Random polynomials and Green functions. (English) Zbl 1097.32012

Bloom, Thomas

Int. Math. Res. Not. 2005, No. 28, 1689-1708 (2005).

Let \(\mu\) be a positive Borel measure on \(\mathbb{C}^n\) such that \(K=\text{supp\,}\mu\) is a regular compact subset of \(\mathbb{C}^n\), \(\mu(K)= 1\), and \(\mu\) satisfies the Bernstein-Markov inequality: for any \(\varepsilon> 0\) there is a constant \(C\) such that for every polynomial \(f\), \(\| f\|_K\leq C(1+\varepsilon)^{\deg(f)}\| f\|_{L^2(\mu)}\).
The main result of the paper is as follows. Let \(\{f_\alpha\}\) be a sequence of random polynomials of increasing lexicographic order \((\alpha\in\mathbb{N}^n)\). Then, with probability 1 (with respect to a Gaussian probability measure on the space of polynomials), \[ \limsup_{\alpha\in\mathbb{N}^n} \Biggl\{{1\over |\alpha|}\log|f_\alpha(z)|\Biggr\}^*= V_K(z),\quad z\in \mathbb{C}^n, \] the Siciak-Zahariuta extremal function of \(K\) (pluricomplex Green function with pole at infinity).

Reviewer: Alexandr Yu. Rashkovsky (Stavanger)

MSC:

32U15	General pluripotential theory
32U35	Plurisubharmonic extremal functions, pluricomplex Green functions
60G99	Stochastic processes

Keywords:

random polynomial; pluricomplex Green function; Markov-Bernstein inequality