Criticality of a randomly-driven front

Consider an advancing `front' $R(t) \in \mathbb{Z}_{\geq 0}$ 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{align*} &\partial_t u_*(t,\xi) = \tfrac12 \partial^2_{\xi} u_*(t,\xi), \quad \xi >r(t), &u_*(t,r(t))=0, &\tfrac{d~}{dt}r(t) = \tfrac12 \partial_\xi u_*(t,r(t)), &t\mapsto r(t) \text{ non-decreasing }, \quad r(0):=0. \end{align*} For a constant initial particles 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, whereasat $\rho \ge 1$ 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.