The Tracy-Widom distribution

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at: lawrence@krubner.com, or follow me on Twitter.

Apparently phase shifts follow a unique distribution:

The Tracy-Widom distribution is an asymmetrical statistical bump, steeper on the left side than the right. Suitably scaled, its summit sits at a telltale value: √2N, the square root of twice the number of variables in the systems that give rise to it and the exact transition point between stability and instability that May calculated for his model ecosystem.

The transition point corresponded to a property of his matrix model called the “largest eigenvalue”: the greatest in a series of numbers calculated from the matrix’s rows and columns. Researchers had already discovered that the N eigenvalues of a “random matrix” — one filled with random numbers — tend to space apart along the real number line according to a distinct pattern, with the largest eigenvalue typically located at or near √2N. Tracy and Widom determined how the largest eigenvalues of random matrices fluctuate around this average value, piling up into the lopsided statistical distribution that bears their names.

Tracy-Widom Distribution vs Gaussian
Olena Shmahalo/Quanta Magazine
Whereas “uncorrelated” random variables such as test scores splay out into the bell-shaped Gaussian distribution, interacting species, financial stocks and other “correlated” variables give rise to a more complicated statistical curve. Steeper on the left than the right, the curve has a shape that depends on N, the number of variables.
When the Tracy-Widom distribution turned up in the integer sequences problem and other contexts that had nothing to do with random matrix theory, researchers began searching for the hidden thread tying all its manifestations together, just as mathematicians in the 18th and 19th centuries sought a theorem that would explain the ubiquity of the bell-shaped Gaussian distribution.

The central limit theorem, which was finally made rigorous about a century ago, certifies that test scores and other “uncorrelated” variables — meaning any of them can change without affecting the rest — will form a bell curve. By contrast, the Tracy-Widom curve appears to arise from variables that are strongly correlated, such as interacting species, stock prices and matrix eigenvalues. The feedback loop of mutual effects between correlated variables makes their collective behavior more complicated than that of uncorrelated variables like test scores. While researchers have rigorously proved certain classes of random matrices in which the Tracy-Widom distribution universally holds, they have a looser handle on its manifestations in counting problems, random-walk problems, growth models and beyond.

“No one really knows what you need in order to get Tracy-Widom,” said Herbert Spohn, a mathematical physicist at the Technical University of Munich in Germany. “The best we can do,” he said, is to gradually uncover the range of its universality by tweaking systems that exhibit the distribution and seeing whether the variants give rise to it too.

So far, researchers have characterized three forms of the Tracy-Widom distribution: rescaled versions of one another that describe strongly correlated systems with different types of inherent randomness. But there could be many more than three, perhaps even an infinite number, of Tracy-Widom universality classes. “The big goal is to find the scope of universality of the Tracy-Widom distribution,” said Baik, a professor of mathematics at the University of Michigan. “How many distributions are there? Which cases give rise to which ones?”

As other researchers identified further examples of the Tracy-Widom peak, Majumdar, Schehr and their collaborators began hunting for clues in the curve’s left and right tails.

Going Through a Phase

Majumdar became interested in the problem in 2006 during a workshop at the University of Cambridge in England. He met a pair of physicists who were using random matrices to model string theory’s abstract space of all possible universes. The string theorists reasoned that stable points in this “landscape” corresponded to the subset of random matrices whose largest eigenvalues were negative — far to the left of the average value of √2N at the peak of the Tracy-Widom curve. They wondered just how rare these stable points — the seeds of viable universes — might be.

To answer the question, Majumdar and David Dean, now of the University of Bordeaux in France, realized that they needed to derive an equation describing the tail to the extreme left of the Tracy-Widom peak, a region of the statistical distribution that had never been studied. Within a year, their derivation of the left “large deviation function” appeared in Physical Review Letters. Using different techniques, Majumdar and Massimo Vergassola of Pasteur Institute in Paris calculated the right large deviation function three years later. On the right, Majumdar and Dean were surprised to find that the distribution dropped off at a rate related to the number of eigenvalues, N; on the left, it tapered off more quickly, as a function of N2.

In 2011, the form of the left and right tails gave Majumdar, Schehr and Peter Forrester of the University of Melbourne in Australia a flash of insight: They realized the universality of the Tracy-Widom distribution could be related to the universality of phase transitions — events such as water freezing into ice, graphite becoming diamond and ordinary metals transforming into strange superconductors.

Because phase transitions are so widespread — all substances change phases when fed or starved of sufficient energy — and take only a handful of mathematical forms, they are for statistical physicists “almost like a religion,” Majumdar said.

Post external references

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    https://www.quantamagazine.org/20141015-at-the-far-ends-of-a-new-universal-law/
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