Large deviations for level sets of branching Brownian motion and Gaussian free fields

Abstract

We study deviation probabilities for the number of high positioned particles in branching Brownian motion, and confirm a conjecture of Derrida and Shi (2016). We also solve the corresponding problem for the two-dimensional discrete Gaussian free field. Our method relies on an elementary inequality for inhomogeneous Galton-Watson processes.

Dedicated to the memory of Professor V.N. Sudakov. The project was partly supported by ANR MALIN; E.A. also acknowledges supports from ANR GRAAL and ANR Liouville. This version corrects a mistake in the original version