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.
Chemical distances for percolation models is a substantially more challenging problem than the question on connectivities. For instance, it is a major challenge to compute the exponent on the chemical distance between (say) the left and right boundaries for the critical planar percolation, conditioned on the existence of an open crossing. It was proved in @cite_4 that the dimension is strictly larger than 1, and it was shown in recent works @cite_13 @cite_30 that the chemical distance is substantially smaller than the length of the lowest open crossing --- indeed it was shown in @cite_30 that the exponent for the chemical distance is strictly less than that of the lowest open crossing.
@cite_13 @cite_4 @cite_30
1515519840 2050316506
We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path. Random systems of curves exhibiting fluctuating features on arbitrarily small scales (δ) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply that typically all the realized curves admit Holder continuous parametrizations with a common exponent and a common random prefactor, which in the scaling limit (δ → 0) remains stochastically bounded. The regularity is used for the construction of scaling limits, formulated in terms of probability measures on the space of closed sets of curves. Under the hypotheses presented here the limiting measures are supported on sets of curves which are Holder continuous but not rectifiable, and have Hausdorff dimensions strictly greater than one. The hypotheses are known to be satisfied in certain two dimensional percolation models. Other potential applications are also mentioned.