Month: April 2013

The characteristic polynomial ${\chi}$ of a sequence of complex numbers ${z_1,\ldots,z_n}$ is defined by

$$\chi(z):=\prod_{k=1}^n(z-z_k).$$

If ${z_1,\ldots,z_n}$ are now random, then one may ask about the roots ${z_1',\ldots,z_n'}$ of the average characteristic polynomial

$$\overline{\chi}(z):=\mathbb{E}(\chi(z))=\mathbb{E}\left(\prod_{k=1}^n(z-z_k)\right) =\prod_{k=1}^n(z-z_k').$$

Note that ${z_1',\ldots,z_n'}$ are not necessarily real or distinct even if ${z_1,\ldots,z_n}$ are real and distinct. By expanding we get

$$\overline{\chi}(z)= z^n+\sum_{k=1}^n\frac{(-1)^k}{k!}z^{n-k} \sum_{i_1\neq\cdots\neq i_k}\mathbb{E}(z_{i_1}\cdots z_{i_k}).$$

(Vieta's formulas: the coefficient of a polynomial are elementary symmetric functions of the roots, in honor of François Viète (1540-1603)). Now, if ${z_1,\ldots,z_n}$ are independent with common mean ${m}$ then

$$\overline{\chi}(z) = z^n+\sum_{k=1}^n\frac{(-1)^k}{k!}z^{n-k} \frac{n!}{(n-k)!}m^k =(z-m)^n.$$

In this case ${z_1'=\cdots=z_n'=m}$, and

$$\frac{1}{n}\sum_{k=1}^n\delta_{z_k'}=\delta_m.$$

In contrast, note that by the law of large numbers,

$$\frac{1}{n}\sum_{k=1}^n\delta_{z_k} \underset{n\rightarrow\infty}{\overset{a.s.}{\longrightarrow}}\mu$$

where ${\mu}$ is the common law of the ${z_k'}$. How about the case where ${z_1,\ldots,z_n}$ are dependent? Let us consider for instance the case where ${z_1,\ldots,z_n}$ are from the Gaussian Unitary Ensemble i.e. the eigenvalues of a ${n\times n}$ Gaussian Hermitian random matrix with density proportional to ${H\mapsto\exp(-n\mathrm{Tr}(H^2))}$. In this case, it is well known that ${\frac{1}{n}\sum_{k=1}^n\delta_{z_k}}$ tend to the Wigner semi-circle law as ${n\rightarrow\infty}$. On the other hand, it is well known that ${\overline{\chi}(z)}$ is the ${n}$-th monic Hermite polynomial, and that ${\frac{1}{n}\sum_{k=1}^n\delta_{z_k'}}$ also tends to the semi-circle law as ${n\rightarrow\infty}$. Beyond the GUE, there are very nice answers when ${z_1,\ldots,z_n}$ follow a determinental point process, explored by Adrien Hardy in arXiv:1211.6564. One may ask the same for permanental point processes.

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