  We study the continuous time version of the random walk pinning model, where
conditioned on a continuous time random walk Y on Z^d with jump rate \rho>0,
which plays the role of disorder, the law up to time t of a second independent
random walk X with jump rate 1 is Gibbs transformed with weight e^{\beta
L_t(X,Y)}, where L_t(X,Y) is the collision local time between X and Y up to
time t. As the inverse temperature \beta varies, the model undergoes a
localization-delocalization transition at some critical \beta_c>=0. A natural
question is whether or not there is disorder relevance, namely whether or not
\beta_c differs from the critical point \beta_c^{ann} for the annealed model.
In Birkner and Sun [BS09], it was shown that there is disorder irrelevance in
dimensions d=1 and 2, and disorder relevance in d>=4. For d>=5, disorder
relevance was first proved by Birkner, Greven and den Hollander [BGdH08]. In
this paper, we prove that if X and Y have the same jump probability kernel,
which is irreducible and symmetric with finite second moments, then there is
also disorder relevance in the critical dimension d=3, and
\beta_c-\beta^{ann}_c is at least of the order e^{-C(\zeta)\rho^{-\zeta}},
C(\zeta)>0, for any \zeta>2. Our proof employs coarse graining and fractional
moment techniques, which have recently been applied by Lacoin [L09] to the
directed polymer model in random environment, and by Giacomin, Lacoin and
Toninelli [GLT09] to establish disorder relevance for the random pinning model
in the critical dimension. Along the way, we also prove a continuous time
version of Doney's local limit theorem [D97] for renewal processes with
infinite mean.
