具有独立同分布根的多项式的热流与重复微分
Heat flow and repeated differentiation of polynomials with i.i.d. roots
- Paderborn University（帕德博恩大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究随机多项式（根独立同分布）在全纯热流和重复微分下零点分布的极限，证明其弱收敛并给出显式传输映射，且可推广至非对称情形。
AI中文摘要：
在全纯热流或重复微分下，多项式的零点如何演化？我们研究了具有独立同分布根 $z_1,\dots,z_n\sim\mu_0$ 的随机多项式的这两种演化，其中 $\mu_0$ 在复平面中具有有界紧支撑密度。对于足够小的时间 $t>0$ 的热流，并假设 $\mu_0$ 的 Stieltjes 变换是 Lipschitz 连续的，我们证明了经验根分布的猜想弱收敛，实际上是几乎必然收敛。我们还确定了在 $\lfloor tn \rfloor$ 次微分后，对于旋转不变初始分布的猜想弱极限。两个极限分布都由显式传输映射下的前推给出：热流将圆律变换为椭圆律，而微分将幸存质量移向原点。共同的证明策略关键依赖于提取一个因子后的递推恒等式，以及 Stieltjes 变换和集中不等式。它允许推广到对称分布之外，并量化了极限分布 $\mu_0$ 在 $o(n)$ 次微分或时间 $t_n\to 0$ 的热流下得以保留的陈述。
英文摘要：
How do the zeros of a polynomial evolve under the holomorphic heat flow or repeated differentiation? In this work, we develop a unified probabilistic proof to three conjectures on the evolution of polynomial zeros under holomorphic heat flow and repeated differentiation. We study these two evolutions for random polynomials with i.i.d.~roots $z_1,\dots,z_n\simμ_0$, where $μ_0$ on $\mathbb C$ satisfies suitable $2+δ$ logarithmic moment conditions. For the heat flow of small time $t>0$ and Lipschitz continuous Stieltjes transform of $μ_0$, we identify the limit distribution $μ_t$ as explicit push-forward of $μ_0$ under a transport map. For rotationally invariant $μ_0$, we characterize the limit after $\lfloor tn\rfloor$ differentiations through its radial quantiles. For instance, the heat flow evolves the circular law into the elliptic and semicircle law, while differentiation moves surviving mass towards the origin. We also show that $o(n)$ derivatives preserve the initial distribution $μ_0$. In fact, all convergences hold almost surely. The common proof strategy crucially relies on recursion identities from leaving out a root, and concentration inequalities, which lead to self-consistent equations for Stieltjes transforms. This brings a method familiar from random matrix theory to polynomial evolutions. Moreover, it allows for generalizations beyond rotational symmetric distributions and quantitative stability estimates for $o(n)$ differentiations and vanishing heat-flow times.
