Consider critical site percolation on Zd with d ≥ 2. Cerf (Ann. Probab. 43 (2015) 2458-2480) pointed out that from classical work by Aizenman, Kesten and Newman (Comm. Math. Phys. 111 (1987) 505-532) and Gandolfi, Grimmett and Russo (Comm. Math. Phys. 114 (1988) 549-552) one can obtain that the two-arms exponent is at least 1/2. The paper by Cerf slightly improves that lower bound. Except for d = 2 and for high d, no upper bound for this exponent seems to be known in the literature so far (not even implicitly). We show that the distance-n two-arms probability is at least cn-(d2+4d-2) (with c > 0 a constant which depends on d), thus giving an upper bound d2+ 4d - 2 for the above mentioned exponent.

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doi.org/10.1214/21-AIHP1153
Annales de l'Institut Henri Poincaré - Probability and Statistics
Stochastics

van den Berg, J, & van Engelenburg, D.G.P. (2021). An upper bound on the two-arms exponent for critical percolation on Zd. Annales de l'Institut Henri Poincaré - Probability and Statistics, 58(1), 1–6. doi:10.1214/21-AIHP1153