论文标题
随机正交多项式具有指数重量的真正根部
Real roots of random orthogonal polynomials with exponential weights
论文作者
论文摘要
我们考虑随机的正交多项式 $$ p_ {n}（x）= \ sum_ {i = 0}^{n}ξ_{i} p_ {i}（x）， $$ 其中$ξ_{0} $，。 。 。 ，$ξ_{n} $是独立的随机变量，均值为零，单位方差且均匀边界$（2+ \ ep_0）$ - 矩，$ \ {p_n \} _ {n = 0}^{\ infty} $是与正常的polynomials的体系，是对$ extenter的polterent fore neace fore neace fore neace foreal neach $ penter line lace y $ n $ w。这类正交多项式包括流行的Hermite和弗洛伊德多项式。我们为全球和本地的$ p_n $的真实根数的领先渐近学数量建立了普遍性。此外，我们发现了计算$ p_n的所有根源的措施的限制。$这是通过在经典的三个术语复发关系的背景下引入对逆利特尔伍德 - 未来理论的应用的新想法来实现p_n。$
We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}ξ_{i}p_{i}(x), $$ where $ξ_{0}$, . . . , $ξ_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and $\{p_n\}_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a general exponential weight $W$ on the real line. This class of orthogonal polynomials includes the popular Hermite and Freud polynomials. We establish universality for the leading asymptotics of the expected number of real roots of $P_n$, both globally and locally. In addition, we find an almost sure limit of the measures counting all roots of $P_n.$ This is accomplished by introducing new ideas on applications of the inverse Littlewood-Offord theory in the context of the classical three term recurrence relation for orthogonal polynomials to establish anti-concentration properties, and by adapting the universality methods to the weighted random orthogonal polynomials of the form $W P_n.$
