{"abstract":[{"text":"We investigate the evolution of the empirical distribution of the complex\nroots of high-degree random polynomials, when the polynomial undergoes the heat\nflow. In one prominent example of Weyl polynomials, the limiting zero\ndistribution evolves from the circular law into the elliptic law until it\ncollapses to the Wigner semicircle law, as was recently conjectured for\ncharacteristic polynomials of random matrices by Hall and Ho, 2022. Moreover,\nfor a general family of random polynomials with independent coefficients and\nisotropic limiting distribution of zeros, we determine the zero distribution of\nthe heat-evolved polynomials in terms of its logarithmic potential.\nFurthermore, we explicitly identify two critical time thresholds, at which\nsingularities develop and at which the limiting distribution collapses to the\nsemicircle law. We completely characterize the limiting root distribution of\nthe heat-evolved polynomials before singularities develop as the push-forward\nof the initial distribution under a transport map. Finally, we discuss the\nresults from the perspectives of partial differential equations (in particular\nHamilton-Jacobi equation and Burgers' equation), optimal transport, and free\nprobability. The theory is accompanied by explicit examples, simulations, and\nconjectures.","lang":"eng"}],"user_id":"113768","date_created":"2025-03-31T07:14:23Z","citation":{"ama":"Hall BC, Ho C-W, Jalowy J, Kabluchko Z. Zeros of random polynomials undergoing the heat flow. <i>arXiv:230811685</i>. Published online 2023.","ieee":"B. C. Hall, C.-W. Ho, J. Jalowy, and Z. Kabluchko, “Zeros of random polynomials undergoing the heat flow,” <i>arXiv:2308.11685</i>. 2023.","mla":"Hall, Brian C., et al. “Zeros of Random Polynomials Undergoing the Heat Flow.” <i>ArXiv:2308.11685</i>, 2023.","apa":"Hall, B. C., Ho, C.-W., Jalowy, J., &#38; Kabluchko, Z. (2023). Zeros of random polynomials undergoing the heat flow. In <i>arXiv:2308.11685</i>.","chicago":"Hall, Brian C., Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. “Zeros of Random Polynomials Undergoing the Heat Flow.” <i>ArXiv:2308.11685</i>, 2023.","short":"B.C. Hall, C.-W. Ho, J. Jalowy, Z. Kabluchko, ArXiv:2308.11685 (2023).","bibtex":"@article{Hall_Ho_Jalowy_Kabluchko_2023, title={Zeros of random polynomials undergoing the heat flow}, journal={arXiv:2308.11685}, author={Hall, Brian C. and Ho, Ching-Wei and Jalowy, Jonas and Kabluchko, Zakhar}, year={2023} }"},"status":"public","author":[{"full_name":"Hall, Brian C.","first_name":"Brian C.","last_name":"Hall"},{"full_name":"Ho, Ching-Wei","first_name":"Ching-Wei","last_name":"Ho"},{"full_name":"Jalowy, Jonas","first_name":"Jonas","last_name":"Jalowy"},{"full_name":"Kabluchko, Zakhar","first_name":"Zakhar","last_name":"Kabluchko"}],"title":"Zeros of random polynomials undergoing the heat flow","type":"preprint","external_id":{"arxiv":["2308.11685"]},"publication":"arXiv:2308.11685","_id":"59187","date_updated":"2025-03-31T07:19:59Z","year":"2023"}