The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph

McKenzie, Theo

Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 399-408.

Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite.

[1] Aizenman, Michael; Warzel, Simone Resonant delocalization for random Schrödinger operators on tree graphs, J. Eur. Math. Soc., Volume 15 (2013) no. 4, pp. 1167-1222 | DOI | Zbl

[2] Alon, Noga; Ganguly, Shirshendu; Srivastava, Nikhil High-girth near-Ramanujan graphs with localized eigenvectors (2019) (https://arxiv.org/abs/1908.03694)

[3] Anantharaman, Nalini Quantum ergodicity on regular graphs, Commun. Math. Phys., Volume 353 (2017) no. 2, pp. 633-690 | DOI | MR | Zbl

[4] Anantharaman, Nalini; Ingremeau, Maxime; Sabri, Mostafa; Winn, Brian Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization, J. Math. Pures Appl., Volume 151 (2021), pp. 28-98 | DOI | MR | Zbl

[5] Anantharaman, Nalini; Le Masson, Etienne Quantum ergodicity on large regular graphs, Duke Math. J., Volume 164 (2015) no. 4, pp. 723-765 | MR | Zbl

[6] Anantharaman, Nalini; Sabri, Mostafa Quantum ergodicity for the Anderson model on regular graphs, J. Math. Phys., Volume 58 (2017) no. 9, 091901, 10 pages | MR | Zbl

[7] Anantharaman, Nalini; Sabri, Mostafa Quantum ergodicity on graphs: from spectral to spatial delocalization, Ann. Math., Volume 189 (2019) no. 3, pp. 753-835 | MR | Zbl

[8] Anantharaman, Nalini; Sabri, Mostafa Recent results of quantum ergodicity on graphs and further investigation, Ann. Fac. Sci. Toulouse, Math., Volume 28 (2019) no. 3, pp. 559-592 | DOI | Numdam | MR | Zbl

[9] Arora, Sanjeev; Bhaskara, Aditya Eigenvectors of random graphs: delocalization and nodal domains (2011) (http://www.cs.princeton.edu/~bhaskara/files/deloc.pdf)

[10] Backhausz, Ágnes; Szegedy, Balázs On the almost eigenvectors of random regular graphs, Ann. Probab., Volume 47 (2019) no. 3, pp. 1677-1725 | MR | Zbl

[11] Bauerschmidt, Roland; Huang, Jiaoyang; Yau, Horng-Tzer Local Kesten–McKay law for random regular graphs, Commun. Math. Phys., Volume 369 (2019) no. 2, pp. 523-636 | DOI | MR | Zbl

[12] Benjamini, Itai; Schramm, Oded Recurrence of distributional limits of finite planar graphs, Electron. J. Probab., Volume 6 (2001) no. 23, pp. 533-545 | MR | Zbl

[13] Brooks, Shimon; Le Masson, Etienne; Lindenstrauss, Elon Quantum ergodicity and averaging operators on the sphere, Int. Math. Res. Not., Volume 2016 (2016) no. 19, pp. 6034-6064 | DOI | MR | Zbl

[14] Brooks, Shimon; Lindenstrauss, Elon Non-localization of eigenfunctions on large regular graphs, Isr. J. Math., Volume 193 (2013) no. 1, pp. 1-14 | DOI | MR | Zbl

[15] Chung, Fan R. K. Laplacians of graphs and Cheeger’s inequalities, Combinatorics, Paul Erdos is Eighty. Vol. 2 (Bolyai Society Mathematical Studies), Volume 2, János Bolyai Mathematical Society, 1996, pp. 157-172 | MR | Zbl

[16] Chung, Fan R. K.; Yau, Shing-Tung Discrete Green’s functions, J. Comb. Theory, Ser. A, Volume 91 (2000) no. 1-2, pp. 191-214 | DOI | MR

[17] Cvetković, Dragoš M.; Rowlinson, Peter; Simic, Slobodan Eigenspaces of graphs, Encyclopedia of Mathematics and Its Applications, 66, Cambridge University Press, 1997 | DOI

[18] De Verdiere, Yves Colin Ergodicité et fonctions propres du laplacien, Commun. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 | DOI | Numdam | Zbl

[19] Dekel, Yael; Lee, James R.; Linial, Nathan Eigenvectors of random graphs: Nodal domains, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (Lecture Notes in Computer Science), Volume 4627, Springer, 2007, pp. 436-448 | DOI | Zbl

[20] Ellis, Robert B. Discrete Green’s functions for products of regular graphs (2003) (https://arxiv.org/abs/math/0309080)

[21] Ganguly, Shirshendu; Srivastava, Nikhil On non-localization of eigenvectors of high girth graphs, Int. Math. Res. Not., Volume 2021 (2021) no. 8, pp. 5766-5790 | DOI | MR | Zbl

[22] Huang, Han; Rudelson, Mark Size of nodal domains of the eigenvectors of a graph, Random Struct. Algorithms, Volume 57 (2020) no. 2, pp. 393-438 | DOI | MR | Zbl

[23] Huang, Jiaoyang; Yau, Horng-Tzer Spectrum of Random $d$-regular Graphs Up to the Edge (2021) (https://arxiv.org/abs/2102.00963)

[24] Ingremeau, Maxime; Sabri, Mostafa; Winn, Brian Quantum ergodicity for large equilateral quantum graphs, J. Lond. Math. Soc., Volume 101 (2020) no. 1, pp. 82-109 | DOI | MR | Zbl

[25] Kottos, Tsampikos; Smilansky, Uzy Quantum chaos on graphs, Phys. Rev. Lett., Volume 79 (1997) no. 24, p. 4794 | DOI

[26] Kottos, Tsampikos; Smilansky, Uzy Periodic orbit theory and spectral statistics for quantum graphs, Ann. Phys., Volume 274 (1999) no. 1, pp. 76-124 | DOI | MR | Zbl

[27] McKenzie, Theo; Mohanty, Sidhanth High-girth near-Ramanujan graphs with lossy vertex expansion (2020) (https://arxiv.org/abs/2007.13630)

[28] Shnirelman, Alexander I. Ergodic properties of eigenfunctions, Usp. Mat. Nauk, Volume 29 (1974) no. 6, pp. 181-182 | MR

[29] Zelditch, Steven Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987) no. 4, pp. 919-941 | MR | Zbl

[30] Zelditch, Steven Quantum ergodicity and mixing of eigenfunctions (2005) (https://arxiv.org/abs/math-ph/0503026)