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 .
Another series of related works is on random walks on random planar maps. This is thanks to the conjectural relation between LQG and random planar maps (note that part of the conjecture has been established in @cite_44 @cite_4 ). Building on ideas from the theory of circle packings @cite_7 , the authors of @cite_49 proved that the uniform infinite planar triangulation and quadrangulation are both almost surely recurrent. In @cite_45 , it was shown that the random walk on the uniform infinite planar quadrangulation is sub-diffusive, where an upper bound of @math on the exponent was given while the conjectured exponent is @math .
@cite_4 @cite_7 @cite_44 @cite_45 @cite_49
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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 . We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time @math , the curvature is asymptotically zero for @math and is strictly positive for @math . We consider an analogue of the Kac random walk on the special orthogonal group SO(N)SO(N), in which at each step a random rotation is performed in a randomly chosen 2-plane of RNRN. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large NN limit. In the special case where the angle of rotation is deterministic, this confirms a conjecture of Rosenthal [Ann. Probab. 22 (1994) 398–423]. Under mild conditions, we also establish a cut-off for convergence of the walk to stationarity under the L2L2 norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and L2L2 norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.