Localization of the Grover Walks on Spidernets and Free Meixner Laws
Abstract
A spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discretetime quantum walks on graphs. We introduce the Grover walk on a spidernet and its onedimensional reduction. We derive an integral representation of the nstep transition amplitude in terms of the free Meixner law which appears as the spectral distribution. As an application we determine the class of spidernets which exhibit localization. Our method is based on quantum probabilistic spectral analysis of graphs.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 September 2013
 DOI:
 10.1007/s002200131742x
 arXiv:
 arXiv:1206.4422
 Bibcode:
 2013CMaPh.322..667K
 Keywords:

 Quantum Physics;
 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 32 pages