Convergence of limit shapes for 2D near-critical first-passage percolation

We 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. For each p is an element of (0, pc), let B(p) be the limit shape in the classical "shape theorem", and let L(p) be the correlation length. We show that as p up arrow pc, the rescaled limit shape L(p)-1B(p) converges to a Euclidean disk. This improves a result of Chayes et al. [J. Stat. Phys. 45 (1986) 933-951]. The proof relies on the scaling limit of near-critical percolation established by Garban et al. [J. Eur. Math. Soc. 20 (2018) 1195-1268], and uses the construction of the collection of continuum clusters in the scaling limit introduced by Camia et al. [Springer Proceedings in Mathematics & Statistics, 299 (2019) 44-89].