Mathematicians crack decades-old percolation problem on phase transitions

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Mathematicians have published a proof resolving a long-standing puzzle in percolation theory, a branch of mathematics concerned with how connectivity emerges in random systems. The problem had remained unsolved for several decades and relates to understanding phase transitions, the points at which a system undergoes a dramatic change in behavior. The breakthrough was reported by Quanta Magazine on August 31, 2026. Percolation theory has broad applications in physics, materials science, and network analysis, making the result significant beyond pure mathematics.

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