2023/07/03 by Barbara Dembin, Dembin, Barbara, Dor Elboim +3
Mathematics · #Stochastic processes and statistical mechanics #Random Matrices and Applications #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.2307.01162
We study first-passage percolation on \mathbb Zd, d≥ 2, with independent weights whose common distribution is compactly supported in (0,∞) with a uniformly-positive density. Given ε>0 and v∈\mathbb Zd, which edges have probability at least ε to lie on the geodesic between the origin and v? It is expected that all such edges lie at distance at most some r(ε) from either the origin or v, but this remains open in dimensions d≥ 3. We establish the closely-related fact that the number of such edges is at most some C(ε), uniformly in v. In addition, we prove a quantitative bound, allowing ε to tend to zero as ‖v‖ tends to infinity, showing that there are at most O(ε-(2d)/(d-1)(log ‖v‖)C) such edges, uniformly in ε and v. The latter result addresses a problem raised by Benjamin-Kalai-Schramm (2003). Our technique further yields a strengthened version of a lower bound on transversal fluctuations due to Licea-Newman-Piza (1996).