On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients

Consider a random polynomial $G_n(z)=\xi_nz^n+...+\xi_1z+\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\to\infty$, to two centered circles and is uniform in the argument as $n\to\infty$. The radii of the circles are $|\xi_0/\xi_\tau|^{1/\tau}$ and $|\xi_\tau/\xi_n|^{1/(n-\tau)}$, where $\xi_\tau$ denotes the coefficient with the maximum modulus.Comment: 8 page

Roots of random polynomials whose coefficients have logarithmic tails

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^n\xi_kz^k$ with i.i.d. coefficients $\xi_0,\ldots,\xi_n$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb{E}\log_+}|\xi_0|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L({\log}|t|)({\log}|t|)^{-\alpha}$ as $t\to\infty$, where $\alpha\geq0$, and $L$ is a slowly varying function. Under this assumption, the structure of complex and real roots of $G_n$ is described in terms of the least concave majorant of the Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}\,du\,dv$.Comment: Published in at http://dx.doi.org/10.1214/12-AOP764 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Universality for zeros of random analytic functions

Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that \E \log (1+|\xi_0|)<\infty. We consider random analytic functions of the form $$G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n} z^k,$$ where $f_{k,n}$ are deterministic complex coefficients. Let $\nu_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\frac 1n \log f_{[tn], n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $\nu_n$ converges weakly to some deterministic measure which is characterized in terms of the Legendre--Fenchel transform of $u$. The limiting measure is universal, that is it does not depend on the distribution of the $\xi_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.Comment: 26 pages, 8 figures, 1 tabl

Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $\xi_1,...,\xi_d$ with covariance matrices $\Sigma_1,...,\Sigma_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of $\xi_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\top\Sigma_i^{-1} \mathbf{x}\leqslant1}$. We show that $$\mathbb{E}\,|\det M|=\frac{d!}{(2\pi)^{d/2}}V_d(\mathcal{E}_1,...,\mathcal{E}_d),$$ where $V_d(\cdot,...,\cdot)$ denotes the {\it mixed volume}. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an analytic expression for the mixed volume of $d$ arbitrary ellipsoids in $\mathbb{R}^d$. As another application, we consider a smooth centered non-degenerate Gaussian random field $X=(X_1,...,X_k)^\top:\mathbb{R}^d\to\mathbb{R}^k$. Using Kac-Rice formula, we obtain the geometric interpretation of the intensity of zeros of $X$ in terms of the mixed volume of location-dispersion ellipsoids of the gradients of $X_i/\sqrt{\mathbf{Var} X_i}$. This relates zero sets of equations to mixed volumes in a way which is reminiscent of the well-known Bernstein theorem about the number of solutions of the typical system of algebraic equations

The political economy of fixed regional investment shares with an illustration for Belgian railway investments.

Many local public goods are allocated by federal governments using fixed regional shares: every region is entitled a fixed share of the total budget for a particular type of public good. This paper compares this fixed regional sharing rule with two alternative allocation rules: first best and common pool allocation. We find that the fixed regional sharing rule performs relatively well if the regional shares are reasonable. Legislative bargaining theory is used to study the determination of the fixed regional shares.