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Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary

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arxiv 1608.00331 v1 pith:4WPYHDII submitted 2016-08-01 cond-mat.dis-nn

Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary

K. S. Tikhonov , A. D. Mirlin This is my paper
classification cond-mat.dis-nn
keywords treecayleyboundarydifferencegraphlocallytree-likewave
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We investigate analytically and numerically eigenfunction statistics in a disordered system on a finite Bethe lattice (Cayley tree). We show that the wave function amplitude at the root of a tree is distributed fractally in a large part of the delocalized phase. The fractal exponents are expressed in terms of the decay rate and the velocity in a problem of propagation of a front between unstable and stable phases. We demonstrate a crucial difference between a loopless Cayley tree and a locally tree-like structure without a boundary (random regular graph) where extended wavefunctions are ergodic.

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