Quantum graphs and randommatrix theory
Abstract
For simple connected graphs with incommensurate bond lengths and with unitary symmetry we prove the BohigasGiannoniSchmit (BGS) conjecture in its most general form. Using supersymmetry and taking the limit of infinite graph size, we show that the generating function for every (P,Q) correlation function for both closed and open graphs coincides with the corresponding expression of randommatrix theory. We show that the classical PerronFrobenius operator is bistochastic and possesses a single eigenvalue +1. In the quantum case that implies the existence of a zero (or massless) mode of the effective action. That mode causes universal fluctuation properties. Avoiding the saddlepoint approximation we show that for graphs that are classically mixing (i.e. for which the spectrum of the classical PerronFrobenius operator possesses a finite gap) and that do not carry a special class of bound states, the zero mode dominates in the limit of infinite graph size.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 July 2015
 DOI:
 10.1088/17518113/48/27/275102
 arXiv:
 arXiv:1407.1148
 Bibcode:
 2015JPhA...48A5102P
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Mathematical Physics
 EPrint:
 35 pages