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Analysis of Off-Critical Percolation Clusters by Schramm-Loewner Evolution

Jamali, Tayyeb | 2011

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 42040 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Rouhani, Shahin
  7. Abstract:
  8. Recently, a new tool in the study of two-dimensional continuous phase transition was provided by Schramm-Loewner evolution. The main part of SLE is a conformal map which relates growth process of a two-dimensional simple curve to one-dimensional motion on the real axis (so-called Loewner driving function). Time evolution of this map and Loewner driving function is connected via the Loewner differential equation. It turns out that for a certain class of stochastic and conformally invariant curves in two dimensions, the driving function shows Brownian motion in one dimension. Strength point of SLE comes from this fact that all the geometrical properties of such curves is described through a single parameter of the Brownian motion, called, the diffusion constant. In the study of physical systems, it is important to find out how the driving functions look like when the system deviates from the critical point because in real life there are a number of sources that may drive a system away from it. The goal of this project is to answer this question about percolation model. For this purpose, the behavior of Loewner driving function, U_t, for cluster boundary of site percolation on the triangular lattice in sub-critical phase p shows a scaling behavior –(p_c-p) t^((ζ+1)/2ζ) with a superdiffusive fluctuation whereas, beyond the crossover time, the driving function U_t undergoes a normal diffusion but with the drift velocity proportional to (p_c-p)^ζ. Simulation results in ζ=1.34±0.04. Since the value of critical exponent of correlation length in percolation model is ν=4/3=1.33 ̅, the exact value of ζ has been conjectured to be ν=4/3. Moreover, it is conjectured from simulation results that as p approaches its critical value p_c=1/2, the crossover time diverges in the form of (p_c-p)^(-2ν).
  9. Keywords:
  10. Percolation ; Schramm-Loewner Evolution (SLE)

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