Generic-case complexity, decision problems in group theory and random   walks
  We give a precise definition of ``generic-case complexity'' and show that for a very large class of finitely generated groups the classical decision problems of group theory - the word, conjugacy and membership problems - all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs. 
Ilya Kapovich, Alexei Myasnikov, Paul Schupp and Vladimir Shpilrain