2nd Annual Conference of Random Geometric Systems

Invited talks

Scale-invariant random spatial networks (SIRSN) are remarkable random structures providing patterns of random routes that satisfy some very plausible axioms (see Aldous and Ganesan 2013, Aldous 2014) including scale-invariance, as found empirically in online maps. It is not a priori apparent that such networks can exist at all! Nevertheless Aldous (2014) constructed a SIRSN based on a dense dyadic mesh, and WSK and Kahn showed how to build a SIRSN from a dense Poisson line process. But both these constructions involve infinitely long linear paths, and in particular do not easily admit local influences. Can one do better? I shall discuss recent work showing that SIRSN can arise under far less stringent conditions, based on line segments or even stiff fibres.

• Aldous, D J. "Scale-Invariant Random Spatial Networks." Electronic Journal of Probability 19, no. 15 (April 2014): 1-41. https://doi.org/10.1214/EJP.v19-2920
• Aldous, D J, and K Ganesan. "True Scale-Invariant Random Spatial Networks." Proceedings of the National Academy of Sciences of the United States of America 110, no. 22 (December 18, 2013): 8782-85. https://doi.org/10.1073/pnas.1304329110
• Kahn, J. "Improper Poisson Line Process as SIRSN in Any Dimension." Annals of Probability 44, no. 4 (July 2016): 2694-2725. https://doi.org/10.1214/15-AOP1032
• WSK "From Random Lines to Metric Spaces." Annals of Probability 45, no. 1 (2017): 469-517. https://doi.org/10.1214/14-AOP935

When a system of interacting particles is subjected to an external field, then their average one-body density becomes spatially varying. The primary tool for describing equilibrium states is the density functional theory (DFT), a well-established approach in the field of statistical mechanics. If the particles undergo Brownian dynamics, then the response of the one-body density to a time dependent external field is commonly treated using a simple dynamical extension of the equilibrium theory - the classical dynamical density functional theory (DDFT). Although widely used, this approach can often provide only a qualitative description and even fails completely in many situations of interest. However, improving the theory has turned out to be very difficult.
In this talk I will introduce and discuss a recently developed 'superadiabatic-DDFT' which identifies clearly and intuitively how to go beyond standard DDFT and which promises to become an essential method in future liquid-state theories.

The study of hyperuniform states of matter is an emerging multidisciplinary field, influencing and linking developments across the physical sciences, mathematics and biology. A hyperuniform point process in d-dimensional Euclidean space is characterized by an anomalous suppression of large-scale density fluctuations relative to those in typical disordered systems, such as liquids and amorphous solids. As such, the hyperuniformity concept generalizes the traditional notion of long-range order to include not only all perfect crystals and quasicrystals, but also exotic disordered states of matter, thus providing a unified framework to quantitatively categorize such phases of matter. Disordered hyperuniform states have attracted great attention across many fields over the last two decades because they have the character of crystals on large length scales but are isotropic like liquids. I will review briefly the hyperuniformity concept, including generalizations to two-phase media (random sets), scalar fields and vector fields. Then I will give an overview of how hyperuniformity is linked to the Epstein zeta function of number theory, sphere packing problems, eigenvalues of random matrices, nontrivial zeros of the Riemann zeta function, spatial distribution of the prime numbers, free fermions, Laughlin's incompressible quantum states, and photoreceptor mosaics in avian retina. Finally, I will show how to generate disordered hyperuniform two-phase media that are rigorously disordered and hyperuniform.

Preliminary schedule

The conference will take place at the DLR site in Cologne-Porz.

The organisers have reserved hotel rooms at Jaumanns Hotel. During the registration process, you will be able to request us to book a room for you.

If you have any questions, please do not hesitate to send us an email at contact@random-geometric-systems-cologne.de.

Local organisers

• Dr. Michael Klatt
• Dr. Till Kranz
• Prof. Dr. Matthias Sperl