Strict inequalities for some critical exponents in two-dimensional percolation |
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Authors: | Harry Kesten Yu Zhang |
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Institution: | (1) Department of Mathematics, Cornell University, 14853 Ithaca, New York |
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Abstract: | For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL( ):={(x, y):x=r cos , y=r sin for some r 0, or![theta](/content/j71pk3073n822r27/xxlarge952.gif) ![les](/content/j71pk3073n822r27/xxlarge10877.gif) } and ( ):=limp p
c
log(p–p
c
)]–1 log Pcr {itO is connected to by an occupied path inL( )}, then ( ) is strictly decreasing in on 0, 2 ]. Similarly, limn![rarr](/content/j71pk3073n822r27/xxlarge8594.gif) –logn]–1 logP
cr {itO is connected by an occupied path inL( )( ) to the exterior of –n, n]×–n, n] is strictly decreasing in on 0, 2 ]. |
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Keywords: | Percolation critical exponent for percolation probability percolation in sectors strict inequalities |
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