On the critical point of the Random Walk Pinning Model in dimension d=3

Details On the critical point of the Random Walk Pinning Model in dimension d=3 On the critical point of the Random Walk Pinning Model in dimension d=3

2009-11-09

arxiv.org/abs/0911.1661v2

Electron. Journal Probab. 15 (2010), 654-683

Mathematics

Probability

23 pages; v2: added reference [4], where a result similar to Th. 2.8 is proven independently for the continuous-time model

Scientific paper

We consider the Random Walk Pinning Model studied in [3,2]: this is a random walk X on Z^d, whose law is modified by the exponential of \beta 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 \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\ge4 (for d\ge 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.

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