Double Transition in a Model of Oscillating Percolation

Abstract

Two distinct transition points have been observed in a problem of lattice percolation studied using a system of pulsating discs. Sites on a regular lattice are occupied by circular discs whose radii vary sinusoidally within $[0,R_0]$ starting from a random distribution of phase angles. A lattice bond is said to be connected when its two end discs overlap with each other. Depending on the difference of the phase angles of these discs a bond may be termed as dead or live. While a dead bond can never be connected, a live bond is connected at least once in a complete time period. Two different time scales can be associated with such a system, leading to two transition points. Namely, a percolation transition occurs at $R_{0c} =0.908$ when a spanning cluster of connected bonds emerges in the system. Here, information propagates across the system instantly, i.e., with infinite speed. Secondly, there exists another transition point $R_0^* = 0.5907$ where the giant cluster of live bonds spans the lattice. In this case the information takes finite time to propagate across the system through the dynamical evolution of finite size clusters. This passage time diverges as $R_0 \to R_0^*$ from above. Both the transitions exhibit the critical behavior of ordinary percolation transition. The entire scenario is robust with respect to the distribution of frequencies of the individual discs. This study may be relevant in the context of wireless sensor networks.