Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation

Details Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation

2007-05-18

arxiv.org/abs/0705.2648v2

Phys. Rev. Lett. 99, 060603 (2007)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 3 figures. Minor changes. Accepted version in Phys. Rev. Lett

Scientific paper

10.1103/PhysRevLett.99.060603

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d)>0 is the exponent associated to the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n^{-2(\theta(d) + \theta(2))}. For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0,1] has an unusual scaling form given by n^{-\tilde \phi(k/\log n)} where \tilde \phi(x) is a universal large deviation function.

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