The expected Euler characteristic (EEC) method is an integral-geometric method used to approximate the tail probability of the maximum of a random field on a manifold. Noting that the largest eigenvalue of a real-symmetric or Hermitian matrix is the maximum of the quadratic form of a unit vector, we provide EEC approximation formulas for the tail probability of the largest eigenvalue of orthogonally invariant random matrices of a large class. For this purpose, we propose a version of a skew-orthogonal polynomial by adding a side condition such that it is uniquely defined, and describe the EEC formulas in terms of the (skew-)orthogonal polynomials. In addition, for the classical random matrices (Gaussian, Wishart, and multivariate beta matrices), we analyze the limiting behavior of the EEC approximation as the matrix size goes to infinity under the so-called edge-asymptotic normalization. It is shown that the limit of the EEC formula approximates well the Tracy-Widom distributions in the upper tail area, as does the EEC formula when the matrix size is finite.

翻译：预期欧拉特征数方法是一种积分几何方法，用于近似流形上随机场最大值的尾部概率。注意到实对称或厄米矩阵的最大特征值是单位向量二次型的最大值，我们为一类广泛的旋转不变随机矩阵的最大特征值的尾部概率提供了EEC近似公式。为此，我们提出了一个斜正交多项式的版本，通过添加一个边条件使其唯一确定，并基于（斜）正交多项式描述了EEC公式。此外，对于经典随机矩阵（高斯型、威沙特型和多元贝塔型矩阵），我们在所谓的边缘渐近归一化下分析了当矩阵尺寸趋于无穷时EEC近似的极限行为。结果表明，在尾部上端区域，EEC公式的极限能够很好地逼近Tracy-Widom分布，这与矩阵尺寸有限时EEC公式的表现一致。