The free energy of the random walk pinning model

We consider the random walk pinning model. This is a random walk on ${\mathbb{Z}}^d$ whose law is given as the Gibbs measure ${\mu }_{N,Y}^{\beta }$, where the polymer measure ${\mu }_{N,Y}^{\beta }$ is defined by using the collision local time with another simple symmetric random walk $Y$ on ${\mathbb{Z}}^d$ 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 $\beta$ is smaller than the critical point, then $X$ under ${\mu }_{N,Y}^{\beta }$ satisfies the central limit theorem and the invariance principle $P_Y$-almost surely.