A tournament (oriented complete graph) is called
regular if for any vertex v the number of outneighbors of v is
the same as the number of inneighbors. We show that every regular
tournament on n vertices has at least n!/(2+o(1))^n Hamiltonian
cycles, thus answering a question of C. Thomassen and providing a partial
answer to a question posed by E. Friedgut and J. Kahn.
The main ingredient in proving the lower bound is analysis of a
self-avoiding walk on the tournament in question. One key point in this
is the use of Azuma's martingale concentration inequality to say that
various quantities associated with the walk evolve in predictable ways.
