We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall.
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... 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. We investigate random interlacements on ℤd, d ≥ 3. This model, recently introduced in [8], corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times. A nonnegative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d ≥ 3 the vacant set at level u percolates when u is small. This solves an open problem of [8], where this fact has only been established when d ≥ 7. It also completes the proof of the nondegeneracy in all dimensions d ≥ 3 of the critical parameter u* of [8]. © 2008 Wiley Periodicals, Inc. We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters ( ) and ( ). The non-negative parameter ( ) measures the amount of loops, and ( ) plays the role of killing on vertices penalizing ( ( >0 )) or favoring ( ( <0 )) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed ( ) and large enough ( ), there exists an infinite cluster in the loop percolation on ( Z ^d ). In the present article, we show a non-trivial phase transition on the integer lattice ( Z ^d ) ( (d 3 )) for ( =0 ). More precisely, we show that there is no loop percolation for ( =0 ) and ( ) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of ( ), namely, for ( =0 ) and any sub-critical value of ( ), the probability of one-arm event decays at most polynomially. For (d 5 ), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For ( ) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For (d=3 ) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity ( ).  We introduce a model of random interlacements made of a countable collection of doubly infinite paths on ℤ d ,d > 3. A nonnegative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder (ℤ Nℤ) d-1 x ℤ by simple random walk, or the set of points visited by simple random walk on the discrete torus (ℤ Nℤ) d at times of order u N d . In particular we study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite connected translation invariant random subset of ℤ d . We introduce a critical value u * such that the vacant set percolates for u u * . Our main results show that u * is finite when d > 3 and strictly positive when d > 7. In this paper, we are interested in the loop cluster model on @math for @math . It is a long range model with two parameters @math and @math , where the non-negative parameter @math measures the amount of loops, and @math plays the role of killing on vertices penalizing ( @math ) or favoring ( @math large balls in the infinite cluster are finally very regular in the sense of Sapozhnikov2014 , which implies that large balls are finally very good in the sense of BarlowMR2094438 . By BarlowMR2094438 and BarlowHamblyMR2471657 , we have Harnack's inequality and Gaussian type estimate for simple random walks on the infinite cluster for all @math .