## Abstract

Consider an advancing ‘front’ \({R(t) \in \mathbb{Z}_{\geqq 0}}\) and particles performing independent continuous time random walks on \({ (R(t), \infty) \cap \mathbb{Z}}\). Starting at \({R(0)=0}\), whenever a particle attempts to jump into \({R(t)}\) the latter instantaneously moves \({k \ge 1}\) steps to the right, *absorbing all* particles along its path. We take *k* to be the minimal random integer such that exactly *k* particles are absorbed by the move of *R*, and view the particle system as a discrete version of the Stefan problem

For a constant initial particle density \({u_{*}(0,\xi)=\rho {\bf 1}_{\{\xi > 0\}}}\), at \({\rho < 1}\) the particle system and the PDE exhibit the same diffusive behavior at large time, whereas at \({\rho \ge 1}\) the PDE explodes instantaneously. Focusing on the critical density \({ \rho=1 }\), we analyze the large time behavior of the front *R*(*t*) for the particle system, and obtain both the scaling exponent of *R*(*t*) and an explicit description of its random scaling limit. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is *sensitive* to the amount of initial local fluctuations. Further, the scaling limit demonstrates an interesting oscillation between instantaneous super- and sub-critical phases. Our method is based on a novel monotonicity as well as PDE-type estimates.

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## Acknowledgments

We thank Vladas Sidoravicius for introducing one of us (A.D.) to questions about critical behavior of the 1d-MDLA. Dembo’s research was partially supported by the NSF Grant DMS-1613091, whereas Tsai’s research was partially supported by a Graduate Fellowship fromthe Kavli Institute for Theoretical Physic (KITP) and by a Junior Fellow award from the Simons Foundation. Some of this work was done during the KITP program “New approaches to non-equilibrium and random systems: KPZ integrability, universality, applications and experiments” supported in part by the NSF Grant PHY-1125915.

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Dembo, A., Tsai, LC. Criticality of a Randomly-Driven Front.
*Arch Rational Mech Anal* **233**, 643–699 (2019). https://doi.org/10.1007/s00205-019-01365-w

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DOI: https://doi.org/10.1007/s00205-019-01365-w