  We consider the continuous time version of the Random Walk Pinning Model
(RWPM), studied in [5,6,7]. 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)=\int_0^t
\ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta
L_t(X,Y)), beta in 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 larger or equal to three 3, the presence of disorder makes the
phase transition at least of second order. This, in dimension larger or equal
to 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.
