  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.
