  We consider the model of random interlacements on transient graphs, which was
first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039-2087] for the
special case of ${\mathbb{Z}}^d$ (with $d\geq3$). In Sznitman [Ann. of Math.
(2) (2010) 171 2039-2087], it was shown that on ${\mathbb{Z}}^d$: for any
intensity $u>0$, the interlacement set is almost surely connected. The main
result of this paper says that for transient, transitive graphs, the above
property holds if and only if the graph is amenable. In particular, we show
that in nonamenable transitive graphs, for small values of the intensity u the
interlacement set has infinitely many infinite clusters. We also provide
examples of nonamenable transitive graphs, for which the interlacement set
becomes connected for large values of u. Finally, we establish the monotonicity
of the transition between the "disconnected" and the "connected" phases,
providing the uniqueness of the critical value $u_c$ where this transition
occurs.
