Law of large numbers for the discriminant of random polynomials

Abstract

Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $\Delta(f_n)$ denote the discriminant of $f_n$, that is $\Delta(f_n) = A^{2n-2}\prod_{i < j} (\alpha_j - \alpha_i)^2$ where $A$ is the leading coefficient of $f_n$ and $\alpha_1,\ldots\alpha_n$ are its roots. We prove that with high probability $$|\Delta(f_n)| = n^{2n} e^{-{\sf D}_\ast n(1+o(1))}$$ as $n\to \infty$, for some explicit universal constant ${\sf D}_\ast>0$. A key step in the proof is an analytic representation for the logarithm of the discriminant, which captures both the distributional reciprocal symmetry of the random roots and the cancellations this symmetry induces.