We discuss the distribution of commuting distances and its relation to income. Using data from Denmark, the UK and the USA, we show that the commuting distance is (i) broadly distributed with a slow decaying tail that can be fitted by a power law with exponent g % 3 and (ii) an average growing slowly as a power law with an exponent less than one that depends on the country considered. The classical theory for job search is based on the idea that workers evaluate the wage of potential jobs as they arrive sequentially through time, and extending this model with space, we obtain predictions that are strongly contradicted by our empirical findings. We propose an alternative model that is based on the idea that workers evaluate potential jobs based on a quality aspect and that workers search for jobs sequentially across space. We also assume that the density of potential jobs depends on the skills of the worker and decreases with the wage. The predicted distribution of commuting distances decays as 1/r 3 and is independent of the distribution of the quality of jobs. We find our alternative model to be in agreement with our data. This type of approach opens new perspectives for the modelling of mobility.

The process of urbanization is one of the most important phenomenon of our societies and it is only recently that the availability of massive amounts of geolocalized historical data allows us to address quantitatively some of its features. Here, we discuss how the number of buildings evolves with population and we show on different datasets (Chicago, 1930(Chicago, − 2010 London, 1900 London, − 2015 New York City, 1790 Paris, 1861 Paris, − 2011 that this 'fundamental diagram' evolves in a possibly universal way with three distinct phases. After an initial pre-urbanization phase, the first phase is a rapid growth of the number of buildings versus population. In a second regime, where residences are converted into another use (such as offices or stores for example), the population decreases while the number of buildings stays approximatively constant. In another subsequent phase, the number of buildings and the population grow again and correspond to a re-densification of cities. We propose a stochastic model based on these simple mechanisms to reproduce the first two regimes and show that it is in excellent agreement with empirical observations. These results bring evidences for the possibility of constructing a minimal model that could serve as a tool for understanding quantitatively urbanization and the future evolution of cities.Keywords: Statistical Physics , Urban change , City growth INTRODUCTIONUnderstanding urbanization and the evolution of urban system is a long-standing problem tackled by geographers, historians, and economists and has been abundantly discussed in the literature but still represents a widely debated problem (see for example [1]). The term urbanization has been used in the literature with various definitions, and depending has been considered as a continuous or an intermittent process. In particular, urbanization measured by the fraction of individuals (in a country for example) living in urban areas describes a continuous process that gradually increased in many countries with a quick growth since the middle of the 19 th century until reaching values around 80% in most european countries ([2]). Another definition has been introduced by [3] and presented by [4] as a theory of differential urbanization where it is assumed that in general we observe the three regimes of urbanization, polarization reversal and counter-urbanization, and that are characterized by a gross migration which favors the larger, intermediate, and small-sized cities, respectively.Another approach to study urban changes is presented in the stages of urban development proposed by [5]. According to this model, the city has a life cycle going from an early growing phase to an older phase of stability or decline, and four main intermediate phases of development are identified. The first one called urbanization consists of a concentration of the population in the city core by migration of the people from outer rings. The second phase of suburbanization is characterized by a population growth of the urban agglomeratio...

We analyze the coalescing model where a 'primary' colony grows and randomly emits secondary colonies that spread and eventually coalesce with it. This model describes population proliferation in theoretical ecology, tumor growth, and is also of great interest for modeling urban sprawl. Assuming the primary colony to be always circular of radius r(t) and the emission rate proportional to r(t)^{θ}, where θ>0, we derive the mean-field equations governing the dynamics of the primary colony, calculate the scaling exponents versus θ, and compare our results with numerical simulations. We then critically test the validity of the circular approximation for the colony shape and show that it is sound for a constant emission rate (θ=0). However, when the emission rate is proportional to the perimeter, the circular approximation breaks down and the roughness of the primary colony cannot be discarded, thus modifying the scaling exponents.