Random Polynomials in Several Complex Variables (2112.00880v2)

Abstract: We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials ${p_j}$ with i.i.d. complex random variable coefficients ${a_j}$ where ${p_j}$ form an orthonormal basis for a Bernstein-Markov measure on a compact set $K\subset {\bf C}^d$. Here $m_n$ is the dimension of $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\bf C}^d$. We consider more general bases ${p_j}$, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow $H_n(z):=\sum_{j=1}^{m_n} a_{nj}p_{nj}(z)$; i.e., we have an array of basis polynomials ${p_{nj}}$ and random coefficients ${a_{nj}}$. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of $\frac{1}{n}\log |H_n|$ in $L^1_{loc}({\bf C}^d)$ to the (weighted) extremal plurisubharmonic function for $K$. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.