Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation |
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Authors: | Grégory Schehr and Satya N Majumdar |
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Institution: | (1) Laboratoire de Physique Théorique (UMR du CNRS 8627), Université de Paris-Sud, 91405 Orsay Cedex, France;(2) Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bat. 100, 91405 Orsay Cedex, France |
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Abstract: | We study various statistical properties of real roots of three different classes of random polynomials which recently attracted
a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis,
i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by
an integer d, of large degree n, one finds that the probability of no real root in the interval 0,1] decays as a power law n
−θ(d) where θ(d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n
−2(θ(2)+θ(d)). For Weyl polynomials and Binomial polynomials, this probability decays respectively like
and
where θ
∞ is such that
in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval a,b] has a scaling form given by
where N
ab
is the mean number of real roots in a,b] and
a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors,
and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields
exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has
a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some
of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, 2007). |
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Keywords: | Persistence probability Random polynomials Zero crossings |
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