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 with 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.

• Received 21 May 2007

DOI:https://doi.org/10.1103/PhysRevLett.99.060603