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Studying Continuum Percolation Models using Gap Scaling Approach

Kamkar, Marziyeh | 2024

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 58148 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Moghimi Araghi, Saman
  7. Abstract:
  8. Percolation is a fundamental topic in statistical physics with applications in modeling natural phenomena such as the spread of diseases and forest fires. It describes a geometric phase transition in which a system evolves from small-scale connectivity to extensive connectedness. At the percolation threshold, the system undergoes a continuous phase transition, exhibiting critical behavior. A recent study introduced a novel method for determining the percolation threshold by measuring the maximum change in the size of the largest cluster during the gradual increase of a control parameter. This maximum change, known as the ”gap,” signals the occurrence of the phase transition. The study also defined six critical exponents, referred to as ”gap exponents,” and analyzed fluctuations in three key quantities – the percolation threshold, the gap, and the size of the largest cluster – using the central limit theorem and extreme-value statistics. In addition to lattice-based models, continuum percolation models, where points are distributed in continuous space, have also been investigated. In these models, random points are scattered within a square, and penetrable objects such as disks or sticks are drawn around them. Percolation occurs when overlapping objects form a path connecting opposite sides of the square. This study focuses on applying gap scaling to such models to determine the percolation threshold, compute gap exponents, and analyze scaling functions associated with fluctuation distributions in continuum percolation for disk and stick models
  9. Keywords:
  10. Percolation Theory ; Gap Exponents ; Continuum Percolation ; Universal Gap Scaling ; Percolation Threshold ; Geometrical Phase Transform

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