  We study Conformal Loop Ensemble (CLE$_{\kappa}$) in doubly connected
domains: annuli, the punctured disc, and the punctured plane. We restrict
attention to CLE$_{\kappa}$ for which the loops are simple, i.e. $\kappa\in
(8/3,4]$. In the paper "Conformal Loop Ensemble" by Sheffield and Werner,
simple CLE in the unit disc is introduced and constructed as the collection of
outer boundaries of outermost clusters of the Brownian loop soup. For simple
CLE in the unit disc, any fixed interior point is almost surely surrounded by
some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of
points that are not surrounded by any loop, almost surely has Lebesgue measure
zero. In the current paper, simple CLE in an annulus is constructed similarly:
it is the collection of outer boundaries of outermost clusters of the Brownian
loop soup conditioned on the event that there is no cluster disconnecting the
two components of the boundary of the annulus. Simple CLE in the punctured disc
can be viewed as simple CLE in the unit disc conditioned on the event that the
origin is in the gasket. Simple CLE in the punctured plane can be viewed as
simple CLE in the whole plane conditioned on the event that both the origin and
infinity are in the gasket. We construct and study these three kinds of CLEs,
along with the corresponding exploration processes.
