Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits

@article{Smirnov2001CriticalPI,
  title={Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits},
  author={Stanislav Smirnov},
  journal={Comptes Rendus De L Academie Des Sciences Serie I-mathematique},
  year={2001},
  volume={333},
  pages={239-244}
}
  • S. Smirnov
  • Published 1 August 2001
  • Mathematics
  • Comptes Rendus De L Academie Des Sciences Serie I-mathematique
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References

SHOWING 1-10 OF 12 REFERENCES
Values of Brownian intersection exponents, I: Half-plane exponents
Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). For
Critical Exponents, Conformal Invariance and Planar Brownian Motion
In this review paper, we first discuss some open problems related to two-dimensional self-avoiding paths and critical percolation. We then review some closely related results (joint work with Greg
Holder Regularity and Dimension Bounds for Random Curves
Random systems of curves exhibiting fluctuating features on arbitrarily small scales (δ) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the
The dimension of the planar Brownian frontier is 4/3
In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is
Conformal invariance in two-dimensional percolation
The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the
143. Two-dimensional Brownian Motion and Harmonic Functions
Two . dimensional Brownian Motion and Harmonic Functions
  • Mathematics
1. The purpose of this paper is to investigate the properties of two-dimensional Brownian motions’ and to apply the results thus obtained to the theory of harmonic functions in the Gaussian plane.
Path Crossing Exponents and the External Perimeter in 2D Percolation
2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of
Critical percolation in finite geometries
The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These
The geometry of critical percolation and conformal invariance, in: STATPHYS
  • (Xiamen,
  • 1995
...
...