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Large deviation bounds for the volume of the largest cluster in 2D critical percolation

   
@article{ECP3438,
	author = {Demeter Kiss},
	title = {Large deviation bounds for the volume of the largest cluster in 2D critical percolation},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {19},
	year = {2014},
	keywords = {critical percolation; critical cluster; moment bounds},
	abstract = {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 Borgs, Chayes, Kesten and Spencer  [BCKS99]. Furthermore, under some general assumptions similar to those  in [BCKS99], we derive a similar upper bound in dimensions $d > 2$.},
	pages = {no. 32, 1-11},
	issn = {1083-589X},
	doi = {10.1214/ECP.v19-3438},    
        url = {http://ecp.ejpecp.org/article/view/3438}}