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.

Accepted:

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Theo McKenzie ^{1}

@article{CRMATH_2022__360_G4_399_0, author = {Theo McKenzie}, title = {The necessity of conditions for graph quantum ergodicity and {Cartesian} products with an infinite graph}, journal = {Comptes Rendus. Math\'ematique}, pages = {399--408}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, year = {2022}, doi = {10.5802/crmath.316}, language = {en}, }

TY - JOUR AU - Theo McKenzie TI - The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph JO - Comptes Rendus. Mathématique PY - 2022 SP - 399 EP - 408 VL - 360 PB - Académie des sciences, Paris DO - 10.5802/crmath.316 LA - en ID - CRMATH_2022__360_G4_399_0 ER -

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

[2] High-girth near-Ramanujan graphs with localized eigenvectors (2019) (https://arxiv.org/abs/1908.03694)

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

[4] 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] Quantum ergodicity on large regular graphs, Duke Math. J., Volume 164 (2015) no. 4, pp. 723-765 | MR | Zbl

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

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

[8] 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] Eigenvectors of random graphs: delocalization and nodal domains (2011) (http://www.cs.princeton.edu/~bhaskara/files/deloc.pdf)

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

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

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

[13] Quantum ergodicity and averaging operators on the sphere, Int. Math. Res. Not., Volume 2016 (2016) no. 19, pp. 6034-6064 | DOI | MR | Zbl

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

[15] 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] Discrete Green’s functions, J. Comb. Theory, Ser. A, Volume 91 (2000) no. 1-2, pp. 191-214 | DOI | MR

[17] Eigenspaces of graphs, Encyclopedia of Mathematics and Its Applications, 66, Cambridge University Press, 1997 | DOI

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

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

[21] 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] 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] Spectrum of Random $d$-regular Graphs Up to the Edge (2021) (https://arxiv.org/abs/2102.00963)

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

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

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

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

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

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

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

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