Random walks on trees

Abstract

Random walks or Brownian motions appear as a useful tool in algorithm analysis. Recently P. Flajolet ([2]) obtained a complete and detailed analysis of the two stacks problem with the help of properties of simple random walks on lattices. G. Louchard ([7], [8]) proved that the Brownian motion permits to give easily asymptotic results on the complexity of manipulation algorithms for sorted tables, dictonaries and priority queues. In [4], J. Françon and the author proved that random walks on some homogeneous trees can be analysed with simple combinatorial technics : generating functions, continued and multicontinued fractions, orthogonal polynomials, theorem of Darboux etc.. In this paper we show that random walks on more general trees can be related to random walks on N and that on general Cayley graphs (i.e. graphs corresponding to finitely generated groups with relations between the generators) the asymptotic behavior of the random walks can be obtained using proporties of the Brownian motions on Riemannian manifolds and a simple criteria can be given in terms of γ(n) the number of different words was length is less than or equal to n .