Random walk, cluster growth, and the morphology of urban conglomerations

We propose a new model of cluster growth according to which the probability that a new unit is placed in a point at a distance r from the city center is a Gaussian with mean equal to the cluster radius and variance proportional to the mean, modulated by the local density ρ(r). The model is analytically solvable in d=2 dimensions, where the density profile varies as a complementary error function. The model reproduces experimental observations relative to the morphology of cities, determined via an original analysis of digital maps with a very high spatial resolution, and helps understanding the emergence of vehicular traffic.