Dixon's theorem and random synchronization

A transformation monoid on a set Ω is called synchronizing if it contains an element of rank 1 (that is, mapping the whole of Ω to a single point). In this paper, I tackle the question: given n and k, what is the probability that the submonoid of the full transformation monoid Tn generated by k random transformations is synchronizing? The question has some similarities with a similar question about the probability that the subgroup of Sn generated by k random permutations is transitive. For k = 1, the answer is 1/n; for k = 2, Dixon’s Theorem asserts that it is 1− o(1) as n →∞ (and good estimates are now known). For our synchronization question, for k = 1 the answer is also 1/n; I conjecture that for k = 2 it is also 1− o(1). Following the technique of Dixon’s theorem, we need to analyse the maximal non-synchronizing submonoids of Tn. I develop a very close connection between transformation monoids and graphs, from which we obtain a description of non-synchronizing monoids as endomorphism monoids of graphs satisfying some very strong conditions. However, counting such graphs, and dealing with the intersections of their endomorphism monoids, seems difficult.