Bloom, Thomas Random polynomials and Green functions. (English) Zbl 1097.32012 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) Cited in 13 Documents MSC: 32U15 General pluripotential theory 32U35 Plurisubharmonic extremal functions, pluricomplex Green functions 60G99 Stochastic processes Keywords:random polynomial; pluricomplex Green function; Markov-Bernstein inequality PDF BibTeX XML Cite \textit{T. Bloom}, Int. Math. Res. Not. 2005, No. 28, 1689--1708 (2005; Zbl 1097.32012) Full Text: DOI