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Collision local time of transient random walks and intermediate phases in interacting stochastic systems. (English) Zbl 1228.60054
Summary: In a companion paper [the authors, Probab. Theory Relat. Fields 148, No. 3–4, 403–456 (2010; Zbl 1243.60027)], a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words according to a renewal process. We apply this LDP to prove that the radius of convergence of the generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on $$\mathbb Z^d$$, $$d\geq 1$$, both starting from the origin, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds when the random walk is transient but not strongly transient. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.

MSC:
 60G50 Sums of independent random variables; random walks 60F10 Large deviations 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82D60 Statistical mechanics of polymers
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