Lensing of ultra-high energy cosmic rays in turbulent magnetic fields

We consider the propagation of ultra high energy cosmic rays through turbulent magnetic fields and study the transition between the regimes of single and multiple images of point-like sources. The transition occurs at energies around $E_c\simeq Z~41 {\rm EeV}(B_{rms}/5 \mu{\rm G}) (L/ 2 {\rm kpc})^{3/2}\sqrt{50 {\rm pc}/L_c}$, where $L$ is the distance traversed by the CR's with electric charge $Ze$ in the turbulent magnetic field of root mean square strength $B_{rms}$ and coherence length $L_c$. We find that above $2 E_c$ only sources located in a fraction of a few % of the sky can reach large amplifications of its principal image or start developing multiple images. New images appear in pairs with huge magnifications, and they remain amplified over a significant range of energies. At decreasing energies the fraction of the sky in which sources can develop multiple images increases, reaching about 50% for $E>E_c/2$. The magnification peaks become however increasingly narrower and for $E<E_c/3$ their integrated effect becomes less noticeable. If a uniform magnetic field component is also present it would further narrow down the peaks, shrinking the energy range in which they can be relevant. Below $E\simeq E_c/10$ some kind of scintillation regime is reached, where many demagnified images of a source are present but with overall total magnification of order unity. We also search for lensing signatures in the AGASA data studying two-dimensional correlations in angle and energy and find some interesting hints.