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Bi-infinite geodesics are fundamental objects of interest in planar firstpassage percolation. A longstanding conjecture states that under mildconditions there are almost surely no bigeodesics, however the result has notbeen proved in any case. For the exactly solvable model of directed lastpassage percolation on $\mathbb{Z}^2$ with i.i.d. exponential passage times, westudy the corresponding question and show that almost surely the onlybigeodesics are the trivial ones, i.e., the horizontal and vertical lines. Theproof makes use of estimates for last passage time available from theintegrable probability literature to study coalescence structure of finitegeodesics, thereby making rigorous a heuristic argument due to Newman.

Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation