Large deviations for the $q$-deformed polynuclear growth

TL;DR

This paper investigates the large deviations of the height function in the $q$-deformed polynuclear growth model, deriving explicit rate functions for upper and lower tails using advanced probabilistic and combinatorial techniques.

Contribution

It provides the first explicit formulas for large deviation rate functions in the $q$-deformed polynuclear growth, including novel methods for lower-tail analysis.

Findings

Upper-tail deviations have speed t with explicit rate function.

Lower-tail deviations have speed t^2 with a variational rate function.

Techniques developed can be applied to other solvable growth models.

Abstract

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.