# Tilings and traces

## TL;DR

This paper investigates the statistical properties of tilings generated by random substitutions, establishing bounds on ergodic averages and applying these to the convergence rates of integrated density of states in aperiodic Schrödinger operators.

## Contribution

It introduces a novel renormalization approach using operator algebras to analyze deviations in ergodic averages for random tilings.

## Key findings

- Bounds on deviations of ergodic averages for random tilings
- Convergence rate estimates for integrated density of states
- Application to aperiodic Schrödinger operators

## Abstract

This paper deals with (globally) random substitutions on a finite set of prototiles. Using renormalization tools applied to objects from operator algebras we establish upper and lower bounds on the rate of deviations of ergodic averages for the uniquely ergodic $\mathbb{R}^d$ action on the tiling spaces obtained from such tilings. We apply the results to obtain statements about the convergence rates for integrated density of states for random Schr\"odinger operators obtained from aperiodic tilings in the construction.