TY - JOUR

T1 - Trivial, critical and near-critical scaling limits of two-dimensional percolation

AU - Camia, Federico

AU - Joosten, Matthijs

AU - Meester, Ronald

N1 - Funding Information:
Research of F. Camia supported in part by a Veni grant of the NWO (Dutch Organization for Scientific Research). Research of R. Meester supported in part by a Vici grant of the NWO (Dutch Organization for Scientific Research).

PY - 2009/10

Y1 - 2009/10

N2 - It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, pc, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to pc. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to pc. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.

AB - It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, pc, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to pc. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to pc. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.

KW - Continuum scaling limit

KW - Massive scaling

KW - Near-critical regime

KW - Percolation

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U2 - 10.1007/s10955-009-9841-y

DO - 10.1007/s10955-009-9841-y

M3 - Article

AN - SCOPUS:70350622056

SN - 0022-4715

VL - 137

SP - 57

EP - 69

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 1

ER -