On maxima of stationary fields

Let $\{X_{\mathbf{n}}:\mathbf{n}\in {\mathbb{Z}}^d\}$ be a weakly dependent stationary field with maxima $M_{A} := \sup{X_{$\mathbf{i}$} : $\mathbf{i}$\in A}$forfinite$A\subset$\mathbb{Z}$^d$and$M_{$\mathbf{n}$} := \sup{X_{$\mathbf{i}$} : $\mathbf{1}$ \leq $\mathbf{i}$ \leq $\mathbf{n}$ }$for$$\mathbf{n}$\in$\mathbb{N}$^d$\mathrm{.}Inageneralsettingweprovethat$P(M_{(n,n,\ldots, n)} \leq v_n) = \exp(- n^d P(X_{$\mathbf{0}$} > v_n , M_{A_n} \leq v_n)) + o(1)$,forsomeincreasingsequenceofsets$A_n$ofsize$ o(n^d)$\mathrm{.}Foraclassoffieldssatisfyingalocalmixingcondition,including$m$-dependentones,thetheoremholdswithaconstantfinite$A$replacing$A_n$. The above results lead to new formulas for the extremal index for random fields.