Recent results of quantum ergodicity on graphs and further investigation

Abstract

We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions.2010 Mathematics Subject Classification. Primary 82B44, 58J51. Secondary 47B80, 60B20.

$$L^p$$ Norms and Support of Eigenfunctions on Graphs

“…Item (iv) follows from (iii) and the continuity of λ → G λ+i0 (v, v), which follows from (2.2) and the continuity of λ → ζ λ+i0 w (u). The latter is proved in [11]. We now prove (v).…”

“…Proof. Claims (i), (ii) are proved in [11,Section 4]. There the case W ≡ 0 was considered, but the proof remains the same : the fact that (T , W) is the universal cover of (G, W ) is responsible for making the Green function of H T algebraic.…”

“…In this paper, we investigate the eigenvectors of graphs which are more general in two respects : we add potentials and consider irregular graphs. A quantum ergodicity result was recently established in this framework [9,10,11], where it is shown essentially that if a sequence of graphs converges in the sense of Benjamini-Schramm 1 to a random tree, and if the spectrum at the limit is purely absolutely continuous almost surely, then most eigenvectors become uniformly distributed in some sense. Our aim here is to provide norm estimates for the eigenfunctions and discuss their support.…”

Quantum ergodicity for large equilateral quantum graphs

“…However, the control we have on the remainder in the limit (2.9) is not good enough to justify (2.16) when a has a small support (for instance, when a is supported on a single edge). Similar difficulties arise for combinatorial graphs [3,6].…”

“…This was among very few theorems of delocalization for the Anderson model, the opposite regime of localization being quite well understood today. More applications appeared in , where the first results of quantum ergodicity for non‐regular graphs were given. We also mention the recent paper which studies quantum ergodicity for sequences of compact hyperbolic surfaces, also in the bounded interval regime.…”