Given any @math and for @math denoting a sample of the two-dimensional discrete Gaussian free field on @math pinned at the origin, we consider the random walk on @math among random conductances where the conductance of edge @math is given by @math . We show that, for almost every @math , this random walk is recurrent and that, with probability tending to 1 as @math , the return probability at time @math decays as @math . In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius @math scales as @math with @math for all @math . Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance @math behaves as @math .
 Suppose that @math is a sequence of finite connected planar graphs, and in each @math a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit @math of such graphs. Assume that the vertex degrees of the vertices in @math are bounded, and the bound does not depend on @math . Then after passing to a subsequence, the limit exists, and is a random rooted graph @math . We prove that with probability one @math is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations. We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain D of the complex plane for which the Riemannian metric tensor at a point z ∈ D is given by exp(h(z)), appropriately normalised. Here h is an instance of the Gaussian Free Field on D and ∈ (0,2) is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian Free Field, and show that it spends Lebesgue-almost all its time in the set of -thick points, almost surely. Similar but deeper results have been independently and simultaneously proved by Garban, Rhodes and Vargas. We study the pioneer points of the simple random walk on the uniform infinite planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of Angel (Geom Funct Anal 13:935–974, 2003) to the quadrangulation case. Our main result is that, up to polylogarithmic factors, n 3 pioneer points have been discovered before the walk exits the ball of radius n in the UIPQ. As a result we verify the KPZ relation (Modern Phys Lett A 3:819–826, 1988) in the particular case of the pioneer exponent and prove that the walk is subdiffusive with exponent less than 1 3. Along the way, new geometric controls on the UIPQ are established. We prove that any distributional limit of nite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform innite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm. We also settle another related problem of Benjamini and Schramm. We show that in any bounded degree, nite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C logT .