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

Full-text: Open access

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

Permanent link to this document
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

Subjects
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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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


Export citation

References

  • Berger, Quentin; Toninelli, Fabio Lucio. On the critical point of the random walk pinning model in dimension $d=3$. Electron. J. Probab. 15 (2010), no. 21, 654–683.
  • Bhattacharya, R. N.; Ranga Rao, R. Normal approximation and asymptotic expansions. Reprint of the 1976 original. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1986. xiv+291 pp. ISBN: 0-89874-690-6
  • Birkner, Matthias. Particle Systems with Locally Dependent Branching: Long-Time Behaviour, Genealogy and Critical Parameters. Dissertation, Johann Wolfgang Goethe-Universität Frankfurt am Main, 2003. http://publikationen.ub.uni-frankfurt.de/volltexte/2003/314/
  • Birkner, Matthias. A condition for weak disorder for directed polymers in random environment. Electron. Comm. Probab. 9 (2004), 22–25 (electronic).
  • Birkner, Matthias. Conditional large deviations for a sequence of words. Stochastic Process. Appl. 118 (2008), no. 5, 703–729.
  • Birkner, Matthias; Greven, Andreas; den Hollander, Frank. Quenched large deviation principle for words in a letter sequence. Probab. Theory Relat. Fields 148 (2010), no. 3/4, 403–456.
  • Birkner, Matthias; Sun, Rongfeng. Annealed vs quenched critical points for a random walk pinning model. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 2, 414–441.
  • Birkner, Matthias; Sun, Rongfeng. Disorder relevance for the random walk pinning model in dimension 3. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 1, 259–293.
  • Bolthausen, Erwin. A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 (1989), no. 4, 529–534.
  • Camanes, A.; Carmona, P. The critical temperature of a directed polymer in a random environment. Markov Process. Related Fields 15 (2009), no. 1, 105–116.
  • Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo. Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9 (2003), no. 4, 705–723.
  • Comets, Francis; Yoshida, Nobuo. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006), no. 5, 1746–1770.
  • Chover, Joshua. A law of the iterated logarithm for stable summands. Proc. Amer. Math. Soc. 17 1966 441–443.
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • Doney, R. A. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 (1997), no. 4, 451–465.
  • Evans, M.R.; and Derrida, B.. Improved bounds for the transition temperature of directed polymers in a finite-dimensional random medium, J. Stat. Phys. 69 (1992), 427–437.
  • Greven, Andreas. A phase transition for the coupled branching process. I. The ergodic theory in the range of finite second moments. Probab. Theory Related Fields 87 (1991), no. 4, 417–458.
  • Greven, A. On phase-transitions in spatial branching systems with interaction. Stochastic models (Ottawa, ON, 1998), 173–204, CMS Conf. Proc., 26, Amer. Math. Soc., Providence, RI, 2000.
  • Greven, A.; den Hollander, F. Phase transitions for the long-time behavior of interacting diffusions. Ann. Probab. 35 (2007), no. 4, 1250–1306.
  • Heyde, C. C. A note concerning behaviour of iterated logarithm type. Proc. Amer. Math. Soc. 23 1969 85–90.
  • den Hollander, Frank. Random polymers. Lectures from the 37th Probability Summer School held in Saint-Flour, 2007. Lecture Notes in Mathematics, 1974. Springer-Verlag, Berlin, 2009. xiv+258 pp. ISBN: 978-3-642-00332-5
  • Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman. Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp.
  • Kallenberg, Olav. Stability of critical cluster fields. Math. Nachr. 77 (1977), 7–43.
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2.
  • Monthus, C.; Garel, T.. Freezing transition of the directed polymer in a 1+d random medium: Location of the critical temperature and unusual critical properties. Phys. Rev. E 74 (2006), 011101.
  • Seneta, Eugene. Regularly varying functions. Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976. v+112 pp.
  • Spitzer, Frank. Principles of random walks. Second edition. Graduate Texts in Mathematics, Vol. 34. Springer-Verlag, New York-Heidelberg, 1976. xiii+408 pp.