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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 $$d > 2$$.

##### MSC:
 82B43 Percolation 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B27 Critical phenomena in equilibrium statistical mechanics
##### Keywords:
critical percolation; critical cluster; moment bounds
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