Paul Federbush
A random walk on the permutation group, some formal
long-time asymptotic relations
(27K, LaTeX)

ABSTRACT.  We consider the group of permutations of the vertices of a lattice. A  
random walk
is generated by unit steps that each interchange two nearest neighbor  
vertices
of the lattice. We study the heat equation on the permutation group,  
using the
Laplacian associated to the random walk. At t=0 we take as initial  
conditions
a probability distribution concentrated at the identity. A natural  
conjecture
for the probability distribution at long times is that it is  
'approximately' a product
of Gaussian distributions for each vertex. That is, each vertex  
diffuses independently
of the others. We obtain some formal asymptotic results in this  
direction. The
problem arises in certain ways of treating the Heisenberg model of  
ferromagnetism
in statistical mechanics.