Quantum ergodicity for large equilateral quantum graphs

Maxime Ingremeau, Mostafa Sabri, Brian Winn

Research output: Contribution to journalArticlepeer-review


Consider a sequence of finite regular graphs converging, in the sense of Benjamini–Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant α) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case α = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it has an explicit C1 density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations.

Original languageEnglish (US)
Pages (from-to)82-109
Number of pages28
JournalJournal of the London Mathematical Society
Issue number1
StatePublished - Feb 1 2020


  • 34B45
  • 58J51 (primary)
  • 81Q10 (secondary)

ASJC Scopus subject areas

  • General Mathematics


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