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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]

Jason Miller (mathematician) studies random planar geometry, including Schramm–Loewner evolution, the Gaussian free field, Liouville quantum gravity and random planar maps. His Cambridge research profile verifies these interests.^[ref]

His Random Planar Geometry by Jason Miller (2020) develops discrete-to-continuum connections through trees, maps and random curves. The trivial bijection between planar maps and quadrangulations is a concrete example: vertices of one type encode original map vertices and the other type encodes faces. Recovering the map from this bipartition explains why the correspondence is reversible.^[ref]

Liouville quantum gravity gives a random geometry in two dimensions whose area is formally weighted by $e^{\gamma h}$, where $h$ is a Gaussian free field. Since the field is a random distribution rather than a pointwise function, this exponential needs regularization. For $0<\gamma <2$, a standard circle-average construction on a planar domain iswith the normalization of the field fixed as in the source. Here $h_{\epsilon }$ is the average around a circle of radius $\epsilon$; the limit is a random measure, not a pointwise density.^[ref]

For example, adding a constant $c$ to the field multiplies the area measure by $e^{\gamma c}$ without changing its coordinate domain. This separates random area from ordinary Euclidean area. The construction relates scaling exponents of random sets measured in these two geometries; it does not constitute a model of four-dimensional spacetime. Constructing an associated random distance requires more than defining the area measure alone.

Nathanaël Berestycki works in probability theory, connecting stochastic motion with random geometry. His University of Vienna homepage identifies research on Brownian motion, the Gaussian free field, Liouville quantum gravity and interacting particle systems.^[ref]

With Christophe Garban and Arnab Sen, he introduced a topology for the coalescing Brownian flow and obtained convergence from coalescing random walks under finite variance of the increments. The state-space choice makes the continuum limit accessible without stronger moment assumptions previously needed in other descriptions.^[ref]

A useful distinction is between convergence of finitely many trajectories and convergence of a whole random flow. Starting particles at many locations creates a shared pattern of meetings and mergers. A continuum description has to preserve the consistency of those histories, rather than merely fit the distribution of one endpoint.