On Gaussian multiplicative chaos and random geometry

This thesis explores three interconnected topics: Gaussian multiplicative chaos (GMC), random planar maps, and random geometric objects in dimensions three and higher. The theory of GMC constructs random measures by formally exponentiating a log-correlated Gaussian field scaled by a parameter $\gamma \geq 0$. These measures undergo a phase transition at $\gamma_c = \sqrt{2d}$, where $d \geq 1$ is the ambient dimension. The regime $\gamma < \gamma_c$ is subcritical, $\gamma = \gamma_c$ is critical, and $\gamma > \gamma_c$ is supercritical. First, we investigate supercritical GMC measures. Specifically, we analyse the local structure of $\star$-scale invariant fields around their extremal points and gain a deeper understanding of the freezing phenomenon in supercritical GMC. We also establish the universality of supercritical GMC: for general convolution approximations of log-correlated Gaussian fields, the supercritical chaos measures converge to a nontrivial limit, which depends on the regularisation only through a multiplicative constant. Additionally, we rigorously prove that subcritical GMC satisfies the multifractal formalism. Next, we study random planar maps and their connection to Liouville quantum gravity (LQG), a canonical model of fractal surfaces with random geometry. We focus on the Smith embedding of planar maps, defined via discrete harmonic functions on the set of primal and dual vertices. We show that if a sequence of embedded planar maps satisfies an invariance principle assumption, then the a priori embedding is asymptotically close to an affine transformation of the Smith embedding. As a consequence, we establish the convergence of the Smith embedding of mated-CRT maps to LQG. Finally, we extend to higher dimensions certain random geometric objects originally introduced in the two-dimensional setting. Specifically, we construct the higher-dimensional analogue of the Liouville Brownian motion and analyse its spectral dimension. Furthermore, in arbitrary even dimensions, we construct the higher-dimensional analogue of the quantum cone.