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To add this additional randomness, mark every place where the corners of the triangles meet with a point — what mathematicians call a vertex.

Color each of the vertices white or black at random. One of the most basic is: Can you move far across the surface by only traveling on black vertices?

MA3D9 Geometry of curves and Surfaces

This process, of moving along connected vertices of the same color, is known as a percolation. It provides a mathematical way to study the physical phenomenon of the same name, in which a fluid flows through a porous medium.

Percolation will be easy or not depending on the weight of the coin: If the coin is weighted heavily toward black vertices, percolation will be nearly guaranteed; if it favors white vertices, percolation will almost certainly be impossible. Bernardi, Curien and Miermont study the case in between these extremes — the tipping point in the weight of the coin where percolation switches from almost impossible to nearly guaranteed.

Physicists are interested in phase transitions because many of the most important phenomena in nature occur on the cusp.

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Mathematicians are interested in phase transitions, too, because important mathematical features often emerge at precisely those points. In the new paper, the three mathematicians prove that similar wild behaviors occur right at the percolation phase transition.

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They show that at that critical threshold a unique geometric shape emerges — unique, but also universal. The first part of the paper determines how to weight a coin so that the coloring of the vertices sits on the threshold between percolation and no percolation. Confirming intuition, they prove that the critical value is a perfectly fair coin — one that gives a 50 percent chance for black and a 50 percent chance for white. The second part examines that interesting stuff. Because the underlying surface is chosen at random, and the process of coloring the vertices is random, the largest cluster on one surface will always be different from the largest cluster on another.

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But the mathematicians prove that across all surfaces and all possible ways of coloring the vertices on those surfaces, the largest clusters have traits in common. The first thing they prove is an exact probability distribution for the sizes of the largest black clusters across all surfaces. People rely on email to manage and organize their lives — from navigating multiple accounts, business documents, [and] travel itineraries [to] shopping.

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Colloids and Surfaces A: Physicochemical and Engineering Aspects

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