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Bertrand Duplantier works on random geometry and statistical physics. With Scott Sheffield, he gave a probabilistic construction of a Liouville quantum gravity area measure and a relation between Euclidean and random-geometry scaling exponents.^[ref]

The construction illustrates why a random field cannot always be manipulated as an ordinary pointwise function. A Gaussian free field first receives a spatial regularization, such as its average around a small circle. Exponentiating this smoothed field requires a scale-dependent normalization before taking a limit. Merely writing $e^{\gamma h}$ without specifying that procedure omits an essential mathematical step.

Their result links how a set scales under ordinary area with how it scales under the random area measure. It turns a prediction from mathematical physics into a statement about precisely defined random objects, rather than identifying the two notions of dimension by analogy alone.^[ref]

By Bertrand Duplantier, Scott Sheffield.

Abstract and construction of the quantum area measure: regularized exponential of circle averages of a Gaussian free field for $0<\gamma <2$. MIT author manuscript record: dspace.mit.edu/entities/publication/7adfc26d-9fb8-4348-8662-cd5ae52cc71f .