On the Critical Point of the Random Walk Pinning Model in Dimension d=3

摘要

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 $mathbb{Z}^d$, 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 $eta$ exceeds a certain critical value $eta_c$, 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 $eta_c$ 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 $d$ 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.