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The expected Euler characteristic (EEC) method is an integral-geometricmethod used to approximate the tail probability of the maximum of a randomfield on a manifold. Noting that the largest eigenvalue of a real-symmetric orHermitian matrix is the maximum of the quadratic form of a unit vector, weprovide EEC approximation formulas for the tail probability of the largesteigenvalue of orthogonally invariant random matrices of a large class. For thispurpose, we propose a version of a skew-orthogonal polynomial by adding a sidecondition such that it is uniquely defined, and describe the EEC formulas interms of the (skew-)orthogonal polynomials. In addition, for the classicalrandom matrices (Gaussian, Wishart, and multivariate beta matrices), we analyzethe limiting behavior of the EEC approximation as the matrix size goes toinfinity under the so-called edge-asymptotic normalization. It is shown thatthe limit of the EEC formula approximates well the Tracy-Widom distributions inthe upper tail area, as does the EEC formula when the matrix size is finite.

Expected Euler characteristic method for the largest eigenvalue: (Skew-)orthogonal polynomial approach