The Effect of Disorder on the Free-Energy for the Random Walk Pinning Model: Smoothing of the Phase Transition and Low Temperature Asymptotics

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in (Berger and Toninelli (Electron. J. Probab., to appear) and Birkner and Sun (Ann. Inst. Henri Poincar, Probab. Stat. 46:414-441, 2010; arXiv:0912.1663). Given a fixed realization of a random walk Y on Z(d) with jump rate rho (that plays the role of the random medium), we modify the law of a random walk X on Z(d) with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L-t(X,Y) = integral(t)(0)1(Xs=Ys) ds: the weight of the path under the new measure is exp (beta L-t(X,Y)), beta is an element of R. As beta increases, the system exhibits a delocalization/localization transition: there is a critical value beta(c) , such that if beta >beta(c) the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d >= 3, the presence of disorder makes the phase transition at least of second order. This, in dimension d >= 4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.