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Quantum graphs and random-matrix theory. (English) Zbl 1334.81046
The celebrated conjecture of O. Bohigas et al. [Phys. Rev. Lett. 52, No. 1, 1–4 (1984; Zbl 1119.81326)] states that the spectral fluctuation properties of a Hamiltonian quantum system that is classically chaotic (mixing) coincide with those of the random matrix ensemble in the same symmetry class. In the paper under review, the Bohigas-Giannoni-Schmit conjecture is proved for simple connected graphs with incommensurate bond lengths and with unitary symmetry. Using supersymmetry and taking the limit of infinite graph size, the authors show that the generating function for every $$(P,Q)$$ correlation function for both closed and open graphs coincides with the corresponding expression of random matrix theory. They show that the classical Perron-Frobenius operator is bistochastic and possesses a single eigenvalue $$+1$$. In the quantum case this implies the existence of a zero (or massless) mode of the effective action. That mode causes universal fluctuation properties. Avoiding the saddle-point approximation the authors show that for graphs that are classically mixing (i.e., for which the spectrum of the classical Perron-Frobenius 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.

##### MSC:
 81Q50 Quantum chaos 15A18 Eigenvalues, singular values, and eigenvectors 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices 81Q60 Supersymmetry and quantum mechanics 60B20 Random matrices (probabilistic aspects) 05C80 Random graphs (graph-theoretic aspects) 47B80 Random linear operators
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