Zeros of random polynomials undergoing the heat flow
B.C. Hall, C.-W. Ho, J. Jalowy, Z. Kabluchko, ArXiv:2308.11685 (2023).
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Author
Hall, Brian C.;
Ho, Ching-Wei;
Jalowy, Jonas;
Kabluchko, Zakhar
Abstract
We investigate the evolution of the empirical distribution of the complex
roots of high-degree random polynomials, when the polynomial undergoes the heat
flow. In one prominent example of Weyl polynomials, the limiting zero
distribution evolves from the circular law into the elliptic law until it
collapses to the Wigner semicircle law, as was recently conjectured for
characteristic polynomials of random matrices by Hall and Ho, 2022. Moreover,
for a general family of random polynomials with independent coefficients and
isotropic limiting distribution of zeros, we determine the zero distribution of
the heat-evolved polynomials in terms of its logarithmic potential.
Furthermore, we explicitly identify two critical time thresholds, at which
singularities develop and at which the limiting distribution collapses to the
semicircle law. We completely characterize the limiting root distribution of
the heat-evolved polynomials before singularities develop as the push-forward
of the initial distribution under a transport map. Finally, we discuss the
results from the perspectives of partial differential equations (in particular
Hamilton-Jacobi equation and Burgers' equation), optimal transport, and free
probability. The theory is accompanied by explicit examples, simulations, and
conjectures.
Publishing Year
Journal Title
arXiv:2308.11685
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Cite this
Hall BC, Ho C-W, Jalowy J, Kabluchko Z. Zeros of random polynomials undergoing the heat flow. arXiv:230811685. Published online 2023.
Hall, B. C., Ho, C.-W., Jalowy, J., & Kabluchko, Z. (2023). Zeros of random polynomials undergoing the heat flow. In arXiv:2308.11685.
@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} }
Hall, Brian C., Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. “Zeros of Random Polynomials Undergoing the Heat Flow.” ArXiv:2308.11685, 2023.
B. C. Hall, C.-W. Ho, J. Jalowy, and Z. Kabluchko, “Zeros of random polynomials undergoing the heat flow,” arXiv:2308.11685. 2023.
Hall, Brian C., et al. “Zeros of Random Polynomials Undergoing the Heat Flow.” ArXiv:2308.11685, 2023.