Percolation models with long-range correlations via isomorphism theorems

• Applicant: Professor Dr. Alexander Drewitz
• Subject Area: Mathematics
• Term: from 2018 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 410738796

• Final Report Year: 2024

Final Report Abstract

This research project aimed to use isomorphism theorems to explore percolation models with long-range correlations, focusing on the level sets of the Gaussian free field (GFF) on metric graphs and the vacant set of random interlacements. For the vacant set of random interlacements, we successfully demonstrated the positivity of the critical parameter across a broad range of transient graphs, including specific examples of prefractals. This result, combined with prior findings, extends the understanding of the non-triviality of the phase transition in this percolation problem to encompass such graphs. Concerning the metric graph GFF, we derived an explicit formula for the law of the capacity of the level sets. This formula was instrumental in establishing that the percolation phase transition is of second order and enabled the determination of the critical exponent for the percolation function. Subsequent work delved deeper into determining various critical exponents, primarily in transient graphs with a volume dimension of at most three. These findings align with predictions from the physics literature. Notably, these critical exponents are all rational functions of the volume growth exponent and the exponent governing the decay of Green’s function, thus illustrating their anticipated universality.

Publications

• Cluster capacity functionals and isomorphism theorems for Gaussian free fields. Probability Theory and Related Fields, 183(1-2), 255-313.
  Drewitz, Alexander; Prévost, Alexis & Rodriguez, Pierre-François
  
• Critical exponents for a percolation model on transient graphs. Inventiones mathematicae, 232(1), 229-299.
  Drewitz, Alexander; Prévost, Alexis & Rodriguez, Pierre-François
  
• Generating Galton–Watson trees using random walks and percolation for the Gaussian free field. The Annals of Applied Probability, 34(3).
  Drewitz, Alexander; Gallo, Gioele & Prévost, Alexis
  
• Geometry of Gaussian free field sign clusters and random interlacements. Probability Theory and Related Fields, 192(1-2), 625-720.
  Drewitz, Alexander; Prévost, Alexis & Rodriguez, Pierre-François
  
• Percolation for two-dimensional excursion clouds and the discrete Gaussian free field. Electronic Journal of Probability, 29(none).
  Drewitz, A.; Elias, O.; Prévost, A.; Tykesson, J. & Viklund, F.