This paper concerns a specific weighted Bergman space equipped with a logarithmic-type weight: \(\begin{aligned} A^{p, \alpha }_{\log }=\left\{ f\in H(\mathbb {D}): \int _{\mathbb {D}} |f(z)|^p (1-|z|)^{-1}\left( \log \frac{e}{1-|z|}\right) ^{-(1+\alpha )}dA(z)<\infty \right\} , \end{aligned}\) where \(p,\alpha \in (0,\infty )\) . These spaces turn out to exhibit characteristics reminiscent of both the Hardy spaces and the Bergman spaces. We illustrate some of these attributes by establishing a Littlewood-type theorem for random analytic functions within \(A^{p, \alpha }_{\log }\) , and compare the results with those in the Hardy and Bergman spaces. Along the way, several embedding problems with independent interest are solved. The second theme addressed in this paper is to study zero sets of random log-Bergman functions for which a key step involves characterizing those random analytic functions whose zero sets satisfy a logarithmic variant of the classical Blaschke condition almost surely.
