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

Quentin Berger (1) , Fabio Toninelli (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 ICJ - Institut Camille Jordan

We consider the Random Walk Pinning Model studied in [Birkner-Sun 2008] and [Birkner-Greven-den Hollander 2008]: this is a random walk X on Z^d, whose law is modified by the exponential of beta times 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 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 larger or equal to 4 (for d strictly larger than 4, the result was proven also in [Birkner-Greven-den Hollander 2008]). Here, we consider the open case of the marginal dimension d=3, and we prove non-coincidence of the critical points.

Mots clés: Pinning Models; Random Walk; Fractional Moment; Method; Marginal Disorder

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]

Quentin Berger, Fabio Toninelli. On the critical point of the Random Walk Pinning Model in Dimension d=3. Electronic Journal of Probability, 2010, 15 (21), pp.654-683. ⟨10.1214/EJP.v15-761⟩. ⟨hal-01426326v1⟩