Abstract：

First-passage percolation is the study of the geometry obtained from a random perturbation of Euclidean geometry. In the discrete setting, one assigns random, independent and identically distributed, lengths to the edges of the lattice Z^d and studies the resulting geodesics - paths of minimal length between points. Given eps>0 and v in Z^d, which edges have probability at least eps to lie on the geodesic between the origin and v? It is expected that all such edges lie at distance at most some r(eps) 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(eps), uniformly in v. We further provide quantitative bounds for eps tending to zero as ||v|| tends to infinity. This addresses a problem raised by Benjamini–Kalai–Schramm (2003). Joint work with Barbara Dembin and Dor Elboim.

About the Speaker:

Ron Peled

Tel Aviv University

I am a full professor in the School of Mathematical Sciences of Tel Aviv University. My research interests are in Probability Theory, Statistical Physics and related fields.

In the 2022-2024 academic years I am visiting Princeton University and the Institute for Advanced Study.