A sharp transition for the two-dimensional Ising percolation

We show that the percolation transition for the two-dimensional Ising model is sharp. Namely, we show that for every reciprocal temperature >0, there exists a critical valueh_c () of external magnetic fieldh such that the following two statements hold.

(i)	Ifh>hc (), then the percolation probability (i.e., the probability that the origin is in the infinite cluster of + spins) with respect to the Gibbs state ,h for the parameter (,h) is positive.
(ii)	Ifhhc (), then the connectivity function ,h+(0,x) (the probability that the origin is connected by + spins tox with respect to ,h) decays exponentially as |x|.

We also shows that the percolation probability is continuous in (,h) except on the half line {(, 0); _c}.