论文标题
多项式和随机矩阵的热流猜想
The heat flow conjecture for polynomials and random matrices
论文作者
论文摘要
当多项式本身根据热流量发展时，我们研究了度$ n $多项式的根的演变。我们提出了一个总体猜想，以实现此演变的巨大限制。具体而言，我们建议（1）限制根分布的对数电势应根据某个一阶非线性PDE演变，并且（2）（2）一般时间的限制根分布应为在一定的显式传输图下的初始分布的推动力。这些结果应该在足够小的时间内得出，也就是说，直到奇异性开始形成。 我们提供三条推理以支持我们的猜想。首先，从随机矩阵的角度来看，猜想在某些随机矩阵模型的特征多项式的第二刻中由变形定理支持。其次，从动力学系统的角度来看，猜想是通过对时间的第二个导数的计算来支持的，该衍生物相对于时间的时间，这在奇点形成之前正式很小。第三，从PDE的角度来看，猜想是由多项式的经验根分布的对数电势所满足的确切pDE \支持的，多项式的经验根分布的对数电势将其正式收敛为所需的PDE作为$ n \ rightarrow \ infty。 最后，我们严格地验证了猜想是否处于全体形态时刻的水平。
We study the evolution of the roots of a polynomial of degree $N$, when the polynomial itself is evolving according to the heat flow. We propose a general conjecture for the large-$N$ limit of this evolution. Specifically, we propose (1) that the log potential of the limiting root distribution should evolve according to a certain first-order, nonlinear PDE, and (2) that the limiting root distribution at a general time should be the push-forward of the initial distribution under a certain explicit transport map. These results should hold for sufficiently small times, that is, until singularities begin to form. We offer three lines of reasoning in support of our conjecture. First, from a random matrix perspective, the conjecture is supported by a deformation theorem for the second moment of the characteristic polynomial of certain random matrix models. Second, from a dynamical systems perspective, the conjecture is supported by the computation of the second derivative of the roots with respect to time, which is formally small before singularities form. Third, from a PDE perspective, the conjecture is supported by the exact PDE\ satisfied by the log potential of the empirical root distribution of the polynomial, which formally converges to the desired PDE as $N\rightarrow \infty.$ We also present a "multiplicative" version of the the conjecture, supported by similar arguments. Finally, we verify rigorously that the conjectures hold at the level of the holomorphic moments.
