PIVOTAL, CLUSTER, AND INTERFACE MEASURES FOR CRITICAL PLANAR PERCOLATION

被引:52
作者
Garban, Christophe [1 ,2 ]
Pete, Gabor [3 ]
Schramm, Oded
机构
[1] Ecole Normale Super Lyon, CNRS, F-69364 Lyon, France
[2] UMPA, F-69364 Lyon, France
[3] Tech Univ Budapest, Inst Math, H-1111 Budapest, Hungary
基金
加拿大自然科学与工程研究理事会;
关键词
NEAR-CRITICAL PERCOLATION; SCALING LIMITS; INTERSECTION EXPONENTS; INFINITE CLUSTER; SLE; PROOF; TREES;
D O I
10.1090/S0894-0347-2013-00772-9
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
This work is the first in a series of papers devoted to the construction and study of scaling limits of dynamical and near-critical planar percolation and related objects like invasion percolation and the Minimal Spanning Tree. We show here that the counting measure on the set of pivotal points of critical site percolation on the triangular grid, normalized appropriately, has a scaling limit, which is a function of the scaling limit of the percolation configuration. We also show that this limit measure is conformally covariant, with exponent 3/4. Similar results hold for the counting measure on macroscopic open clusters (the area measure) and for the counting measure on interfaces (length measure). Since the aforementioned processes are very much governed by pivotal sites, the construction and properties of the "local time"-like pivotal measure are key results in this project. Another application is that the existence of the limit length measure on the interface is a key step towards constructing the so-called natural time-parametrization of the SLE6 curve. The proofs make extensive use of coupling arguments, based on the separation of interfaces phenomenon. This is a very useful tool in planar statistical physics, on which we included a self-contained Appendix. Simple corollaries of our methods include ratio limit theorems for arm probabilities and the rotational invariance of the two-point function. © 2013 American Mathematical Society.
引用
收藏
页码:939 / 1024
页数:86
相关论文
共 54 条
[41]  
Pommerenke C., 1975, UNIVALENT FUNCTIONS
[42]   Proof of the Van den Berg-Kesten conjecture [J].
Reimer, D .
COMBINATORICS PROBABILITY & COMPUTING, 2000, 9 (01) :27-32
[43]   Scaling limits of loop-erased random walks and uniform spanning trees [J].
Schramm, O .
ISRAEL JOURNAL OF MATHEMATICS, 2000, 118 (1) :221-288
[44]   ON THE SCALING LIMITS OF PLANAR PERCOLATION [J].
Schramm, Oded ;
Smirnov, Stanislav ;
Garban, Christophe .
ANNALS OF PROBABILITY, 2011, 39 (05) :1768-1814
[45]   Quantitative noise sensitivity and exceptional times for percolation [J].
Schramm, Oded ;
Steif, Jeffrey E. .
ANNALS OF MATHEMATICS, 2010, 171 (02) :619-672
[46]   Contour lines of the two-dimensional discrete Gaussian free field [J].
Schramm, Oded ;
Sheffield, Scott .
ACTA MATHEMATICA, 2009, 202 (01) :21-137
[47]   SCHRAMM'S PROOF OF WATTS' FORMULA [J].
Sheffield, Scott ;
Wilson, David B. .
ANNALS OF PROBABILITY, 2011, 39 (05) :1844-1863
[48]   EXPLORATION TREES AND CONFORMAL LOOP ENSEMBLES [J].
Sheffield, Scott .
DUKE MATHEMATICAL JOURNAL, 2009, 147 (01) :79-129
[49]   Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits [J].
Smirnov, S .
COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, 2001, 333 (03) :239-244
[50]  
Smirnov S, 2001, MATH RES LETT, V8, P729