On maxima of stationary fields

Let $\{X_{\mathbf{n}} : \mathbf{n}\in\mathbb{Z}^d\}$ be a weakly dependent stationary field with maxima $M_{A} := \sup\{X_{\mathbf{i}} : \mathbf{i}\in A\}$ for finite $A\subset\mathbb{Z}^d$ and $M_{\mathbf{n}} := \sup\{X_{\mathbf{i}} : \mathbf{1} \leq \mathbf{i} \leq \mathbf{n} \}$ for $\mathbf{n}\in\mathbb{N}^d$. In a general setting we prove that $P(M_{(n,n,\ldots, n)} \leq v_n) = \exp(- n^d P(X_{\mathbf{0}} > v_n , M_{A_n} \leq v_n)) + o(1)$, for some increasing sequence of sets $A_n$ of size $o(n^d)$. For a class of fields satisfying a local mixing condition, including $m$-dependent ones, the theorem holds with a constant finite $A$ replacing $A_n$. The above results lead to new formulas for the extremal index for random fields.