  In this paper we construct a unitary matrix model that captures the
asymptotic growth of Young diagrams under $q$-deformed Plancherel measure. The
matrix model is a $q$ analog of Gross-Witten-Wadia (GWW) matrix model. In the
large $N$ limit the model exhibits a third order phase transition between
no-gap and gapped phases, which is a $q$-deformed version of the GWW phase
transition. We show that the no-gap phase of this matrix model captures the
asymptotic growth of Young diagrams equipped with $q$-deformed Plancherel
measure. The no-gap solutions also satisfies a differential equation which is
the $q$-analogue of the automodel equation. We further provide a droplet
description for these growing Young diagrams. Quantising these droplets we
identify the Young diagrams with coherent states in the Hilbert space. We also
elaborate the connection between moments of Young diagrams and the infinite
number of commuting Hamiltonians obtained from the large $N$ droplets and
explicitly compute the moments for asymptotic Young diagrams.
