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We extend the peeling exploration introduced in arxiv:1506.01590 to thesetting of Boltzmann planar maps coupled to a rigid $O(n)$ loop model. Its lawis related to a class of discrete Markov processes obtained by confining randomwalks to the positive integers with a new type of boundary condition. As anapplication we give a rigorous justification of the phase diagram of the modelpresented in arXiv:1106.0153. This entails two results pertaining to theso-called fixed-point equation: the first asserts that any solution determinesa well-defined model, while the second result, contributed by Chen in theappendix, establishes precise existence criteria.  A scaling limit for the exploration process is identified in terms of a newclass of positive self-similar Markov processes, going under the name ofricocheted stable processes. As an application we study distances onloop-decorated maps arising from a particular first passage percolation processon the maps. In the scaling limit these distances between the boundary and amarked point are related to exponential integrals of certain L\'evy processes.The distributions of the latter can be identified in a fairly explicit formusing machinery of positive self-similar Markov processes.  Finally we observe a relation between the number of loops that surround amarked vertex in a Boltzmann loop-decorated map and the winding angle of asimple random walk on the square lattice. As a corollary we give acombinatorial proof of the fact that the total winding angle around the originof a simple random walk started at $(p,p)$ and killed upon hitting $(0,0)$normalized by $\log p$ converges in distribution to a Cauchy random variable.

The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)