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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@article{CRMATH_2022__360_G4_399_0, author = {McKenzie, Theo}, 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}, number = {G4}, year = {2022}, doi = {10.5802/crmath.316}, language = {en}, url = {http://www.numdam.org/articles/10.5802/crmath.316/} }

TY - JOUR AU - McKenzie, Theo 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 IS - G4 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.316/ DO - 10.5802/crmath.316 LA - en ID - CRMATH_2022__360_G4_399_0 ER -

%0 Journal Article %A McKenzie, Theo %T The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph %J Comptes Rendus. Mathématique %D 2022 %P 399-408 %V 360 %N G4 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.316/ %R 10.5802/crmath.316 %G en %F CRMATH_2022__360_G4_399_0

McKenzie, Theo. The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph. Comptes Rendus. Mathématique, Volume 360 (2022) no. G4, pp. 399-408. doi : 10.5802/crmath.316. http://www.numdam.org/articles/10.5802/crmath.316/

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