The study of random analytic functions has a rich history, and the past research focused on the study of individual models, such as the Kac polynomials and special entire Gaussian analytic functions. This direction has exhibited extensive contact with statistical physics and random matrices. In this talk, we shall report recent progresses along a less investigated direction: the study of Banach spaces of random analytic functions. In other words, instead of individual functions, we look at them collectively, and study the metric aspect of them. This leads naturally to the interaction between randomization and the Banach space structure. In particular, we report Littlewood-type theorems for four classical Banach spaces: Hardy, Bergman, Dirichlet and Fock. The methodology is an interplay between probability, complex analysis, and functional analysis.

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