Comptes Rendus
Physique mathématique, Théorie spectrale
The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph
Theo McKenzie1
1 Evans Hall, University of California, Berkeley, CA, USA
Comptes Rendus. Mathématique, Volume 360 (2022), 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.316
Theo McKenzie 1

1 Evans Hall, University of California, Berkeley, CA, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     language = {en},
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Theo McKenzie. The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 399-408. doi : 10.5802/crmath.316. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.316/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[19] Yael Dekel; James R. Lee; Nathan Linial 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] Robert B. Ellis Discrete Green’s functions for products of regular graphs (2003) (https://arxiv.org/abs/math/0309080)

[21] Shirshendu Ganguly; Nikhil Srivastava 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] Han Huang; Mark Rudelson 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] Jiaoyang Huang; Horng-Tzer Yau Spectrum of Random d-regular Graphs Up to the Edge (2021) (https://arxiv.org/abs/2102.00963)

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

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

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

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

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

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

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

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