Scale-invariant universal crossing probability in one-dimensional diffusion-limited coalescence

TL;DR

This paper derives exact, scale-invariant crossing probabilities for a 1+1 dimensional diffusion-limited coalescence system, revealing universal behavior in off-equilibrium reaction-diffusion dynamics.

Contribution

It provides the first exact analytical results for crossing probabilities in diffusion-limited coalescence, demonstrating their universal scaling form.

Findings

Crossing probability is a universal function of L^2/Dt.

Results hold for both periodic and free boundary conditions.

The crossing probability exhibits scale invariance in the finite-size scaling limit.

Abstract

The crossing probability in the time direction is defined for an off-equilibrium reaction-diffusion system as the probability that the system of size L is still active at time t, in the finite-size scaling limit. Exact results are obtained for the diffusion-limited coalescence problem in 1+1 dimensions with periodic and free boundary conditions using empty interval methods. The crossing probability is a scale-invariant universal function of an effective aspect ratio, L^2/Dt, which is the natural scaling variable for this strongly anisotropic system.