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Percolation Inequalities and Decision Trees
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The use of decision trees for percolation inequalities started with the celebrated O'Donnell--Saks--Schramm--Servedio (OSSS) inequality. We prove decision tree generalizations of the Harris--Kleitman (HK), van den Berg--Kesten (vdBK), and other inequalities. These inequalities are then applied to estimate the connection probabilities in Bernoulli bond percolation on general graphs.
Forward citations
Cited by 4 Pith papers
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Super-Brownian limits and the $k$-point function for high-dimensional percolation
High-dimensional critical percolation clusters rescale to super-Brownian excursion, verifying the 1984 Aizenman–Newman k-point conjecture under lace-expansion hypotheses.
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Critical long-range percolation II: Low effective dimension
In the long-range low-dimensional regime of percolation, the cluster volume tail and k-point functions are determined up to constants, yielding the hyperscaling identities delta=(d+alpha)/(d-alpha) and d_f=(d+alpha)/2.
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Critical long-range percolation I: High effective dimension
In the regime d > min{6, 3alpha}, critical long-range percolation clusters have an n^{-1/2} volume tail and integrated superprocess scaling limits, switching from super-Levy (alpha < 2) to super-Brownian (alpha >= 2);...
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Critical long-range percolation III: The upper critical dimension
For long-range percolation with d=3α<6, the critical volume tail is ~(log n)^{1/4}/√n, the critical two-point function is ~||x-y||^{-d+α}, and superprocess scaling limits hold with explicit logarithmic corrections.
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