article published yes László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Torben H Krüger author 3020C786-F248-11E8-B48F-1D18A9856A870000-0002-4821-3297 Dominik J Schröder author 408ED176-F248-11E8-B48F-1D18A9856A870000-0002-2904-1856 LaEr department Random matrices, universality and disordered quantum systems project IST Austria Open Access Fund project For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner–Dyson–Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969). https://research-explorer.ista.ac.at/download/6185/8771/2020_CommMathPhysics_Erdoes.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Springer Nature2020 eng 0010-3616 1432-0916 1809.03971 00052948300000110.1007/s00220-019-03657-4 3781203-1278 https://research-explorer.ista.ac.at/record/6179 Erdös, László, et al. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” <i>Communications in Mathematical Physics</i>, vol. 378, Springer Nature, 2020, pp. 1203–78, doi:<a href="https://doi.org/10.1007/s00220-019-03657-4">10.1007/s00220-019-03657-4</a>. Erdös, L., Krüger, T. H., &#38; Schröder, D. J. (2020). Cusp universality for random matrices I: Local law and the complex Hermitian case. <i>Communications in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00220-019-03657-4">https://doi.org/10.1007/s00220-019-03657-4</a> L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality for random matrices I: Local law and the complex Hermitian case,” <i>Communications in Mathematical Physics</i>, vol. 378. Springer Nature, pp. 1203–1278, 2020. Erdös, László, Torben H Krüger, and Dominik J Schröder. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” <i>Communications in Mathematical Physics</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s00220-019-03657-4">https://doi.org/10.1007/s00220-019-03657-4</a>. L. Erdös, T.H. Krüger, D.J. Schröder, Communications in Mathematical Physics 378 (2020) 1203–1278. Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I: Local law and the complex Hermitian case. <i>Communications in Mathematical Physics</i>. 2020;378:1203-1278. doi:<a href="https://doi.org/10.1007/s00220-019-03657-4">10.1007/s00220-019-03657-4</a> Erdös L, Krüger TH, Schröder DJ. 2020. Cusp universality for random matrices I: Local law and the complex Hermitian case. Communications in Mathematical Physics. 378, 1203–1278. 61852019-03-28T10:21:15Z2026-04-08T13:55:03Z