Geodesic switches and exceptional times in dynamical Brownian last passage percolation

2025/10/31 by Bhatia, Manan
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.27589

Abstract

We consider Brownian last passage percolation evolving dynamically via a discrete resampling procedure. Using Γ_(0,0)^(n,n),r to denote a geodesic from (0,0) to (n,n) at time r, we prove that the expected total number of coarse-grained changes (or "switches") accumulated by Γ_(0,0)^(n,n),r away from its endpoints during a time interval [s,t] is at most n^5/3+o(1)(t-s); we expect the exponent 5/3 to be tight. Using the above estimate, we establish that the set \mathscrT of exceptional times at which a non-trivial bi-infinite geodesic exists a.s. has Hausdorff dimension at most 1/2. Further, for any fixed direction θ, we show that the set \mathscrT^θ⊆ \mathscrT of times at which a non-trivial bi-infinite geodesic directed along θ exists a.s. has Hausdorff dimension equal to 0.

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