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]
Liouville Quantum Gravity and KPZ by Bertrand Duplantier and Scott Sheffield. abstract, regularized area measure and KPZ relation
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 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]
Liouville Quantum Gravity and KPZ by Bertrand Duplantier and Scott Sheffield. abstract and quantum scaling formulation
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