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Logarithmic Divergences of the Percolation Theory

Bonyadi, Behroz | 2025

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 58255 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Faraji Astaneh, Amin; Rouhani, Shahin
  7. Abstract:
  8. Investigating the 2−dimensional critical percolation still faces some challenges. According to the general belief the LCFT with central charge c = 0 should work, but in practice it isn’t clear where the logarithmic divergence appears in critical percolation. In the recent years numerical and Conformal bootstrap methods have led to a significant progress in the general theory of LCFTs. Yet finding explicit examples where the correlation functions exhibit logarithmic divergences and the explanation of such divergences in terms of the observables of the lattice has remained a challenge. Another difficulty in the percolation theory is that first the Q−state Potts model is solved on a lattice and then the Q → 1 limit is taken, which could be a delicate task and the fields defined in this approach aren’t well defined in the Q → 1 limit either. An alternative approach is to note that in the continuum limit, the behavior of CFT correlation functions resembles that of the cluster connectivity probabilities such that it is assumed that there is a field of scaling dimension equal to 5/96 that measures the cluster density at each point. This method allows the calculation of 4-point correlation functions both in the bulk and situations where a boundary exists and determines the logarithmic divergences. Using the OPE of the fields we find an expression for t similar to what was done for T. We perform certain transformations on the 4-point function and show that two of the critical exponents of dynamical percolation on complete graphs are observed which could be just an accident or an actual relation. We also apply this method to the gap dynamics in Site percolation and get an approximate critical exponent which has 99.06% correlation with the simulation results
  9. Keywords:
  10. Percolation Theory ; Conformal Fields Theory ; Logarithmic Divergences

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