Several fractals that arise as the scaling limit of statistical physics models come equipped with a natural measure. This measure can often be defined equivalently in multiple ways: axiomatically, via Minkowski content, or as the limit of counting measure for the discrete model. We study such measures for the Schramm-Loewner evolution, percolation pivotal points, and various fractals in the geometry of Liouville quantum gravity. Based on joint works with subsets of Bernardi, Lawler, Li, and Sun.
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