Criticality of a Randomly-Driven Front

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

$$\begin{aligned}&\partial_{t} u_{*}(t,\xi) = \frac{1}{2} \partial^{2}_{\xi} u_{*}(t,\xi), \quad \xi >r(t), \\ &u_{*}(t,r(t))=0, \\ &\frac{{\rm d} }{{\rm d}t}r(t) = \frac{1}{2} \partial_\xi u_{*}(t,r(t)), \\ &t \mapsto r(t)\ \hbox{non-decreasing}, \quad r(0):=0.\end{aligned} $$

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.