We study various geometrical aspects of the propagation of particles obeying fractional statistics in the physical setting of the quantum Hall system. We find a discrete set of zeros for the two-particle kernel in the lowest Landau level; these arise from a combination of a two-particle Aharonov-Bohm effect and the exchange phase related to fractional statistics. The kernel also shows short-distance exclusion statistics, for instance, in a power law behavior as a function of the initial and final positions of the particles. We employ the one-particle kernel to compute impurity-mediated tunneling amplitudes between different edges of a finite-sized quantum Hall system and find that they vanish for certain strengths and locations of the impurity scattering potentials. We show that even in the absence of scattering, the correlation functions between different edges exhibit unusual features for a narrow enough Hall bar.