Liouville Quantum Duality and Random Planar Maps

We consider models of block-weighted random planar maps in which possibly decorated maps are decomposed canonically into blocks, each block receiving the weight $u$. These maps present a transition at some critical value $u=u_{cr}$ above which the maps degenerate into Brownian trees. We show that the enumerative properties and critical exponents of the maps at $u=u_{cr}$ and those for $u<u_{cr}$ are connected by duality relations which are precisely those expected in the context of the Liouville quantum gravity description of random surfaces. We illustrate this result by various instances of block-weighted maps: random planar quadrangulations decomposed into simple blocks, Hamiltonian cycles on cubic or bicubic planar maps decomposed into irreducible blocks, and meandric systems.