Large deviations for the $q$-deformed polynuclear growth

In this paper, we study large time large deviations for the height function $\mathfrak{h}(x,t)$ of the $q$-deformed polynuclear growth introduced in ABW22 [arXiv:2108.06018]. We show that the upper-tail deviations have speed $t$ and derive an explicit formula for the rate function $\Phi_+(\mu)$. On the other hand, we show that the lower-tail deviations have speed $t^2$ and express the corresponding rate function $\Phi_-(\mu)$ in terms of a variational problem. Our analysis relies on distributional identities between the height function $\mathfrak{h}$ and two important measures on the set of integer partitions: the Poissonized Plancherel measure and the cylindric Plancherel measure. Following a scheme developed in DT21 [arXiv:1910.09271], we analyze a Fredholm determinant representation for the $q$-Laplace transform of $\mathfrak{h}(x,t)$, from which we extract exact Lyapunov exponents and through inversion the upper-tail rate function $\Phi_+$. The proof of the lower-tail large deviation principle is more subtle and requires several novel ideas which combine classical asymptotic results for the Plancherel measure and log-concavity properties of Schur polynomials. Techniques we develop to characterize the lower-tail are rather flexible and have the potential to generalize to other solvable growth models.