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Application of Conformal Field Theory in Abelian Sandpile Model

Azimi Tafreshi, Nahid | 2010

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 40659 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Moghimi-Araghi, Saman; Rouhani, Shahin
  7. Abstract:
  8. The theory of self-organized criticality is originally introduced by Bak, Tang and Wiesenfeld as a general mechanism that can explain the behaviour of complex systems which naturally organize themselves into a critical state. They defined the sandpile model as an example of slowly driven and dissipative complex system to explain the concept of self-organized criticality. From the definition of the model, extensive work has been done on this model. Thanks to the Abelian property of the model, many statistical results have been derived exactly. Other properties of the model such as critical exponents and dynamical behaviors have been also studied using the mapping with some statistical models and by numerical simulations. The sandpile model has been investigated in the continuum limit and from a field theoretical perspective. It has been shown that the Abelian sandpile model can be described by a specific logarithmic conformal field theory, with central charge c = −2. In this thesis, we study critical properties of some variations of the model. At first, we define a new version with continuous variables and we show that the model can be mapped to the original model, so the general properties of the two models are identical. The effect of very small mass on the height probabilities, different boundary conditions and corresponding operators with changing of boundary condition are some problems investigated in this model. We also insert some anisotropies in toppling rules and with concepts of statistical mechanics and also from the conformal field theory point of view, we show some anisotropies change the critical behavior of the model. Then, we check the universality properties of the original model by computing some of its properties on the honeycomb lattice. Exact expressions for unit height correlation functions in presence of boundaries and for different boundary conditions are derived. Also, we study the statistics of the boundaries of avalanche waves by using the theory of SLE and suggest that these curves are conformally invariant and described by SLE2.
  9. Keywords:
  10. Conformal Fields Theory ; Self Organized Criticality ; Abelian Sandpile Model

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