TY - JOUR
T1 - Universality of Nodal Count Distribution in Large Metric Graphs
AU - Alon, Lior
AU - Band, Ram
AU - Berkolaiko, Gregory
N1 - Publisher Copyright:
© 2022 Taylor & Francis Group, LLC.
PY - 2022
Y1 - 2022
N2 - An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph’s first Betti number β. We study the distribution of the nodal surplus values in the countably infinite set of the graph’s eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.
AB - An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph’s first Betti number β. We study the distribution of the nodal surplus values in the countably infinite set of the graph’s eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.
KW - Quantum graphs
KW - nodal count
KW - quantum chaos
UR - http://www.scopus.com/inward/record.url?scp=85133428925&partnerID=8YFLogxK
U2 - 10.1080/10586458.2022.2092565
DO - 10.1080/10586458.2022.2092565
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AN - SCOPUS:85133428925
SN - 1058-6458
JO - Experimental Mathematics
JF - Experimental Mathematics
ER -