20,332 research outputs found
Periodic-Orbit Theory of Universality in Quantum Chaos
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory,
that full classical chaos is paralleled by quantum energy spectra with
universal spectral statistics, in agreement with random-matrix theory. For
dynamics from all three Wigner-Dyson symmetry classes, we calculate the
small-time spectral form factor as power series in the time .
Each term of that series is provided by specific families of pairs of
periodic orbits. The contributing pairs are classified in terms of close
self-encounters in phase space. The frequency of occurrence of self-encounters
is calculated by invoking ergodicity. Combinatorial rules for building pairs
involve non-trivial properties of permutations. We show our series to be
equivalent to perturbative implementations of the non-linear sigma models for
the Wigner-Dyson ensembles of random matrices and for disordered systems; our
families of orbit pairs are one-to-one with Feynman diagrams known from the
sigma model.Comment: 31 pages, 17 figure
Large cycles and a functional central limit theorem for generalized weighted random permutations
The objects of our interest are the so-called -permutations, which are
permutations whose cycle length lie in a fixed set . They have been
extensively studied with respect to the uniform or the Ewens measure. In this
paper, we extend some classical results to a more general weighted probability
measure which is a natural extension of the Ewens measure and which in
particular allows to consider sets depending on the degree of the
permutation. By means of complex analysis arguments and under reasonable
conditions on generating functions we study the asymptotic behaviour of
classical statistics. More precisely, we generalize results concerning large
cycles of random permutations by Vershik, Shmidt and Kingman, namely the weak
convergence of the size ordered cycle length to a Poisson-Dirichlet
distribution. Furthermore, we apply our tools to the cycle counts and obtain a
Brownian motion central limit theorem which extends results by DeLaurentis,
Pittel and Hansen.Comment: 24 pages, 3 Figure
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