Combinatorics

Percolation theory

Physics

Neuroscience

Continuum percolation theory

Topology (electrical circuits)

Critical dimension

Excursion

Mathematics

Biology

Statistical physics

Gaussian

Critical exponent

Political science

Percolation critical exponents

Directed percolation

Acoustics

Geometry

Condensed matter physics

Percolation threshold

Lattice (music)

Gaussian free field

Percolation (cognitive psychology)

Quantum mechanics

Electrical resistivity and conductivity

Abelian sandpile model

Scaling

Inventiones mathematicae

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.

Critical exponents for a percolation model on transient graphs