Asymptotic behavior of a selfinteracting random walk

We consider a simple one-dimensional random walk with the statistical weight of each sample path given by $\pi_t(\omega)=\exp\{-\beta\sum_{0\leq i<j\le n}V(|\omega_j-\omega_i|)\}$, where $\beta$ has the meaning of negative temperature, and $V$ is a nonnegative decreasing function with finite support. We show that for $\beta>\beta_0$ the distribution of $\omega_n$ is concentrated in the area $\{|\omega_n|>c\,n\}$, where $c=c(\beta)>0$, and for $\beta<0$ every sample path becomes localized, in the sense that $\omega_n$ never leaves some fixed interval.