# Archive ouverte HAL – Stochastic growth in time-dependent environments

### Guillaume Barraquand 1 Pierre Le Doussal 1 Alberto Rosso 2

#### Guillaume Barraquand, Pierre Le Doussal, Alberto Rosso. Stochastic growth in time-dependent environments. Physical Review E , American Physical Society (APS), 2020, 101 (4), ⟨10.1103/PhysRevE.101.040101⟩. ⟨hal-02565202⟩

We study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance $c(t)$ depending on time. We find that for $c(t)\propto t^{-\alpha}$ there is a transition at $\alpha=1/2$. When $\alpha>1/2$, the solution saturates at large times towards a non-universal limiting distribution. When $\alpha<1/2$ the fluctuation field is governed by scaling exponents depending on $\alpha$ and the limiting statistics are similar to the case when $c(t)$ is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential (2) An exactly solvable discretization, the log-gamma polymer model (3) Numerical simulations.

• 1. Champs Aléatoires et Systèmes hors d'Équilibre
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques