Critical exponents for a percolation model on transient graphs

Abstract

We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.

arXiv comment46 pages, 1 figure

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Cited by in corpus (11)

• Equality of critical parameters for percolation of Gaussian free field level-sets
• On the radius of Gaussian free field excursion clusters
• Cluster capacity functionals and isomorphism theorems for Gaussian free fields
• On the cost of the bubble set for random interlacements
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• Percolation for the Gaussian free field on the cable system: counterexamples
• Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$
• First passage percolation, local uniqueness for interlacements and capacity of random walk
• Crossing exponent in the Brownian loop soup
• A Ray-Knight theorem for $\nabla \varphi$ interface models and scaling limits
• An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields