  A meandric system of size $n$ is the set of loops formed from two arc
diagrams (non-crossing perfect matchings) on $\{1,\dots,2n\}$, one drawn above
the real line and the other below the real line. A uniform random meandric
system can be viewed as a random planar map decorated by a Hamiltonian path
(corresponding to the real line) and a collection of loops (formed by the
arcs). Based on physics heuristics and numerical evidence, we conjecture that
the scaling limit of this decorated random planar map is given by an
independent triple consisting of a Liouville quantum gravity (LQG) sphere with
parameter $\gamma=\sqrt 2$, a Schramm-Loewner evolution (SLE) curve with
parameter $\kappa=8$, and a conformal loop ensemble (CLE) with parameter
$\kappa=6$.
  We prove several rigorous results which are consistent with this conjecture.
In particular, a uniform meandric system admits loops of nearly macroscopic
graph-distance diameter with high probability. Furthermore, a.s., the uniform
infinite meandric system with boundary has no infinite path. But, a.s., its
boundary-modified version has a unique infinite path whose scaling limit is
conjectured to be chordal SLE$_6$.
