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Continuum Scaling Limit of Critical Percolation

Ghodratipour, Nahid | 2011

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 43909 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Alishahi, Kasra; Rouhani, Shahin
  7. Abstract:
  8. Percolation is a simple probabilistic model which exhibits a phase transition. Here, we study this critical model from properties of random curves which in the scaling limit, appear as features seen on the macroscopic scale, in situations where the microscopic scale is taken to zero. Among the principal questions are the construction of the scaling limit, and the discription of some of the emergent properties, in particular the behavior under conformal maps Over the past few years, SLE has been developed as a valuable new tool to study the random paths of the scaling limit of two-dimensional critical models, and it is believed that SLE is the conformally invariant scaling limit of these models. This belief allowed physicists to predict (unrigorously) and calculate many of the properties of these critical models, in particular exact values of critical exponents. Part of the importance of these properties is due to the prediction that large-scale features near the critical point are universal. The goal of this thesis is to study the geometric aspects of conformally invariant random paths, obtained in the scaling limit of critical site percolation on triangular lattice, which is the only standard 2D critical percolation model that convergence of said limit is known. We explain first the proof of Smirnov’s theorem in convergence and conformally invariance of crossing probabilities, and then relying on this theorem, the existence, uniqueness and conformal invariance of the continuum scaling limit. In particular, we explain that the critical site percolation exploration path on triangular lattice converges to the trace of chordal SLE6
  9. Keywords:
  10. Phase Transition ; Critical Phenomena ; Critical Exponent ; Schramm-Loewner Evolution (SLE) ; Percolation Theory ; Conformal Invariance ; Universality Class

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