Real roots of Random Polynomials: Universality close to accumulation points

Abstract

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent, identically distributed coefficients c_i, as long as the second moment σ=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to the previously reported abrupt) and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value μ_n = E(c_i) scaled as μ_n\sim n^{-1/2}.

Some minor mistakes that crept through into publication have been removed. 10 pages, 12 eps figures. This version contains all updates, clearer pictures and some more thorough explanations

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• High-Dimensional Random Fields and Random Matrix Theory
• Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation
• Condensation of the roots of real random polynomials on the real axis