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SUMMARY:INI-RIMS joint seminar: The multi-target problem on Cartesian\, he
 xagonal and triangular lattices in homogeneous and heterogeneous environme
 nts - Luca Giuggioli (University of Bristol)
DTSTART:20231025T080000Z
DTEND:20231025T090000Z
UID:TALK202918@talks.cam.ac.uk
DESCRIPTION:In many complex systems the emergence of spatio-temporal patte
 rns depends on the interaction between pairs of individuals\, agents or su
 bunits comprising the whole system. Theoretical predictions of such patter
 ns rely upon quantifying when and where interaction events might occur. Ev
 en in simple scenarios when the dynamics are Markovian\, it has been chall
 enging to obtain estimates of encounter statistics between individuals due
  to the lack of a mathematical formalism to represent the occurrence of mu
 ltiple random processes at the same time\, the so-called splitting probabi
 lity. With the help of a resolution of a hundred year old&nbsp\; problem o
 n lattice random walks\, I will present such formalism and develop a gener
 al theory that allows to quantify the spatio-temporal dynamics of interact
 ions when a token of information is transferred upon co-location or proxim
 ity. Spatial discretisation is key to develop such a theory bypassing the 
 need to solve unwieldy boundary value problems\, giving predictions that a
 re either fully analytical or obtained through the simple inversion of a g
 enerating function. The formalism is also applied to hexagonal and triangu
 lar lattices for which an analytical exact description of the occupation p
 robability has recently been derived. In heterogeneous environments\, a mo
 dification of the original formalism has been developed to quantify the in
 teractions of a lattice random walk with the so-called inert heterogeneiti
 es. Such heterogeneities may represent locations with long-range connectio
 ns to distant sites\, areas with different diffusivity\, or when permeable
  or impenetrable barriers are present. In this latter context\, in the spa
 ce-time continuous limit\, a new fundamental equation that go beyond the d
 iffusion and the Smoluchowski equation in the presence of permeable barrie
 rs has been found. Some applications of the formalism such as animal thigm
 otaxis\, i.e. the tendency of an animal to remain close to the boundaries 
 of a confining domain\, and the search of a promoter region on DNA by tran
 scription factors\, a prototypical two-particle coalescent process\, will 
 also be presented.
LOCATION:Seminar Room 1\, Newton Institute
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