Random Geometry and Quantum Spacetime: From scale-invariant random geometries and asymptotic safety to random hyperbolic surfaces and JT gravity

This thesis is driven by a central question: "What can we learn from random geometries about the structure of quantum spacetime?" In Chapter 2, we provide a partial review of the mathematical foundation of this thesis, random geometry. In Chapter 3, we use a construction coming from random geometry called Mating of Trees to build scale-invariant random geometries that appear in Liouville Quantum Gravity and have the potential to implement the UV fixed point predicted by Asymptotic Safety in two and three dimensions. In Chapter 4 we explore the random geometry formulation of JT gravity and how our understanding of random critical maps yields the discovery of a new family of deformations of JT gravity. Furthermore, the connection between JT gravity and matrix models leads us to delve deeper into the link between discrete geometry and hyperbolic surfaces, building upon the geometry of metric maps and irreducible metric maps in Chapter 5.