Real roots of Random Polynomials: Universality close to accumulation points

Details Real roots of Random Polynomials: Universality close to accumulation points Real roots of Random Polynomials: Universality close to accumulation points

2003-09-04

arxiv.org/abs/math-ph/0309014v4

Journal of Physics A: Mathematical and General, 2004, volume 37, issue 4, pages 1231 - 1239

Physics

Mathematical Physics

Some minor mistakes that crept through into publication have been removed. 10 pages, 12 eps figures. This version contains all

Scientific paper

10.1088/0305-4470/37/4/011

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 \sigma=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 \mu_n = E(c_i) scaled as \mu_n\sim n^{-1/2}.

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