Let @math be a discrete Gaussian free field in a two-dimensional box @math of side length @math with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where each vertex is given a weight of @math for some @math . We show that for sufficiently small but fixed @math , with probability tending to @math as @math , all geodesics between vertices of macroscopic Euclidean distances simultaneously have (the conjecturally unique) length exponent strictly larger than 1.
In @cite_4 the chemical distance (i.e., graph distance in the induced open cluster) for the percolation of level sets of two-dimensional DGFF was studied, and in particular [Theorem 1.1] DL16 implies that there exists a path of dimension 1 joining the two boundaries of an annulus which has Liouville FPP weight @math . This can be seen as a complement of our Theorem . In @cite_9 , a non-universality result was proved on the weight exponent for the geodesic among log-correlated Gaussian fields, which suggests the subtlety when attempting to estimate the weight exponent. We would like to point out that the proof of our main result Theorem is expected to be adaptable to general log-correlated Gaussian fields with @math -scale invariant kernels as in @cite_1 . However, in light of @cite_9 there seems to be no reason to expect that the geodesic dimension is universal among log-correlated fields.
@cite_1 @cite_9 @cite_4
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We consider first passage percolation (FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with Liouville quantum gravity (LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF). The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. More precisely, we construct a family of log-correlated Gaussian fields, and show that the FPP distance between two typically sampled vertices (according to the LQG measure) is (N^ 1+ O( ) ), where N is the side length of the box and ( ) can be made arbitrarily small if we tune a certain parameter in our construction. That is, the exponents can be arbitrarily close to 1. Combined with a recent work of the first author and Goswami on an upper bound for this exponent when the underlying field is a GFF, our result implies that such exponent is not universal among the family of log-correlated Gaussian fields. In this paper, we provide general conditions on a one parameter family of random infinite subsets of Zd to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of Cerný and Popov [“On the internal distance in the interlacement set,” Electron. J. Probab. 17(29), 1–25 (2012)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of Antal and Pisztora [“On the chemical distance for supercritical Bernoulli percolation... In this paper, we present a new framework for geo-locating an image utilizing a novel multiple nearest neighbor feature matching method using Generalized Minimum Clique Graphs (GMCP). First, we extract local features (e.g., SIFT) from the query image and retrieve a number of nearest neighbors for each query feature from the reference data set. Next, we apply our GMCP-based feature matching to select a single nearest neighbor for each query feature such that all matches are globally consistent. Our approach to feature matching is based on the proposition that the first nearest neighbors are not necessarily the best choices for finding correspondences in image matching. Therefore, the proposed method considers multiple reference nearest neighbors as potential matches and selects the correct ones by enforcing consistency among their global features (e.g., GIST) using GMCP. In this context, we argue that using a robust distance function for finding the similarity between the global features is essential for the cases where the query matches multiple reference images with dissimilar global features. Towards this end, we propose a robust distance function based on the Gaussian Radial Basis Function (G-RBF). We evaluated the proposed framework on a new data set of 102k street view images; the experiments show it outperforms the state of the art by 10 percent. We consider level-set percolation for the Gaussian free field on ( Z ^ d ), d ≥ 3, and prove that, as h varies, there is a non-trivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h*(d) satisfies h*(d) ≥ 0 for all d ≥ 3 and that h*(3) is finite, see (J Stat Phys 48(5 6):1249–1268, 1987). We prove here that h*(d) is finite for all d ≥ 3. In fact, we introduce a second critical parameter h** ≥ h*, show that h**(d) is finite for all d ≥ 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h > h**. Finally, we prove that h* is strictly positive in high dimension. It remains open whether h* and h** actually coincide and whether h* > 0 for all d ≥ 3.