Electronic Journal of Probability

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

Matthias Birkner, Andreas Greven, and Frank den Hollander

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82D60: Polymers

Random walks collision local time annealed vs. quenched large deviation principle interacting stochastic systems intermediate phase

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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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