Resolution of a hundred year old problem on lattice random walks and its impact on modelling interactions in complex systems

In many complex systems the emergence of spatio-temporal patterns depends on the interaction between pairs of individuals, agents or subunits comprising the whole system. Theoretical predictions of such patterns rely upon quantifying when and where interactions event might occur. Even in simple scenarios when the dynamics are Markovian, it has been challenging to obtain estimates of encounter statistics between individuals due to the lack of a mathematical formalism to represent the occurrence of multiple random processes at the same time. In this talk Luca will present such formalism and develop a general theory that allows to quantify the spatio-temporal dynamics of interactions. Spatial discretisation is key to develop such a theory bypassing the need to solve unwieldy boundary value problems, giving predictions that are either fully analytical or semi-analytical. Luca presents applications of the theory for simple interaction processes such as pathogen transmission, thigmotaxis, and molecular binding/unbinding in DNA target search. The formalism can be extended to study the dynamics of an individual in a heterogeneous environment where the heterogeneities may represent areas with different diffusivity, partially permeable or impenetrable barriers or the existence of long-range connections between distant sites. In such context Luca will also show the appearance, in the spatial continuous limit, of new fundamental equations that go beyond the diffusion and the Smoluchowski equation.