Large Deviations for the Number of Open Clusters per site in Long Range Bond Percolation

1997, v.3, №3, 367-376

ABSTRACT

We consider the long range bond percolation model: for each pair of sites $i,j$ of $Z^d$ there is a bond connecting them; the bonds are open (closed), independently of each other, with a translation invariant probability $p(i,j)$ $[1-p(i,j)].$ It was proved in [M. Aizenman, H. Kesten and C.M. Newman, Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation, Commun. Math. Phys., 1987, 111, 505-532] that $\kappa(p),$ the mean number of open clusters per site, can be obtained as an almost sure limit of ${\bar K}_n,$ the empirical mean number of open clusters per site inside a box of side $n,$ centered at the origin, when $n$ goes to infinity. Using a large deviation theorem proved in [J.L. Lebowitz and R.H. Schonmann, Pseudo-free energies and large deviations for non gibbsian FKG measures. Probab. Theory and Relat. Fields, 1988, v. 77, 49-64], we estimate how fast the probability of very rare events like $\{|{\bar K}_n - \kappa(p)| \ge \epsilon\}$ goes to zero; we prove that this happens exponentially fast.

Keywords: percolation model,free energy,large deviations