The Power of Randomness

How Randomness Can Make Math Easier:

When your path proceeds at random, you don’t have to worry about your past steps. Each step is, in a sense, as free as the first one: Just flip a coin to decide where to go next.

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.

If mathematics were easier, mathematicians might not need to resort to randomness. But most important mathematical questions are too hard for mathematicians to answer directly. “It’s something that may be obvious, but is good to remember, that most of the time if you state a problem in math or theoretical physics, it’s impossible,” said Paul Bourgade, a mathematician at New York University. “We just don’t have the tools to solve it.” In some of these situations, randomness loosens things up just enough to make a solution possible.

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