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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$ onZ^d with jump rate rho (that plays the role of the random medium), we modifythe 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(betaL_t(X,Y)), beta in R. As beta increases, the system exhibits adelocalization/localization transition: there is a critical value beta_c, suchthat 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 quenchedand annealed systems have the same behavior. In this paper we investigate howthe disorder modifies the shape of the free energy curve: (1) We prove that, indimension d larger or equal to three 3, the presence of disorder makes thephase transition at least of second order. This, in dimension larger or equalto 4, contrasts with the fact that the phase transition of the annealed systemis of first order. (2) In any dimension, we prove that disorder modifies thelow temperature asymptotic of the free energy.

The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics