In high-dimensional critical percolation the rescaled k-point connection probability converges to an explicit constant, confirming the Aizenman-Newman conjecture.
and van der Hofstad, R.Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models.Ann
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Convergence of $k$-point functions in high dimensional percolation
In high-dimensional critical percolation the rescaled k-point connection probability converges to an explicit constant, confirming the Aizenman-Newman conjecture.