Properties of Schramm Loewner Evolution and Supercritical Liouville Quantum Gravity Metric Exponents

Schramm–Loewner Evolution (SLE) is a family of random curves in the plane, indexedby a parameter $\kappa \geq 0.$ These non-crossing curves are the fundamental tool used to describe the scaling limits of a plethora of natural probabilistic processes in two dimensions, such as critical percolation interfaces, loop erased random walks, and (in conjecture) self-avoiding walks. Their introduction by Oded Schramm in 1999 was a milestone of modern probability theory. The first part of this thesis will focus mainly on two key properties of SLE; namely, reversibility and topological invariance. For $\gamma \in (0,2)$, $U \subset \mathbb{C}$, and an instance $h$ of the Gaussian free field (GFF) on $U$, the $\gamma$-Liouville quantum gravity (LQG) surface associated with $(U,h)$ is formally described by the Riemannian metric tensor $e^{\gamma h}(dx^2 + dy^2)$ on $U$. It is known that one can define a canonical metric (distance function) $D_h$ on $U$ associated with a $\gamma$-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for $\gamma$-LQG surfaces. We consider a discrete analog of this metric, and show, in the final chapter of the present work, that it has the same distance exponent as in the continuum case.