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 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, the so-called splitting probability. With the help of a resolution of a hundred year old problem on lattice random walks, I will present such formalism and develop a general theory that allows to quantify the spatio-temporal dynamics of interactions when a token of information is transferred upon co-location or proximity. 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 obtained through the simple inversion of a generating function. The formalism is also applied to hexagonal and triangular lattices for which an analytical exact description of the occupation probability has recently been derived. In heterogeneous environments, a modification of the original formalism has been developed to quantify the interactions of a lattice random walk with the so-called inert heterogeneities. Such heterogeneities may represent locations with long-range connections to distant sites, areas with different diffusivity, or when permeable or impenetrable barriers are present. In this latter context, in the space-time continuous limit, a new fundamental equation that go beyond the diffusion and the Smoluchowski equation in the presence of permeable barriers has been found. Time permitting, I will show some applications of the formalism: from transdermal drug delivery and animal thigmotaxis, i.e. the tendency of an animal to remain close to the boundaries of a confining domain, to the search of a promoter region on DNA by transcription factors, a prototypical two-particle coalescent process.

**References**
Giuggioli, Exact spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions: a century after Smoluchowski and Pólya, *Physycal Review X *10:021045 (2020)

Sarvaharman & Giuggioli (2020) Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions, *Physical Review E *102(6):062124 (2020)

Kenkre & Giuggioli, Theory of the spread of epidemics and movement ecology of animals: an interdisciplinary approach with methodologies from physics and mathematics, *Cambridge University Press* (2021)

Das & Giuggioli, *Discrete space-time resetting model: application to first-passage and transmission statistics*, **Journal of Physics A: Mathematical and Theoretical**, *55*(42):424004 (2022)

Kay and Giuggioli, *Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics*, **Physical Review Research **4(3):L032039 (2022)

Giuggioli and Sarvaharman, *Spatio-temporal dynamics of random transmission events: from information sharing to epidemic spread*, **Journal of Physics A: Mathematical and Theoretical **55(37):375005 (2022)

Das & Giuggioli, *Dynamics of lattice random walk within regions composed of different media and interfaces*, **Journal of Statistical Mechanics: Theory and Experiment**, 013201 (2023)

Sarvaharman and Giuggioli, *Particle-environment interactions in arbitrary dimensions: a unifying analytic framework to model diffusion with inert spatial heterogeneities*, under revision, arXiv preprint arXiv:2209.09014.

Marris, Sarvaharman and Giuggioli, *Exact Spatio-Temporal Dynamics of Lattice Random Walks in Hexagonal and Honeycomb Domains*, under revision.