Equivalence Principle

thesis Equivalence principle

A few weeks ago I was lucky enough to be be the Tracy–Widom distribution, just as for present at a lecture on statistical physics by Why does this random matrices. Satya Majumdar, of the CNRS, University Tracy–Widom Majumdar now moved to the punchline. of Paris-Sud. Contrary to prevailing norms, Starting around the year 2000, several Majumdar didn’t use a laptop, and never distribution pop up in physicists and mathematicians discovered showed a single PowerPoint slide. He wrote so many seemingly how to make an exact mapping between out words and equations with chalk on a variants of the Ulam problem and models blackboard. I’m not sure I’ve ever learned so unrelated problems? of the KPZ type, showing that these much in only 30 minutes. problems are entirely equivalent. Hence, Majumdar started with some history moment) of the distribution of fluctuations.” there turns out to be an unexpected link about the growth of bacterial colonies. Seed It was unknown if the universality might between the Tracy–Widom distribution of a new colony on the surface of a nutrient run deeper — to the entire distribution random matrix theories and the physics medium, and it will grow into a vaguely of fluctuations — or might only of irregular growth. It is now known circular blob, yet with an outer edge that be approximate. that a number of discrete models of the is rough and gets rougher with time. Back That was the end of the first part of the KPZ universality class follow the exact in 1961, Murray Eden tried to explain the talk. Majumdar then turned to something Tracy–Widom distribution, as does the origin of this roughness, using a simple, very different: random matrices. continuous KPZ equation itself. linear mathematical model for diffusion Imagine an N × N matrix with the So, that open question about KPZ driven by random noise. That model didn’t entries being random numbers taken from universality is no longer open — the work quantitatively. a Gaussian distribution, and ask: what is universality it describes for a range of Yet Eden helped kick off a study of the distribution of the largest eigenvalue irregular growth processes indeed holds irregular surfaces, growth processes and of such a matrix? Random matrices for the entire probability distribution, not interfaces, which continues today. Surprising were first introduced into physics by only for the second moment. A beautiful progress over the past two decades, Eugene Wigner, and their study has found experiment carried out in 2010 by Majumdar suggested, has researchers an extremely wide range of applications. Kazumasa Takeuchi and Masaki Sano made thinking they’re just about to discover In 1993, Majumdar noted, Craig Tracy and a precise measurement of the fluctuations something truly profound. Unexpected links Harold Widom made a major breakthrough during the irregular growth of drops of keep turning up between problems with no by calculating exactly the probability a liquid crystal and found precisely the obvious connection. distribution of the largest eigenvalue in Tracy–Widom distribution. In 1986, Mehran Kardar, Giorgio Parisi the large N limit. This eigenvalue has All of which leads to a satisfying and Yi-Cheng Zhang modified Eden’s mean value √(2N), and fluctuates over theoretical unification — and also a model by including the lowest order a range of width N−1/6; the precise shape puzzle. There does seem to be a deep nonlinear term. This model — known as the of the distribution is now called the universal connection between many KPZ equation — does accurately describe Tracy–Widom distribution. different processes of the KPZ type. how the irregular fluctuations grow in both So what? Well, Majumdar went Strangely, it is also shared with many other space and time. Specifically, it predicts two on to another famous mathematical things such as random matrices and the exponents detailing how the mean square problem — the Ulam problem, named after distribution of sub-sequences within longer size of the fluctuations grows with time or mathematician Stanislaw Ulam. Consider sequences. What’s going on? Why does this when considering increasingly larger regions the N! permutations of the firstN integers Tracy–Widom distribution pop up in so along the front. {1, 2, 3, ..., N}. For each permutation, list many seemingly unrelated problems? If KPZ applied only to bacteria, it all the possible increasing subsequences Majumdar ended his talk here, would be of marginal importance. But and then find the longest one. ForN = 5, suggesting that something enormously in the 1980s, as Majumdar recounted, in for example, the permutation {1, 3, 4, 2, 5} tantalizing lies just beyond our current experiments and simulations physicists has increasing subsequences such as {1, 5}, view. Several recent studies (that he discovered that the KPZ exponents also {1, 3, 4} and {1, 3, 4, 5}, with the latter mentioned to me after the talk) have fit lots of other irregular growth patterns being the longest. The Ulam problem is to found signs of a peculiar ‘third-order’ arising in models of solid surface growth determine, for any N, and assuming that all phase transition lurking within all of or in the way polymers orient themselves N! permutations are equally probable, the these problems. This in turn appears to over disordered lattices, as well as in distribution of the length lN of the longest be closely linked to another generic phase interface fluctuations of the bacterial type. increasing subsequence. transition — the Gross–Witten–Wadia To a large degree, KPZ seemed to capture Ulam himself originally found that the transition — known from lattice gauge a universal pattern in the emergence average of lN is proportional to √N for large theories of quantum chromodynamics. But of fluctuations and roughness during N. But lN fluctuates about this mean. In 1999, this is still conjecture. irregular growth. mathematicians Jinho Baik, Percy Deift and Surprising and fascinating. I only wish But how ‘universal’ is universal? As Kurt Johansson derived the full distribution the lecture could have lasted another Majumdar stressed, this KPZ ‘universality’ for large N, finding it to be 2√N + N1/6χ, few hours. ❐ referred only to the two exponents with χ being a fixed universal function. The associated with the width (or second surprise — the function turned out, again, to MARK BUCHANAN

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