  The spatial dynamics of criminal activities has been recently studied through
statistical physics methods; however, models and results have been focused on
local scales (city level) and much less is known about these patterns at larger
scales such as at a country level. Here we report on a characterization of the
spatial dynamics of the homicide crimes along the Brazilian territory using
data from all cities (~5000) in a period of more than thirty years. Our results
show that the spatial correlation function in the per capita homicides decays
exponentially with the distance between cities and that the characteristic
correlation length displays an acute increasing trend in the latest years. We
also investigate the formation of spatial clusters of cities via a
percolation-like analysis, where clustering of cities and a phase
transition-like behavior describing the size of the largest cluster as a
function of a homicide threshold are observed. This transition-like behavior
presents evolutive features characterized by an increasing in the homicide
threshold (where the transitions occur) and by a decreasing in the transition
magnitudes (length of the jumps in the cluster size). We believe that our work
sheds new lights on the spatial patterns of criminal activities at large
scales, which may contribute for better political decisions and resources
allocation as well as opens new possibilities for modeling criminal activities
by setting up fundamental empirical patterns at large scales.
