article published yes Andrew J Campbell author 582b06a9-1f1c-11ee-b076-82ffce00dde4 Kyle Luh author Sean O’Rourke author Santiago Arenas-Velilla author Victor Perez-Abreu author LaEr department Random matrices beyond Wigner-Dyson-Mehta project A Laplacian matrix is a real symmetric matrix whose row and column sums are zero. We investigate the limiting distribution of the largest eigenvalues of a Laplacian random matrix with Gaussian entries. Unlike many classical matrix ensembles, this random matrix model contains dependent entries. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics. In particular, after properly shifting and scaling, we show that the largest eigenvalue converges to the Gumbel distribution as the dimension of the matrix tends to infinity. While the largest diagonal entry is also shown to have Gumbel fluctuations, there is a rather surprising difference between its deterministic centering term and the centering term required for the largest eigenvalues. https://research-explorer.ista.ac.at/download/20046/20069/2025_ElectronJourProbab_Campbell.pdf application/pdfno Institute of Mathematical Statistics2025 eng 1083-6489 2211.17175 00154092700002410.1214/25-ejp1366 301-52 Campbell, Andrew J, Kyle Luh, Sean O’Rourke, Santiago Arenas-Velilla, and Victor Perez-Abreu. “Extreme Eigenvalues of Laplacian Random Matrices with Gaussian Entries.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2025. <a href="https://doi.org/10.1214/25-ejp1366">https://doi.org/10.1214/25-ejp1366</a>. A.J. Campbell, K. Luh, S. O’Rourke, S. Arenas-Velilla, V. Perez-Abreu, Electronic Journal of Probability 30 (2025) 1–52. A. J. Campbell, K. Luh, S. O’Rourke, S. Arenas-Velilla, and V. Perez-Abreu, “Extreme eigenvalues of Laplacian random matrices with Gaussian entries,” <i>Electronic Journal of Probability</i>, vol. 30. Institute of Mathematical Statistics, pp. 1–52, 2025. Campbell, Andrew J., et al. “Extreme Eigenvalues of Laplacian Random Matrices with Gaussian Entries.” <i>Electronic Journal of Probability</i>, vol. 30, Institute of Mathematical Statistics, 2025, pp. 1–52, doi:<a href="https://doi.org/10.1214/25-ejp1366">10.1214/25-ejp1366</a>. Campbell, A. J., Luh, K., O’Rourke, S., Arenas-Velilla, S., &#38; Perez-Abreu, V. (2025). Extreme eigenvalues of Laplacian random matrices with Gaussian entries. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/25-ejp1366">https://doi.org/10.1214/25-ejp1366</a> Campbell AJ, Luh K, O’Rourke S, Arenas-Velilla S, Perez-Abreu V. Extreme eigenvalues of Laplacian random matrices with Gaussian entries. <i>Electronic Journal of Probability</i>. 2025;30:1-52. doi:<a href="https://doi.org/10.1214/25-ejp1366">10.1214/25-ejp1366</a> Campbell AJ, Luh K, O’Rourke S, Arenas-Velilla S, Perez-Abreu V. 2025. Extreme eigenvalues of Laplacian random matrices with Gaussian entries. Electronic Journal of Probability. 30, 1–52. 200462025-07-21T08:06:18Z2025-09-30T14:07:19Z