Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

In high dimensional percolation at parameter p < pc, the one-arm probability πp(n) is known to decay exponentially on scale (pc − p)−1/2. We show the same statement for the ratio πp(n)/πpc(n), establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability pc. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at “mesoscopic distance” from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter n box on scale nd−6; this result complements a lower bound of Aizenman [1].

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