Abstract
For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponentβ for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL(ϕ):={(x, y):x=r cosθ, y=r sinθ for some r⩾0, orθ⩽ϕ} andβ(ϕ):=limp↓p c [log(p−p c )]−1 log Pcr {itO is connected to ∞ by an occupied path inL(ϕ)}, thenβ(ϕ) is strictly decreasing inϕ on [0, 2π]. Similarly, limn→∞ [−logn]−1 logP cr {itO is connected by an occupied path inL(ϕ)(ϕ) to the exterior of [−n, n]×[−n, n] is strictly decreasing inϕ on [0, 2π].
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Kesten, H., Zhang, Y. Strict inequalities for some critical exponents in two-dimensional percolation. J Stat Phys 46, 1031–1055 (1987). https://doi.org/10.1007/BF01011155
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DOI: https://doi.org/10.1007/BF01011155