Multirange percolation of words on the square lattice

• Grant number: 25/04550-2
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: July 01, 2025
• End date: June 30, 2027
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Fabio Prates Machado
• Grantee: Otávio Augusto Silva Lima
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/13453-5 - Stochastic systems modeling, AP.TEM

Abstract

The problem percolation of words arises in the 80's with the interest in observingalternating sequences (1, 0, 1, 0, 1, 0, 1, 0, ¿ ) in graphs with randomness in sites. In 1995,Itai Benjamini and Harry Kesten formalized the problem of see all words in a graphin Percolation of Arbitrary Words in {0, 1}¿ and showed some results and counterexamples, as well as presenting problems. One of the problems presented was answered in2022 by Pierre Nolin, Vincent Tassion and Augusto Teixeira in No exceptional wordsfor Bernoulli percolation. This paper uses techniques that allowed results in longrange graphs in the format of the well-known truncation question: is it a model ofinvariant percolation by translations of infinite range that percolates with positiveprobability, can the range be limited and still percolation occurs? In the context ofword percolation we are interested in seeing all binary sequences.