Local Universality of Zeroes of Random Polynomials

Abstract

In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials f = n i=1 ciξiz i and f = n i=1 ciξiz i , where the ξi andξi are iid random variables that match moments to second order, the coefficients ci are deterministic, and the degree parameter n is large. Our results show, under some light conditions on the coefficients ci and the tails of ξi,ξi, that the correlation functions of the zeroes of f andf are approximately the same. As an application, we give some answers to the classical question "How many zeroes of a random polynomials are real ?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions f andf if their log magnitudes log |f |, log |f | are close in distribution, and if some non-concentration bounds are obeyed.

Analysis of the expected density of internal equilibria in random evolutionary multi-player multi-strategy games

“…Last but not least, as the density functions we analyzed here are closely related to Legendre polynomials, and actually, they are of the same form as the Bernstein polynomials, we have made a clear contribution to the longstanding theory of random polynomials [6,15,45,38](see again discussion in introduction). We have derived asymptotic behaviors for the expected real zeros E B of a random Bernstein polynomial, which, to our knowledge, had not been provided before.…”

“…Theorem 5 and Proposition 2, which are deduced from Theorems 3 and 4, are qualitative statements, which tell us how the expected number of internal equilibria per unit length f 2,d in a d-player twostrategy game changes when the number of players d increases. Furthermore, it is important to note that the expected number of real zeros of a random polynomial has been extensively studied, dating back to 1932 with Block and Pólya's seminal paper [6] (see, for instance, [15] for a nice exposition and [45,38] for the most recent progress). Therefore, our results, in Theorems 2, 3 and 4, provide important, novel insights within the theory of random polynomials, but also reveal its intriguing connections and applications to EGT.…”

“…The study of the distribution and expected number of real zeros of a random polynomial is an active research field with a long history dating back to 1932 with Block and Pólya [6], see for instance [15] for a nice exposition and [45,38] for the most recent results and discussions. Consider a random polynomial of the type…”

Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

“…In particular, the results of [TV15] generalized aforementioned ones for real Gaussians to the setting where ζ is a random variable satisfying the moment condition E|ζ| 2+δ < ∞ for some δ > 0. Moreover, Tao and Vu also obtained some variance estimate on N n which leads to a weak law of large numbers for real zeros.…”

“…Then the assertion of Theorem 1.1 is a consequence of Theorem 2.4 which indicates that with high probability (complex) zeros of a random polynomial f n accumulate in the bulk. For non-Gaussian case, we adapt the replacement principle of [TV15] to our setting in order to prove that EN ζ n has universal property in the sense that under mild moment assumptions the lim n→∞…”