We shall discuss the following problem: A fully occupied lattice is subjected to a sequence of random attacks. Each attack removes a site at random and recursively removes all other sites that have less than k nearest neighbors. Random attacks continue till the lattice is empty. The entire sequence of events is characterized by the size of avalanche (total number of sites removed) in each attack a1, a2, a3, ..... Some questions of interest are: (i) the distribution of avalanche sizes, (ii) the size of the last avalanche that empties the lattice, and the nature of fluctuations just before the lattice is emptied. We attempt to answer these questions by numerical simulations as well as exact solutions on a tree and place this work in the broader context of first order phase transitions that are preceded by critical fluctuations.