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 '18 and Balasz-Busani-Seppalainen '19.
For dynamical LPP, we show that such exceptional times are at least very close to existing; namely, we give a subpolynomial lower bound $((1/\log n))$ 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 a dynamics on the related Brownian LPP model, we analyse `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.