Non-Hermitian random matrices with long-range correlations show α-dependent breakdown of the circular law, with spectral radius growing as a power law for α<1 and self-similar density at α=1.
How many zeros of a random polynomial are real?
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abstract
We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve $(1,t,\ldots,t^n)$ projected onto the surface of the unit sphere, divided by $\pi$. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.
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cond-mat.dis-nn 1years
2026 1verdicts
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Spectral properties of non-Hermitian real random matrices with long-range correlations
Non-Hermitian random matrices with long-range correlations show α-dependent breakdown of the circular law, with spectral radius growing as a power law for α<1 and self-similar density at α=1.