FKPP fronts in quenched random media

¹ Fondazione Bruno Kessler - Via Sommarive 18, 38123 Povo (TN), Italy
² Department of Mathematics, University of Trento - Via Sommarive 14, 38123 Povo (TN), Italy
³ Department of Mathematics, University of Maryland - College Park, MD, USA
⁴ Centre d'Analyse et de Mathématique Sociales, Ecole des Hautes Etudes en Sciences Sociales - Paris, France
⁵ Institute of Advanced Study, Hong Kong University of Science and Technology - Hong Kong, PRC
⁶ Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique - 91191, Gif-sur-Yvette, France
⁷ Complexity Science Hub - Metternichgasse 8, 1030, Vienna, Austria

Received: 18 February 2026
Accepted: 18 May 2026

Abstract

We study numerically the evolution of one-dimensional FKPP fronts initiated from steep initial conditions in the presence of a quenched random growth rate. Compared to both the homogeneous case (with velocity v₀) and deterministic disorder, quenched randomness increases the average propagation speed. We show that the velocity shift relative to the homogeneous case scales linearly with the disorder variance , with a universal prefactor —independent of the specific distribution of the disorder— such that , with . Moreover, the front position exhibits diffusive fluctuations across disorder realizations. The corresponding effective diffusion coefficient scales quadratically with , with . These results suggest a universal statistical response of FKPP fronts to quenched heterogeneity.