There are some remarkable bijections that connect the spectrum of classical random matrix ensembles (such as Gaussian and Laguerre unitary ensembles) to certain planar random growth models. I shall review some of these connections and discuss a few results on either side that uses the interplay.

Based on joint works with Jnaneshwar Baslingker, Sudeshna Bhattacharjee and Manjunath Krishnapur.