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

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 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 [3] 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 >= 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.