[Paper Review] Quantitative weak mixing for random substitution tilings

Rodrigo Treviño|arXiv (Cornell University)|Jun 30, 2020

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes quantitative lower bounds on the local dimension of spectral measures for random substitution tilings in higher dimensions, extending Bufetov-Solomyak's work on one-dimensional systems. By analyzing twisted ergodic integrals and using cohomological techniques with Bratteli diagrams and trace cocycles, it proves uniform weak mixing for the $\mathbb{R}^d$ action on tiling spaces, showing spectral measures cannot concentrate at points—providing a quantitative obstruction to eigenfunctions.

ABSTRACT

For $N$ compatible substitution rules on $M$ prototiles $t_1,\dots,t_M$, consider tilings and tiling spaces constructed by applying the different substitution rules at random. These give (globally) random substitution tilings. In this paper I obtain bounds for the growth on twisted ergodic integrals for the $\mathbb{R}^d$ action on the tiling space which give lower bounds on the lower local dimension of spectral measures. For functions with some extra regularity, uniform bounds on the lower local dimension are obtained. The results here extends results of Bufetov-Solomyak to tilings of higher dimensions.

Motivation & Objective

• To extend quantitative weak mixing results from one-dimensional to higher-dimensional random substitution tilings.
• To establish lower bounds on the local dimension of spectral measures for the $\mathbb{R}^d$ action on tiling spaces.
• To prove uniform weak mixing by analyzing twisted ergodic integrals and return vector dynamics.
• To show that spectral measures cannot be Dirac deltas, thus ruling out eigenfunctions with non-trivial spectral concentration.

Proposed method

• Uses twisted ergodic integrals $\mathcal{S}^{\mathcal{T}}_R(f,\lambda)$ to quantify spectral measure concentration.
• Applies the Veech criterion and its cohomological refinement to relate spectral properties to return vector dynamics.
• Employs Bratteli diagrams and inductive limit $C^*$-algebras to model tiling spaces and their trace spaces.
• Analyzes trace cocycles $\Theta^{(k)}_x$ and renormalization maps $\Phi_x$ to track spectral behavior across scales.
• Utilizes the ErdÖs-Kahane method to control the density of times when dynamics avoid lattice points in cohomology.
• Applies differential forms and $d_\eta$-cohomology to study eigenfunction obstructions via $d_\eta f = 0$.

Experimental results

Research questions

• RQ1 Can quantitative weak mixing be established for random substitution tilings in $\mathbb{R}^d$ beyond the one-dimensional case?
• RQ2 What bounds can be placed on the lower local dimension of spectral measures for such systems?
• RQ3 How do twisted ergodic integrals relate to spectral measure concentration and eigenfunction obstructions?
• RQ4 To what extent do return vector dynamics and trace cocycles control spectral properties?
• RQ5 Can uniform lower bounds on local dimension be obtained for functions with extra regularity?

Key findings

• The lower local dimension of spectral measures is bounded below by 2 for functions in $\bar{\Delta}^0_{\mathcal{T}}$, implying $d_f^-(\lambda) \geq 2$.
• For $f \in \bar{\Delta}^0_{\mathcal{T}}$, the spectral measure satisfies $\mu_f(B_r(\lambda)) \leq C' r^2$ for $r < 1/2$, confirming $d_f^-(\lambda) \geq 2$.
• The cohomological equation $d_\lambda u = \omega$ has no solution if $\eta = \sum \eta_k dx_k$ is non-constant, implying trivial $H^0_\eta(\Omega;\mathbb{C})$.
• The system is uniformly weak mixing: no non-trivial $L^2$ eigenfunctions exist, as spectral measures cannot be Dirac deltas.
• The trace cocycle $\Theta^{(k)}_x$ controls the growth of twisted integrals, enabling quantitative bounds on spectral measure decay.
• The density $\mathcal{D}_N^x$ of times when $\Phi_x^{(n)*}[\mathrm{д}](\lambda)$ stays away from lattice points in cohomology is bounded away from 1, preventing spectral concentration.