Abstract: Part 1 (Ritvik Radhakrishnan): We use interpolation with noise to obtain quantitative and unified proofs of the FKG and BK inequalities. Using this we show that in critical Bernoulli percolation on the square lattice the two-arm exponent is strictly larger than twice the one-arm exponent. This method also gives a new proof of a result due to Beffara and Nolin (2011) stating that monochromatic arm exponents are strictly larger than polychromatic arm exponents. This talk is based on joint work with Vincent Tassion.
Part 2 (Loic Gassmann): We show, using a different method, that the strict inequality involving the one-arm and two-arm exponents also holds for critical planar FK-percolation in the continuous phase transition regime (
). The key idea is to show that a good pattern occurs with positive probability at every scale. To emphasize the generality of the method, we also show that, by considering a different pattern, we get that the four-arm exponent is strictly larger than the mixing rate exponent (for 
). This talk is based on a joint work with Ioan Manolescu \url{https://doi.org/10.30757/ALEA.v22-41}.
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