# A Topological Glass

## TL;DR

This paper introduces a model of triangulations with colored nodes exhibiting glassy dynamics, slow approach to equilibrium, and a hierarchical energy landscape similar to spin glasses, revealing complex emergent behavior.

## Contribution

It presents a novel triangulation-based model with glassy properties, analyzing its slow dynamics, unordered steady state, and hierarchical energy landscape.

## Key findings

- System approaches steady state exponentially slowly
- Stationary state is unordered
- Energy landscape has hierarchical structure similar to spin glasses

## Abstract

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with $n$ nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with different numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.