Pedestrians in static crowds are not grains, but game players.

The local navigation of pedestrians is assumed to involve no anticipation beyond the most imminent collisions, in most models. These typically fail to reproduce some key features experimentally evidenced in dense crowds crossed by an intruder, namely, transverse displacements toward regions of higher density due to the anticipation of the intruder's crossing. We introduce a minimal model based on mean-field games, emulating agents planning out a global strategy that minimizes their overall discomfort. By solving the problem in the permanent regime thanks to an elegant analogy with the nonlinear Schrödinger's equation, we are able to identify the two main variables governing the model's behavior and to exhaustively investigate its phase diagram. We find that, compared to some prominent microscopic approaches, the model is remarkably successful in replicating the experimental observations associated with the intruder experiment. In addition, the model can capture other daily-life situations such as partial metro boarding.

Space and time correlations for diffusion models with prompt and delayed birth-and-death events.

Understanding the statistical properties of a collection of individuals subject to random displacements and birth-and-death events is key to several applications in physics and life sciences, encompassing the diagnostic of nuclear reactors and the analysis of epidemic patterns. Previous investigations of the critical regime, where births and deaths balance on average, have shown that highly non-Poissonian fluctuations might occur in the population, leading to spontaneous spatial clustering, and eventually to a "critical catastrophe," where fluctuations can result in the extinction of the population. A milder behavior is observed when the population size is kept constant: the fluctuations asymptotically level off and the critical catastrophe is averted. In this paper, we extend these results by considering the broader class of models with prompt and delayed birth-and-death events, which mimic the presence of precursors in nuclear reactor physics or incubation in epidemics. We consider models with and without population control mechanisms. Analytical or semi-analytical results for the density, the two-point correlation function, and the mean-squared pair distance will be derived and compared with Monte Carlo simulations, which will be used as a reference.