We study Liouville quantum gravity (LQG) surfaces whose law has been reweighted
according to nesting statistics for a conformal loop ensemble (CLE) relative to
marked points
. The idea is to consider
a reweighting by
,
where
and
is the number of CLE loops surrounding the points
for
.
This is made precise via an approximation procedure where as part of
the proof we derive strong spatial independence results for CLE. The
reweighting induces logarithmic singularities for the Liouville field at
with a magnitude
depending explicitly on
.
We define the partition function of the surface, compute it for
, and derive a recursive
formula expressing the
point partition function in terms of lower-order partition functions.
The proof of the latter result is based on a continuum peeling process
previously studied by Miller, Sheffield and Werner in the case
,
and we derive an explicit formula for the generator of a boundary
length process that can be associated with the exploration for general
.
We use the recursive formula to partly characterize for which values of
the
partition function is finite. Finally, we give a new proof for the law of the conformal
radius of CLE, which was originally established by Schramm, Sheffield, and
Wilson.