TY - JOUR
T1 - The scaling limit geometry of near-critical 2D percolation
AU - Camia, Federico
AU - Fontes, Luiz Renato G.
AU - Newman, Charles M.
N1 - Funding Information:
The research of the authors was supported in part by the following sources: for F. C., a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and a Veni grant of the Dutch Science Foundation (NWO); for L. R. G. F., FAPESP project no. 04/07276-2 and CNPq project no. 300576/92-7; for C. M. N., grant DMS-01-04278 of the U.S. NSF. This paper has benefitted from the hospitality shown to various of the authors at a number of venues where the research and writing took place, including the Courant Institute of Mathematical Sciences, the Ninth Brazilian School of Probability at Maresias Beach, Instituto de Matemática e Estatística - USP, and Vrije Universiteit Amsterdam. The authors thank Marco Isopi and Jeff Steif for useful discussions.
PY - 2006/12
Y1 - 2006/12
N2 - We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = pc+λδ1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of "macroscopically pivotal" lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.
AB - We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = pc+λδ1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of "macroscopically pivotal" lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.
KW - Finite size scaling
KW - Minimal spanning tree
KW - Near-critical
KW - Percolation
KW - Scaling limits
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U2 - 10.1007/s10955-005-9014-6
DO - 10.1007/s10955-005-9014-6
M3 - Article
AN - SCOPUS:33845733049
SN - 0022-4715
VL - 125
SP - 1155
EP - 1171
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 5-6
ER -