In this paper with Mo Dick Wong, we initiated a study of the spectral geometry of Liouville quantum gravity, i.e., the study of eigenvalues and eigenfunctions associated to Liouville Brownian motion in a bounded domain \(D\). We were able to prove a Weyl law for the eigenvalues (with spectral dimension two), in the sense that if \(N(\lambda)\) is the eigenfunction counting function then \(N(\lambda)/\lambda \to c_\gamma \mu_\gamma(D)\) as \(\lambda \to \infty\), where \(\mu_\gamma(D)\) denotes the total LQG area measure of \(D\). Surprisingly, the constant \(c_\gamma\) is strictly greater than its Riemannian analogue, a fact for which we don’t have a good explanation beyond a calculation.

This investigation led us to a number of very intriguing conjectures. For instance, we conjecture that the eigenfunctions are delocalised and in fact satisfy quantum (unique) ergodicity: i.e., their \(L^2\) mass is approximately distributed according to the LQG area measure. This is the analogue of celebrated conjectures of Rudnick-Sarnak for hyperbolic surfaces. The first two pictures show a plot of a high-energy eigenfunction, together with a heat map of its values.

Regarding the eigenvalues we conjecture (in agreement with the above and quantum chaos) Gaussian Orthogonal Ensemble spacing statistics, shown in red in the third picture.

Multiplicative chaos and Brownian loop soup

Brownian chaos associated to a single trajectory of Brownian motion with parameter \(\)γ = 1\(\) (credit picture: Antoine Jégo), as constructed by Jégo in this paper. Informally this is the exponential of the square root of the local time of Brownian motion. This construction can be extended to the Brownian loop soup (credit picture: Wendelin Werner and Serban Nacu), as shown in this paper with Elie Aïdékon, Antoine Jégo and Titus Lupu. At the critical intensity \(\)θ = 1/2\(\) for the loop soup, the associated multiplicative chaos describes the hyperbolic cosine of the Gaussian free field, an identity which follows informally from Le Jan’s isomorphism.

Dimers and Imaginary Geometry

Lozenge tiling (or equivalently dimer model of the hexagonal lattice) with tilted boundary conditions (picture credit: Rick Kenyon). With Benoit Laslier and Gourab Ray, we proved in this paper that the height function converges to a multiple of the Gaussian free field. This used a connection to Imaginary geometry with κ = 2 (as illustrated on the second picture, credit Jason Miller). Third picture: (part of) the Temperleyan cycle-rooted spanning forest of a hyperbolic Riemann surface, represented on its universal cover (i.e., the unit disc) modulo the associated Fuchsian group. Using these ideas we proved in a sequence of papers convergence of the height function of the dimer model to a conformally invariant scaling limit.

Liouville Brownian motion

Random planar map (embedded by Mathematica in 3-space, simulation by Nicolas Curien), Gaussian free field superimposed with a piece of the trace of Liouville Brownian motion (simulation by Henry Jackson), circle packing of a random triangulation (simulation by Jason Miller). The full video of Liouville Brownian motion in the GFF landscape is available here.

Condensation of random walks and Wulff crystals

Simulations of a random walk in two dimensions which has been penalised by a factor exp(−β|∂Rt|), where ∂Rt is the boundary of the range, and β is a positive factor, to be thought of as an inverse temperature. As increases, the distribution favours increasingly condensed configurations. In this paper, it is proved that the diameter of the shape in two dimensions is (t/β)1/3.

Brunet-Derrida particle systems

A model for natural selection in a population dynamics: the Brunet-Derrida particle system; the closely related model of branching Brownian motion with killing wall moving at the critical speed √2; and the Bolthausen-Sznitman coalescent which describes the scaling limit of the genealogy of particles, as predicted by Brunet, Derrida and coauthors and proved rigourously here.

Multidimensional Brunet-Derrida particle systems

Multidimensional Brunet-Derrdia system, introduced in this paper with Lee Zhuo Zhao. The fittest particles survive at each generation or branching event, where fitness is measured according to a given score function. The video on the left corresponds to the score being the Euclidean norm. We showed that the cloud of particles eventually travels in a random direction at positive speed. The second picture shows a zoom of this cloud of particles, showing the particles spread out in a direction orthogonal to the limiting speed. We were able to establish this feature in the case of a linear score function, with a nontrivial scaling.

The last picture is a still frame from the video below of the Brownian bees process, corresponding to the case where particles are fit if they are close to the centre of mass of the system (whose trajectory you can follow on the video). The yellow circle on the picture is the circle of radius j0,1/√2, where jα,n is the nth zero of the Bessel function of the first kind Jα(x). I conjectured at an Oberwolfach meeting in 2013 that particles are confined within this disc in the large N limit. This conjecture was proved in 2020 by J. Berestycki, E. Brunet, J. Nolen, and S. Penington in this paper and this paper.