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.

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

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Cited by in corpus (28)

• Quantum simulation and computing with Rydberg-interacting qubits
• Local Kesten--McKay law for random regular graphs
• Local semicircle law for random regular graphs
• Eigenstate Thermalization Hypothesis for Wigner Matrices
• Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices
• Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
• Short geodesic loops and $L^p$ norms of eigenfunctions on large genus random surfaces
• Fluctuations in local quantum unique ergodicity for generalized Wigner matrices
• Random band matrices in the delocalized phase, I: Quantum unique ergodicity and universality
• Pseudo-differential calculus on homogeneous trees
• Universality for a class of random band matrices
• Quantum ergodicity for the Anderson model on regular graphs
• The Eigenvector Moment Flow and local Quantum Unique Ergodicity
• Around quantum ergodicity
• Spectrum of Random $d$-regular Graphs Up to the Edge
• Entropy of eigenfunctions on quantum graphs
• Microlocal lifts and quantum unique ergodicity on $GL(2,{\mathbb{Q}}_p)$
• Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations
• Quantum ergodicity for quantum graphs without back-scattering
• Rényi and Tsallis entropies related to eigenfunctions of quantum graphs
• On quantum percolation in finite regular graphs
• Localization of eigenvectors of non-Hermitian banded noisy Toeplitz matrices
• $L^p$ norms and support of eigenfunctions on graphs
• Pointwise Weyl law for graphs from quantized interval maps
• Quantum Unique Ergodicity for Cayley graphs of quasirandom groups
• Quantum Ergodicity for compact quotients of ${\mathrm{SL}}_d(\mathbb{R})/\mathrm{SO}(d)$ in the Benjamini-Schramm limit
• Eigenvectors of the square grid plus GUE
• Eigenstates and spectral projection for quantized baker's map