  Isoperimetric profile describes the minimal boundary size of a set with a
prescribed volume. Itai Benjamini conjectured that the isoperimetric profile of
the giant component in supercritical percolation experiences an averaging
effect and satisfies the law of large numbers. This conjecture was settled by
Biskup-Louidor-Procaccia-Rosenthal for 2D percolation, and later resolved by
Gold for higher-dimensional lattices.
  However, more refined properties of the isoperimetric profile, such as
fluctuations and large deviations, remain unknown. In this paper, we determine
the large deviation probabilities of the isoperimetric constant in 2D
supercritical percolation, answering the question by
Biskup-Louidor-Procaccia-Rosenthal. Interestingly, while the large deviation
probability is of surface order in the entire upper tail regime, a phase
transition occurs in the lower tail regime, exhibiting both surface and volume
order large deviations.
