Abstract: Consider Bernoulli first-passage percolation on the triangular lattice in which sites have and
passage times with probability
and
, respectively. First, I will discuss a result in the subcritical regime, which is based on https://arxiv.org/abs/2104.01211. Let
denote the correlation length, and let
denote the limit shape in the classical shape theorem. I will show that the re-scaled limit shape
converges to a Euclidean disk, as
tends to
from below. The proof relies on the scaling limit of near-critical percolation established by Garban, Pete and Schramm (2018) and the construction of the collection of continuum clusters introduced by Camia, Conijn and Kiss (2019). Next, I will review some recent results and problems in the critical and supercritical cases.