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Does the directed distance exponent in spanning-tree-decorated planar maps remain universal across all critical weighted planar map families in the same universality class?

Related: KPZ universality conjecture for random planar maps, Brownian map universality theorem of Le Gall and Miermont, mating-of-trees theorem of Duplantier-Miller-Sheffield

The paper on spanning-tree-decorated planar maps establishes an exact exponent and scaling limit for directed distances in a particular model, and claims universality within a certain class. However, a genuinely open problem is whether this universality extends to all critical Boltzmann planar maps decorated by a wired or free uniform spanning tree, especially those converging to Liouville quantum gravity surfaces with differing matter central charges. The difficulty is that universality in random planar maps is notoriously subtle: while the Brownian map captures many geometric universality classes, the interaction between the spanning tree structure and the internal metric of the map creates correlations that resist standard coupling or absolute continuity arguments. Existing techniques rely heavily on bijections specific to certain map families, and extending these to broader classes requires new combinatorial machinery that does not yet exist. Resolving this would settle a fundamental question about the robustness of KPZ-type exponents for correlated geometric observables on random surfaces, would clarify the scope of mating-of-trees theory as a universal framework, and would provide tools for computing critical exponents in other decorated random planar map models such as those carrying the Ising model or percolation clusters.

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