Exponential last passage percolation (LPP) is a canonical planar directed model of random geometry in the KPZ universality class where the Euclidean metric is distorted by i.i.d. noise. One can also consider a dynamical version of LPP, where the noise is resampled at a constant rate, thereby gradually altering the underlying geometry. In fact, LPP is known to be noise sensitive in the sense that running the dynamics for a microscopic amount of time leads to a macroscopic change in the geometry. In this talk, we shall discuss the question of the existence of exceptional times in dynamical LPP at which bi-infinite geodesics exist. For static LPP, bi-infinite geodesics almost surely do not exist as was shown in Basu–Hoffman–Sly (2018) and Balasz–Busani–Seppalainen (2019).

For dynamical LPP, we show that such exceptional times are at least very close to existing; namely, we give a subpolynomial lower bound (Ω(1/log n)) on the probability that there is an exceptional time t \in [0,1] at which the origin lies on a geodesic of length $n$. In the other direction, for dynamics on the related Brownian LPP model, we analyze "geodesic switches" to establish that the corresponding set of exceptional times almost surely has Hausdorff dimension at most 1/2: we expect the correct dimension to be 0, as can be gathered by an intuitive non-rigorous argument.