Random geometry and the KPZ universality class

Abstract

This webpage is designed to convey extra information about our research project at Universidad Carlos III de Madrid on random geometry and KPZ. This work combines two ideas: the study of random metrics, flat on average, and the KPZ universality class, which we study in a covariant manner (in other words: we can study it in any 2D geometry).

Balls and geodesics in random metrics

Balls are the circles of our (random) metric. You can define the ball of radius R around the origin by finding the points whose distance to the origin is equal to R. When you plot them in Euclidean space, they don't look regular, but wiggly. Nonetheless, it can fit between two Euclidean circles. The difference between their radii constitute a measure of the roughness of the ball. This rougness scales as a power law of the Euclidean radius of the ball. The exponent is 1/3.

Geodesics are the straight lines of our (random) metric. But when we plot them in Euclidean space, they look wiggly. There are, nonetheless, a few differences between geodesics and straight lines. To start with, there may be many geodesics joining two points. But only one of them is the "minimizing" one, the one whose length is minimal. This minimal length can be obtained in the following way. Draw balls of increasing radii starting from the initial and final points. When the two balls touch for the first time, we have the middle point of the minimizing geodesic. The maximal vertical deviation of this middle point scales as a power law of the distance between the two points. The exponent is 2/3.

Covariant KPZ in circular geometry

The following video shows the action of our covariant KPZ equation acting on a certain initial interface on a plane.

Growth in non-Euclidean metrics

Let's start with non-random surfaces, where we make balls grow.

We write the surface as r (x₁,x₂)=(x₁, x₂, f (x₁,x₂))
The components of the metric tensor are given by: g_ij= r'_xi· r'_xj

We have chosen a few surfaces to illustrate the general idea:

A paraboloid: f (x₁, x₂) = x₁² + x₂². This surface has positive curvature.

A saddle: f (x₁, x₂) = x₁² - x₂². Surface with negative curvature.

An egg-crate: f (x₁, x₂) = sin(x₁) + sin(x₂).

Profiles of the balls in random metrics

Growh of individual balls is similar to growth of the individual sample of the covariant KPZ equation shown previously. We study the fluctuations of the intersection between two balls to check if we obtain the same result as with the maximal deviation of the geodesics. The simulation searches the coordinates of the intersection of two balls which grow at the same speed on a random metric, initially separated a given distance. In order to optimize the simulation, once the intersection point has been found, we separate the two balls further and rotate them an arbitrary angle. This makes the same effect as launching two more fresh balls.

Another geometric model: first passage percolation

Let's consider a lattice in which each link has an associated (random) traversing time. We can find the minimal time which it takes us to reach each point from the center. The following videos show the time of arrival function for a single realization (and the average of many). Rotational symmetry is restorated at large distances.

Analysis of the arrival times

Instead of studying the profiles of the balls as we did initially, we now consider the times of arrival from the center to each point on the plane. In the first video we align our points, but in the second we use a different algorithm for the distribution of the angles, which removes unwanted correlations.