Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

First-passage percolation studies the geometry obtained from a random perturbation of Euclidean geometry. In the discrete planar setting, one assigns random, independent and identically distributed, lengths to the edges of the lattice Z^2 and studies the resulting geodesics - paths of minimal length between points. While the physics literature presents an elaborate picture for the behavior of the model, placing it in the KPZ universality class, mathematical progress remains rather limited. Aside from the random geometry perspective, the model also enjoys close ties with disordered spin systems. The talk will give an introduction to planar first-passage percolation, followed by a description of recent progress, joint with Barbara Dembin and Dor Elboim, on the problem of coalescence of geodesics. Our result shows, under mild assumptions, that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result further implies the first quantitative progress on a problem of Benjamini-Kalai-Schramm (2002), which asks to prove that the midpoint of a geodesic is unlikely to lie on the straight line connecting the endpoints of the geodesic.



Tel Aviv University, Member, School of Mathematics