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DY: Fachverband Dynamik und Statistische Physik
DY 28: Networks: From Topology to Dynamics I (joint session SOE/DY)
DY 28.10: Talk
Wednesday, March 20, 2024, 17:30–17:45, TC 006
Crossword puzzle percolation — •Alexander K. Hartmann — University of Oldenburg, Germany
Games are a popular subject, also for physicists. Many games have a lattice or network representation. A crossword puzzle consists of black (blocked) and white sites, the latter can be empty or occupied with letters. A word is known in the puzzle if a complete horizontal or vertical segment of white sites, usually between two black sites, is occupied (periodic boundary conditions are used, words may also take a full column or row).
Here the crossword puzzle is considered as percolation problem: Two known words are connected if they are perpendicular to each other and share one occupied site. A configuration is considered as percolating if there exists a path of connected words around the system, in either direction.
Numerical simulations for two-dimensional crosswords up to size 1000× 1000 are performed. For uncorrelated occupation with probability p for the white sites, percolation transitions at critical thresholds pc, depending on the fraction of black sites, are found. The results are analyzed by finite-size scaling and indicate that the problem is in the universality class of standard two-dimensional percolation. This changes, when the real game case is considered where full words are known with a probability pw(x) which depends on the fraction x of already known letters in the word, introducing correlations. The universality class depends on the shape of pw(x).
Keywords: percolation; phase transitions; simulations; critical exponents; correlations