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# Quantum dynamics versus random walks

Assume L is the generator of a random walk on a graph (V,E) where V is the set of vertices and E is the set of edges. The hopping probabilities to the vertices so that define the random walk. For example, if every edge has d neighbours and for every vertex (w,v), we get a symmetric random walk on a regular graph. The operator is selfadjoint and has norm 1. Let be a wave which is localized initially at a vertex v. The quantum evolution should be compared with the random walk : while is the probability that the walker starting at the vertex v returns to v in n steps, is the probability that the wave returns back after n steps of the quantum evolution. Opposite to the irreversible random walk, the discrete time quantum evolution is invertible in the sense that the pair determines . The Fourier coefficients of are easy to compute and determine the measures of with respect to L. In many examples of regular infinite graphs, one does not know the spectral type. Aperiodic graphs defined by aperiodic tilings of (see [18, 17]) are examples, where the spectral type is unknown. The spectrum of a graph can be be pure point like in the case of a finite graph or certain self-similar graphs, it can be absolutely continuous like for or for Cayley graphs of infinite Abelian discrete groups, it can also be singular continuous as has been pointed out recently in [26].

In order to relate the return probability of the random walk with the Fourier coefficients, we consider a one parameter family of operators . Denote by the Fourier coefficients of the spectral measure on the circle with respect to the operator .

A measure is called uniformly -continuous if for all intervals [a,b).

Next: Numerical experiments Up: A remark on quantum Previous: Spectral properties and the

Oliver Knill
Tue Aug 18, 1998