Lattice random walks and their applications to random search in a multi-target environment: from foraging processes to infection transmission
Probability Seminar
24th January 2025, 3:30 pm – 4:30 pm
Fry Building, 2.04
A random or stochastic search is a general mathematical representation of the so-called reactive interactions (probability non-conserving), that is those processes whereby an agent moving with some degree of randomness, at specified locations may collect a resource or may get trapped or cease to exist. When multiple target locations or reactive centres exist, it has been challenging to obtain theoretical predictions of when and where reactive interaction events occur, even in simple scenarios when the movement dynamics are Markovian. The challenge can be ascribed to the lack of a generalisation to multiple targets of the so-called renewal formalism for the first-passage probability, that is the probability to reach a specific location for the first time. With the help of a resolution of a hundred year-old problem on lattice random walks, such formalism has now been developed and has allowed the construction of a general theory to quantify the spatio-temporal dynamics of reactive interactions in a multi-target environment with predictions that are either fully analytical or obtained through the simple inversion of a generating function. The formalism is valid independently of the topology and I will show results for hypercubic, hexagonal and triangular lattices. The theory has also been extended to spatially heterogeneous environments that is in the presence of inert interactions (probability conserving) interactions. If time allows, I will also show the application of the spatially heterogeneous formalism to study the dynamics on some networks as well as a generalisation when the movement statistics is correlated, that is for the one-step non-Markov walk.
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