Non-existence of bi-infinite polymer Gibbs measures on Z^2

To each vertex x\in  Z^2 assign a positive weight \omega_x. A geodesic between two ordered points on the lattice is an up-right path maximizing the cumulative weight along itself.  A bi-infinite geodesic is an infinite path taking up-right steps on the lattice and such that for every two points on the path, its restriction to between the points is a geodesic. Assume the weights across the lattice are  i.i.d.,  does there exist a bi-infinite geodesic with some positive probability?

In the case the weights are Exponentially distributed, we answer this question in the negative. We show an analogous result for the positive-temperature variant of this model.

Joint work with Marton Balazs and Timo Seppalainen.