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Friday, July 12, 2019

How Randomness Can Make Math Easier | Mathematics - Quanta Magazine

Kevin Hartnett, senior writer at Quanta Magazine covering mathematics and computer science explains, Randomness would seem to make a mathematical statement harder to prove. In fact, it often does the opposite.

 Randomness is an underappreciated mathematical tool.
Photo: tostphoto
Of all the tools available to the mathematician, randomness would seem to offer little benefit. Math traffics in logic and rigor. Its broad goals are to find order and structure in a vast sea of objects. It’s precisely because the mathematical world isn’t random that the whole enterprise of mathematics is possible.

Yet the recent Quanta article “Random Surfaces Hide an Intricate Order” concerned a new proof in which randomness made all the difference. The result involves checkerboard-like patterns drawn atop geometric spaces that are built at random. The authors of the proof found that the randomness in the geometric space made the checkerboard patterns easier to describe. “It’s a bit surprising that adding randomness enables you to do more than you can” without it, said Nicolas Curien, a mathematician at Paris-Sud University and a co-author of the work.

As it turns out, randomness is helpful in mathematics in many ways...

Mathematicians try to exploit this fact. There’s a conjectural relationship, known as the KPZ formula, that tells mathematicians how to convert a result about the random grid into a result for the deterministic one, or vice versa. “In theory it means you’re free to compute on either” the random or deterministic side, said Olivier Bernardi, a mathematician at Brandeis University and a co-author of the recent paper. This new work is consistent with previous (much harder to prove) results about percolation on a regular grid, validating the KPZ formula.
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Source: Quanta Magazine