Electronic Journal of Probability

On the Critical Point of the Random Walk Pinning Model in Dimension d=3

Quentin Berger and Fabio Toninelli

Full-text: Open access

Abstract

We consider the Random Walk Pinning Model studied in [Birkner-Sun 2008] and [Birkner-Greven-den Hollander 2008]: this is a random walk $X$ on $\mathbb{Z}^d$, whose law is modified by the exponential of beta times the collision local time up to time $N$ with the (quenched) trajectory $Y$ of another $d$-dimensional random walk. If $\beta$ exceeds a certain critical value $\beta_c$, the two walks stick together for typical $Y$ realizations (localized phase). A natural question is whether the disorder is relevant or not, that is whether the quenched and annealed systems have the same critical behavior. Birkner and Sun proved that $\beta_c$ coincides with the critical point of the annealed Random Walk Pinning Model if the space dimension is $d=1$ or $d=2$, and that it differs from it in dimension $d$ larger or equal to $4$ (for $d$ strictly larger than $4$, the result was proven also in [Birkner-Greven-den Hollander 2008]). Here, we consider the open case of the marginal dimension $d=3$, and we prove non-coincidence of the critical points.

Article information

Source
Electron. J. Probab. Volume 15 (2010), paper no. 21, 654-683.

Dates
Accepted: 17 May 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819806

Digital Object Identifier
doi:10.1214/EJP.v15-761

Mathematical Reviews number (MathSciNet)
MR2650777

Zentralblatt MATH identifier
1226.82027

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 60K37: Processes in random environments

Keywords
Pinning Models Random Walk Fractional Moment Method Marginal Disorder

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Berger, Quentin; Toninelli, Fabio. On the Critical Point of the Random Walk Pinning Model in Dimension d=3. Electron. J. Probab. 15 (2010), paper no. 21, 654--683. doi:10.1214/EJP.v15-761. https://projecteuclid.org/euclid.ejp/1464819806


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