The free energy of the random walk pinning model

We consider the random walk pinning model. This is a random walk on Zd whose law is given as the Gibbs measure μN,Yβ, where the polymer measure μN,Yβ is defined by using the collision local time with another simple symmetric random walk Y on Zd up to time N. Then, at least two definitions of the phase transitions are known, described in terms of the partition function and the free energy. In this paper, we will show that the two critical points coincide and give an explicit formula for the free energy in terms of a variational representation. Also, we will prove that if β is smaller than the critical point, then X under μN,Yβ satisfies the central limit theorem and the invariance principle PY-almost surely.