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Large deviation bounds for the volume of the largest cluster in 2D critical percolation. (English) Zbl 1302.82055
Summary: Let \(M_n\) denote the number of sites in the largest cluster in site percolation on the triangular lattice inside a box side length \(n\). We give lower and upper bounds on the probability that \(M_n / \mathbb{E} M_n > x\) of the form \(\exp(-Cx^{2/\alpha_1})\) for \(x \geq 1\) and large \(n\) with \(\alpha_1 = 5/48\) and \(C>0\). Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by C. Borgs et al. [Commun. Math. Phys. 224, No. 1, 153–204 (2001; Zbl 1038.82035)]. Furthermore, under some general assumptions similar to those in [loc. cit.], we derive a similar upper bound in dimensions \(d > 2\).

82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B27 Critical phenomena in equilibrium statistical mechanics
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