Collision local time of transient random walks and intermediate phases in interacting stochastic systems

M. Birkner, A. Greven, W.Th.F. Hollander, den

Research output: Book/ReportReportAcademic

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Abstract

In a companion paper, 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 moment generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on Zd, d >= 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.
Original languageEnglish
Place of PublicationEindhoven
PublisherEurandom
Number of pages29
Publication statusPublished - 2010

Publication series

NameReport Eurandom
Volume2010016
ISSN (Print)1389-2355

Fingerprint

Local Time
Stochastic Systems
Random walk
Collision
Large Deviation Principle
Interacting Diffusions
Directed Polymers
Radius of convergence
Moment generating function
Renewal Process
Random Environment
Empirical Process
Branching process
Long-time Behavior
Continuous Time
Discrete-time
Strictly
Imply

Cite this

Birkner, M., Greven, A., & Hollander, den, W. T. F. (2010). Collision local time of transient random walks and intermediate phases in interacting stochastic systems. (Report Eurandom; Vol. 2010016). Eindhoven: Eurandom.
Birkner, M. ; Greven, A. ; Hollander, den, W.Th.F. / Collision local time of transient random walks and intermediate phases in interacting stochastic systems. Eindhoven : Eurandom, 2010. 29 p. (Report Eurandom).
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Birkner, M, Greven, A & Hollander, den, WTF 2010, Collision local time of transient random walks and intermediate phases in interacting stochastic systems. Report Eurandom, vol. 2010016, Eurandom, Eindhoven.

Collision local time of transient random walks and intermediate phases in interacting stochastic systems. / Birkner, M.; Greven, A.; Hollander, den, W.Th.F.

Eindhoven : Eurandom, 2010. 29 p. (Report Eurandom; Vol. 2010016).

Research output: Book/ReportReportAcademic

TY - BOOK

T1 - Collision local time of transient random walks and intermediate phases in interacting stochastic systems

AU - Birkner, M.

AU - Greven, A.

AU - Hollander, den, W.Th.F.

PY - 2010

Y1 - 2010

N2 - In a companion paper, 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 moment generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on Zd, d >= 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.

AB - In a companion paper, 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 moment generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on Zd, d >= 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.

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Birkner M, Greven A, Hollander, den WTF. Collision local time of transient random walks and intermediate phases in interacting stochastic systems. Eindhoven: Eurandom, 2010. 29 p. (Report Eurandom).