Quantum ergodicity on large regular graphs

Abstract

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.

Statement of theorem 1.7 has been corrected. Some comments, remarks and references added. 30 pages

References in corpus (4)

• Local semicircle law and complete delocalization for Wigner random matrices
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Eigenvectors of the discrete Laplacian on regular graphs - a statistical approach
• The Eigenvector Moment Flow and local Quantum Unique Ergodicity

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• Random band matrices in the delocalized phase, I: Quantum unique ergodicity and universality
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• The Eigenvector Moment Flow and local Quantum Unique Ergodicity
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• Microlocal lifts and quantum unique ergodicity on $GL(2,{\mathbb{Q}}_p)$
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• Quantum Unique Ergodicity for Cayley graphs of quasirandom groups
• Pointwise Weyl law for graphs from quantized interval maps
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• Quantum Ergodicity for compact quotients of ${\mathrm{SL}}_d(\mathbb{R})/\mathrm{SO}(d)$ in the Benjamini-Schramm limit