= Bertrand Duplantier
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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.<Liouville Quantum Gravity and KPZ by Bertrand Duplantier and Scott Sheffield>[abstract, regularized area measure and KPZ relation]{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.<Liouville Quantum Gravity and KPZ by Bertrand Duplantier and Scott Sheffield>[abstract and quantum scaling formulation]{ref}
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