## Electronic Journal of Probability

### Collision Local Time of Transient Random Walks and Intermediate Phases in Interacting Stochastic Systems

#### Abstract

In a companion paper (M. Birkner, A. Greven, F. den Hollander, Quenched LDP for words in a letter sequence, <em>Probab. Theory Relat. Fields</em> <strong>148</strong>, no. 3/4 (2010), 403-456), 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 $\mathbb{Z}^d$, $d\geq1$<em></em>, 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 <em>intermediate phase</em> for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 20, 552-586.

Dates
Accepted: 11 January 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820189

Digital Object Identifier
doi:10.1214/EJP.v16-878

Mathematical Reviews number (MathSciNet)
MR2786642

Zentralblatt MATH identifier
1228.60054

Rights

#### Citation

Birkner, Matthias; Greven, Andreas; den Hollander, Frank. Collision Local Time of Transient Random Walks and Intermediate Phases in Interacting Stochastic Systems. Electron. J. Probab. 16 (2011), paper no. 20, 552--586. doi:10.1214/EJP.v16-878. https://projecteuclid.org/euclid.ejp/1464820189

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