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

• Published in: Electronic journal of probability Vol. 15; no. none; pp. 654 - 683
• Main Authors: Berger, Quentin, Toninelli, Fabio
• Format: Journal Article
• Language: English
• Published: Institute of Mathematical Statistics (IMS) 01.01.2010
• Subjects: Mathematical Physics; Mathematics; Probability; Marginal Disorder; Fractional Moment Method; Method; Random Walk; Pinning Models; Fractional Moment
• ISSN: 1083-6489; 1083-6489
• DOI: 10.1214/EJP.v15-761

We consider the Random Walk Pinning Model studied in [3] and [2]: this is a random walk X on Z d , whose law is modified by the exponential of β times L N (X , Y), the collision local time up to time N with the (quenched) trajectory Y of another d-dimensional random walk. If β exceeds a certain critical value β 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 [3] proved that β 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 ≥ 4 (for d ≥ 5, the result was proven also in [2]). Here, we consider the open case of the marginal dimension d = 3, and we prove non-coincidence of the critical points.