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

TL;DR

This paper investigates the critical behavior of the Random Walk Pinning Model in three dimensions, demonstrating that disorder affects the phase transition differently than in lower or higher dimensions.

Contribution

It proves that in three dimensions, the quenched and annealed critical points do not coincide, addressing an open problem in the marginal case.

Findings

Critical points differ in dimension d=3.

Disorder relevance is confirmed in d=3.

Results extend understanding of phase transitions in random walk models.

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 \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 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\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.