  Building on our previous work [Phys.Rev.D82,085016(2010)], we show in this
paper how a Brownian motion on a short scale can originate a relativistic
motion on scales that are larger than particle's Compton wavelength. This can
be described in terms of polycrystalline vacuum. Viewed in this way, special
relativity is not a primitive concept, but rather it statistically emerges when
a coarse graining average over distances of order, or longer than the Compton
wavelength is taken. By analyzing the robustness of such a special relativity
under small variations in the polycrystalline grain-size distribution we
naturally arrive at the notion of doubly-special relativistic dynamics. In this
way, a previously unsuspected, common statistical origin of the two frameworks
is brought to light. Salient issues such as the role of gauge fixing in
emergent relativity, generalized commutation relations, Hausdorff dimensions of
representative path-integral trajectories and a connection with Feynman
chessboard model are also discussed.
