C.-L. Yao: Asymptotics for near-critical first-passage percolation on the triangular lattice

Abstract: Consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability p and 1-p, respectively. First, I will discuss a result in the subcritical regime, which is based on https://arxiv.org/abs/2104.01211. Let L(p) denote the correlation length, and let B(p) denote the limit shape in the classical shape theorem. I will show that the re-scaled limit shape B(p)/L(p) converges to a Euclidean disk, as p tends to p_c 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.


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