2025/04/16 by Manan Bhatia, Bhatia, Manan
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2504.12293
It is believed that, under very general conditions, bi-infinite geodesics (or bigeodesics) do not exist for planar first and last passage percolation (LPP) models. However, if one endows the model with a natural dynamics, thereby gradually perturbing the geometry, then it is plausible that there could exist a non-trivial set \mathscrT of exceptional times at which such bigeodesics exist. For dynamical exponential LPP, we show that \mathscrT is "very close" to being non-trivial; namely, we obtain an Ω( 1/log n) lower bound on the probability that there exists a random time t∈ [0,1] at which a non-trivial geodesic of length n passes through the origin at its midpoint; note that if the above probability were Ω(1), then it would imply the non-triviality of \mathscrT. We conjecture that, even if \mathscrT≠ ∅, it a.s. has Hausdorff dimension exactly zero.