Some metric properties of planar Gaussian free field

In this thesis we study the properties of some metrics arising from two-dimensional Gaussian free field (GFF), namely the Liouville first-passage percolation (Liouville FPP), the Liouville graph distance and ,an effective resistance metric. ,In Chapter 1, we define these metrics as well as discuss the motivations for studying them. ,Roughly speaking, Liouville FPP is the shortest path metric in a planar domain $D$ where the length of a path $P$ is given by $\int_P \e^{\gamma h(z)}|dz|$ where $h$ ,is the GFF on $D$ and $\gamma > 0$. In Chapter 2, we present an upper bound on the expected Liouville FPP distance between two typical ,points for small values of $\gamma$ (the ,\emph{near-Euclidean} regime). A similar upper bound is derived in Chapter 3 for the Liouville graph distance which is, roughly, the minimal number of Euclidean balls with comparable Liouville quantum gravity (LQG) measure whose union contains a continuous ,path between two endpoints. Our bounds seem to be in disagreement with Watabiki's prediction (1993) on the random metric of Liouville quantum gravity in this ,regime. The contents of these two chapters are based on a joint work with Jian Ding. ,In Chapter 4, we derive some asymptotic estimates for effective resistances on a ,random network which is defined as follows. Given any $\gamma>0$ and for $\eta=\{\eta_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we equip the edge $(u, v)$ with conductance ,$\e^{\gamma(\eta_u + \eta_v)}$. The metric structure of effective resistance plays a crucial role in our proof of the main result in Chapter 4. ,The primary motivation behind this metric is to understand the random walk on $\Z^2$ where the edge $(u, ,v)$ has weight $\e^{\gamma(\eta_u + \eta_v)}$. Using the estimates from Chapter 4 we show in Chapter 5 that for almost every $\eta$, this random walk is recurrent and that, with probability tending to 1 as $T\to \infty$, the return probability at time $2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius $N$ scales as $N^{\psi(\gamma)+o(1)}$ with $\psi(\gamma)>2$ for ,all $\gamma>0$. The contents of these chapters are based on a joint work with Marek Biskup and Jian Ding .