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

We consider the RandomWalk 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 LN (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 βc , 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 βc 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.