euclid: 16 March 2015

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Zentralblatt MATH identifier

Primary: 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

large random graphs Laplacian eigenfunctions quantum ergodicity semiclassical measures


Anantharaman, Nalini; Le Masson, Etienne. Quantum ergodicity on large regular graphs. Duke Math. J. 164 (2015), no. 4, 723--765. doi:10.1215/00127094-2881592. https://projecteuclid.org/euclid.dmj/1426512106

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