Random geometry and the Kardar–Parisi–Zhang universality class
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| License | CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/38781 (DOI) | |
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Transcript: English(auto-generated)
00:05
Welcome to the video abstract of our paper Random Metrics and the Kargil Barisichan Universality class. In the next few minutes, we will attempt to show you some of our results. And more importantly, we will try to whet your appetite
00:21
to read the full paper. Euclid is the first hero of our story. It is difficult to overemphasize the beauty of classical geometry, circumstances, and straight lines, mathematics with ruler and compass. But what will happen if a space itself becomes irregular?
00:45
Then the circles and the lines themselves become rough. Now we will call the lines geodesics. And we will call the circles walls of these new irregular manifold.
01:01
But since we don't want our experiments to destroy the rainforest, we will do the simulations on a computer. Here we are, computer-generated geodesics and balls. Please forgive us for the missing technical details at this stage. Let us say that this represents a randomly generated
01:22
two-dimensional manifold with only local correlators. Now we compute the balls of different radii around a certain point. A useful analogy is to imagine the manifold as a map of a certain country and the balls as a set of places where you can reach in a certain time. We wonder how the roughness of those balls
01:40
scales with the ball size. Do the balls of all sizes keep that rough look? No, they look smoother as they grow. Roughness can be seen to scale as the radius raised to the power one third. Such a simple exponent should have a simple explanation. A growth exponent of one third is not novel in kinetic roughening. Indeed, it is one of the hallmarks of the so-called
02:03
Carter-Parise-Chang universality class far from equilibrium growth. This KPZ class shows up in very different phenomena, fire propagation, biological growth, turbulence in liquid crystals. In fact, KPZ is a very special class. A huge variety of integrable models and experiments
02:21
have made research as conjecture, but it involves not only quality of scaling exponents, but also of the full fluctuation statistics. Here we can see a histogram of the radial fluctuations of the balls compared to the well-known Tracy-Wydon probability distribution function connected to the extreme eigenvalues
02:40
of random Hermitian matrices. What about minimizing geodesics? They can be traced by letting two balls grow from different points and marking their intersection. The deviation from the straight line depends with the Euclidean distance between the points as a power law with exponent two thirds.
03:05
We have also found the meaning of that exponent within the KPZ framework. Indeed, it is related to the so-called dynamic exponent, which rules the growth of the lateral correlation length. Among many other observables,
03:21
we have also evaluated the expected times of arrival at points which have a fixed Euclidean distance to the center. Indeed, the fluctuations of those times also grow as a power law of their distance to the center with power one third. They also possess Tracy-Wydon statistics. We hope to have made you want to take a look
03:42
at the full article, which contains many other interesting surprises. As a final remark, we can only say that Euclid rules even when he's drunk.
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