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2010 On the Critical Point of the Random Walk Pinning Model in Dimension d=3
Quentin Berger, Fabio Toninelli
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Electron. J. Probab. 15: 654-683 (2010). DOI: 10.1214/EJP.v15-761


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.


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Quentin Berger. Fabio Toninelli. "On the Critical Point of the Random Walk Pinning Model in Dimension d=3." Electron. J. Probab. 15 654 - 683, 2010.


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

zbMATH: 1226.82027
MathSciNet: MR2650777
Digital Object Identifier: 10.1214/EJP.v15-761

Primary: 82B44
Secondary: 60K35, 60K37, 82B27


Vol.15 • 2010
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