Exponential growth of random determinants beyond invariance (Q2690087)

\par The paper is devoted to the study of the expected absolute values of the determinants of \(N\times N\) symmetric random matrices \(H_N\), mainly the limit \[ \lim_{N\to\infty}\frac 1N\log\mathbb{E}[|\det(H_N)|]. \] The authors assume that the distribution of the matrix \(H_N\) does not have to be invariant by orthogonal conjugacy. \par The paper starts with recalling some history on determinants of random matrices and presenting the main results. Then the authors present the applications to matrix models -- the Wigner matrices, Erdős-Rényi matrices, \(d\)-regular matrices, band matrices, sample covariance matrices and some Gaussian matrices. Next they discuss determinants in the presence of long-range correlations. Additionally, Appendix A contains the extension of the results to products of determinants, whereas the Appendix B presents the transition for the exponential order of determinants of Wigner matrices.