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

Abstract

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 βtimes L_N(X,Y), the collision local time up to time N with the (quenched) trajectory Y of another d-dimensional random walk. If βexceeds 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 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\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.