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Area of Science:

  • Statistical Physics
  • Computational Modeling
  • Game Theory

Background:

  • Percolation theory studies the formation of connected clusters in random systems.
  • Crossword puzzles involve filling letter grids, with word intersections creating dependencies.
  • Existing percolation models do not fully capture the emergent complexities of word-based games.

Purpose of the Study:

  • To introduce and analyze a novel percolation model inspired by crossword puzzle mechanics.
  • To investigate the distinct behaviors of two variants: an independent site occupation (iid) model and a dependent word-solving game model.
  • To characterize the phase transitions and critical phenomena in both model variants.

Main Methods:

  • Development of a two-dimensional lattice model simulating word-solving processes.
  • Implementation of an iid variant where site occupation is independent, with percolation based on solved words.
  • Implementation of a game variant where solving intersecting words is facilitated by existing solutions, potentially causing avalanches.

Main Results:

  • Both model variants exhibit a percolation transition dependent on solving probability.
  • The iid variant aligns with standard two-dimensional percolation universality classes.
  • The game variant displays a nonuniversal critical exponent (ν) for correlation length, influenced by the word-solving acceleration function.

Conclusions:

  • The crossword-inspired percolation model successfully captures distinct behaviors, including avalanches in the game variant.
  • The game variant's critical behavior deviates from standard percolation, highlighting the impact of interdependent word-solving.
  • This model offers new insights into complex systems with emergent properties and interdependent processes.