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 interaction events occur. Even in simple scenarios when the dynamics are Markovian, it has been difficult to obtain estimates of interactions since the analysis has mostly relied upon time-consuming methodologies in the form of stochastic simulations or the numerical solution of challenging boundary value partial differential equations (PDE). Space-time discrete models can instead offer a practical solution given their ability to deal with any type of boundary constraint, but they require knowledge of the analytic solution of the discrete diffusion equation (DDE) in arbitrary dimensions and arbitrary boundary conditions, a problem that has remained open for a long time since the introduction of lattice random walks by Smoluchowski. Armed with exact solution of the DDE, I will introduce a general formalism that allows to quantify inert (probability preserving) and reactive (probability non-preserving) interactions. Example applications include the random movement in hexagonal and triangular lattices, the dynamics of resetting random walks, animal thigmotaxis, the exact prediction of the basic reproduction number, as well as the derivation of a new fundamental PDE beyond the Smoluchowski equation in presence of permeable barriers.
