Luca Cattivelli - Università di Parma #
The two-particle problem on comb-like topologies #
When studying the encounter of two random walkers, in principle one can 
map the problem into a one particle problem, where encountering 
corresponds to reaching a given (set of) point(s). This mapping 
preserves the underlying topology as long as one focuses on infinite 
homogeneous structures. However, for inhomogeneous structures this is no 
longer the case and the new embedding topology may be extremely 
different. Here this peculiar feature is analyzed for the two-particle 
problem defined on combs which is mapped into a one-particle problem 
defined on book graphs.
Through this mapping we can prove the emergence of the finite collision 
property and extend it to the case of particles with any, strictly 
positive, velocities \(v_1,v_2>0\). Combs obtained by using as a base 
self-similar fractals as well as \(d\)-dimensional combs are also 
considered showing that the finite collision property is robust. 
Finally, the mapping introduced here allows to get some insights in the 
finite-size problem, where encounters are also known to be slow.