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Catalytic branching processes via spine techniques and renewal theory
2011, Döring, Leif, Roberts, Matthew I.
In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behavior is then deduced from renewal theorems and induction.
The longest excursion of a random interacting polymer
2011, Köcher, Janine, König, Wolfgang
We consider a random N-step polymer under the influence of an attractive interaction with the origin and derive a limit law -- after suitable shifting and norming -- for the length of the longest excursion towards the Gumbel distribution. The embodied law of large numbers in particular implies that the longest excursion is of order log N long. The main tools are taken from extreme value theory and renewal theory.