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Does the critical 2-LQG metric arise as a limit of natural discrete random planar map models in the same way that subcritical LQG metrics do?

Related: KPZ relation for critical LQG, Brownian map universality theorem, Sheffield peanosphere construction

Liouville Quantum Gravity at the critical parameter value of gamma equal to 2 sits at a phase boundary in the theory of random surfaces and random planar maps. For subcritical values of gamma strictly less than 2, there is a well-developed correspondence between LQG surfaces and scaling limits of decorated random planar maps, where the metric structure of the discrete maps converges to the LQG metric. The open problem is whether an analogous discrete-to-continuum convergence theorem holds at the critical value, meaning whether there exists a natural family of decorated random planar maps whose metric scaling limits converge to the critical 2-LQG metric constructed as the limit of subcritical metrics.

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