Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions

Summary We show a strong type of conditionally mixing property for the Gibbs states ofd-dimensional Ising model when the temperature is above the critical one. By using this property, we show that there is always coexistence of infinite (+ *)-and (–*)-clusters when is smaller than _c andh=0 in two dimensions. It is also possible to show that this coexistence region extends to some non-zero external field case, i.e., for every < _c, there exists someh_c()>0 such that |h|<h_c() implies the coexistence of infinite (*)-clusters with respect to the Gibbs state for (,h).work supported in part by Grant in Aid for Cooperative research no. 03302010, Grant in Aid for Scientific Research no. 03640056 and ISM Cooperative research program (91-ISM,CRP-3)To the memory of Professor Haruo Totoki