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Random polynomials and Green functions. (English) Zbl 1097.32012
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).

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