Scaling limit of the heavy-tailed ballistic deposition model with $p$-sticking

TL;DR

This paper investigates a heavy-tailed ballistic deposition model with probabilistic sticking, revealing a phase transition in the interface's scaling limit as the sticking probability varies from 1 to 0.

Contribution

It introduces a new heavy-tailed variant of the ballistic deposition model and characterizes its scaling limits, highlighting a phase transition based on the sticking probability.

Findings

Identifies a phase transition in the interface growth behavior.

Establishes scaling limits for different sticking probabilities.

Shows the impact of heavy-tailed block heights on interface dynamics.

Abstract

Ballistic deposition is a classical model for interface growth in which unit blocks fall down vertically at random on the different sites of $\mathbb{Z}$ and stick to the interface at the first point of contact, causing it to grow. We consider an alternative version of this model in which the blocks have random heights which are i.i.d. with a heavy (right) tail, and where each block sticks to the interface at the first point of contact with probability $p$ (otherwise, it falls straight down until it lands on a block belonging to the interface). We study scaling limits of the resulting interface for the different values of $p$ and show that there is a phase transition as $p$ goes from $1$ to $0$.