## Abstract

We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) we compute the regularity of their spatial distribution using discrepancy, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid ^{Z2}.

Original language | English |
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Pages (from-to) | 535-541 |

Number of pages | 7 |

Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |

Volume | 379 |

Issue number | 6 |

Early online date | 10 Dec 2014 |

DOIs | |

Publication status | Published - 06 Mar 2015 |

## Keywords

- Laplacian eigenfunctions
- Local extrema
- Quantum chaos
- Universality phenomena

## ASJC Scopus subject areas

- General Physics and Astronomy