Non-universality for Crossword Puzzle Percolation

TL;DR

This paper introduces a crossword-inspired percolation model demonstrating a phase transition, with the game variant showing non-universal critical behavior influenced by the dynamics of solving words.

Contribution

It presents a novel percolation model based on crossword puzzles, revealing non-universal critical exponents in the game variant, unlike standard percolation.

Findings

The iid variant aligns with standard 2D percolation universality class.

The game variant exhibits a non-universal critical exponent ν.

Avalanches of solved words can occur in the game variant.

Abstract

A percolation model inspired by crossword puzzle games is introduced. A game proceeds by solving words, which are segments of sites in a two-dimensional lattice. As test case, the \emph{iid} variant allows for independently occupying sites with letters, only the percolation criterion depends on the existence of solved words. For the \emph{game} variant, inspired by real crossword puzzles, it becomes more likely to solve crossing words which share sites with the already solved words. In this way avalanches of solved words may occur. Both model variants exhibit a percolation transition as function of the a-priori site or word solving probability, respectively. The \emph{iid} variant is in the universality class of standard two-dimensional percolation. The \emph{game} variant exhibits a non-universal critical exponent $\nu$ of the correlation length. The actual value of $\nu$ depends on the function which controls how much solved words accelerate the solved of crossing words.