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Boundary-collision orbit statistics in vertex-splitting rational polygonal and disk-scattering billiards is studied using deterministic and stochastic schemes. On increasing the number of vertices in pseudointegrable polygons, the diffusion exponent, deduced from the mean-square orbit displacement, exhibits a crossover from a ballistic to a superdiffusive regime, characteristic of chaotic Sinai billiards.

Here we review the many aspects and distinct phenomena associated to quantum dynamics on general graph structures. For so, we discuss such class of systems under the energy domain Green's function (G) framework. This approach is particularly interesting because G can be written as a sum over classical-like paths, where local quantum effects are taking into account through the scattering matrix amplitudes (basically, transmission and reflection amplitudes) defined on each one of the graph vertices. Hence, the exact G has the functional form of a generalized semiclassical formula, which through different calculation techniques (addressed in details here) always can be cast into a closed analytic expression. It allows to solve exactly arbitrary large (although finite) graphs in a recursive and fast way. Using the Green's function method, we survey many properties for open and closed quantum graphs as scattering solutions for the former and eigenspectrum and eigenstates for the latter, also considering quasi-bound states. Concrete examples, like cube, binary trees and Sierpiński-like topologies are presented. Along the work, possible distinct applications using the Green's function methods for quantum graphs are outlined.

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The interplay between the regular (piecewise-linear) and irregular (vertex-angle) boundary effects in nonintegrable rational polygonal billiards (of m equal sides) is discussed. Decay dynamics in polygons (of perimeter P(m) and small opening Delta) is analyzed through the late-time survival probability S(m) approximately equal t(-delta). Two distinct slow relaxation channels are established. The primary universal channel exhibits relaxation of regular sliding orbits, with delta=1. The secondary channel is given by delta>1 and becomes open when m>P(m)/Delta. It originates from vertex order-disorder dual effects and is due to relaxation of chaoticlike excitations.