Avoidance Coupling
  We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time. 
Omer Angel, Alexander E. Holroyd, James Martin, David B. Wilson, Peter   Winkler
math.PR