[Paper Review] Almost mixing of all orders and CLT for some $\mathbb{Z}^d$-actions on subgroups of $\mathbb{F}\_p^{\mathbb{Z}^d}$

Guy Cohen, Jean-Pierre Conze|arXiv (Cornell University)|Sep 21, 2016

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper establishes the Central Limit Theorem (CLT) for $Δ^d$-actions on shift-invariant subgroups of $\mathbb{F}_p^{\mathbb{Z}^d}$ by analyzing the scarcity of non-mixing configurations through $S$-unit type equations. Using cumulant methods and bounding solutions to special $\mathcal{D}$-polynomials, it proves that almost mixing of all orders implies CLT validity under various summation schemes, even when full mixing fails.

ABSTRACT

For N d-actions by algebraic endomorphisms on compact abelian groups, the existence of non-mixing configurations is related to "S-unit type" equations and plays a role in limit theorems for such actions. We consider a family of endomorphisms on shift-invariant subgroups of F Z d p and show that there are few solutions of the corresponding equations. This implies the validity of the Central Limit Theorem for different methods of summation.

Motivation & Objective

• To establish the Central Limit Theorem (CLT) for $\mathbb{Z}^d$-actions on shift-invariant subgroups of $\mathbb{F}_p^{\mathbb{Z}^d}$, particularly when full mixing fails.
• To analyze the structure of non-mixing configurations in $\mathbb{Z}^d$-actions via $S$-unit type equations arising from algebraic endomorphisms.
• To show that the scarcity of such configurations—quantified by bounding solutions to special $\mathcal{D}$-polynomials—implies almost mixing of all orders.
• To extend the cumulant method to prove CLT under multiple summation methods (e.g., rectangles, random walks), even in non-connected, totally disconnected group settings.
• To provide a functional CLT for ergodic sums over increasing rectangles under regularity conditions on observables.

Proposed method

• The authors model $\mathbb{Z}^d$-actions via algebraic endomorphisms on shift-invariant subgroups of $\mathbb{F}_p^{\mathbb{Z}^d}$, using Laurent polynomials and formal power series representations.
• They define a class of endomorphisms via multiplication by polynomials $R_j$ in $\mathcal{P}_d$, generating commuting actions on the group.
• They analyze non-mixing configurations by reducing them to solutions of $S$-unit type equations, which are shown to be sparse via counting arguments on special $\mathcal{D}$-polynomials.
• The cumulant method is applied by bounding higher-order cumulants of ergodic sums, relying on the decay of $V_n(f)$, the variation of the observable $f$ on distant coordinates.
• The key technical step involves decomposing and counting solutions to special $\mathcal{R}$-polynomials, showing that the number of non-mixing configurations grows subexponentially.
• Functional CLT is derived using martingale approximations and uniform convergence of approximations $\varphi_n$ to $f$, under the condition $V_n(f) = O(\lambda^n)$ with $\lambda < p^{-1}$.

Experimental results

Research questions

• RQ1 Can the Central Limit Theorem be established for $\mathbb{Z}^d$-actions on $\mathbb{F}_p^{\mathbb{Z}^d}$ subgroups when full mixing fails?
• RQ2 How sparse are non-mixing configurations in such actions, and can their scarcity be quantified via $S$-unit equations?
• RQ3 Does the cumulant method remain effective for proving CLT in non-mixing, totally disconnected group actions?
• RQ4 Under what regularity conditions on observables does the functional CLT hold for ergodic sums over rectangles in $\mathbb{Z}^d$?
• RQ5 Can the CLT be extended to different summation schemes (e.g., random walks) in this algebraic setting?

Key findings

• The number of solutions to the $S$-unit type equations associated with non-mixing configurations is bounded, implying their scarcity in the action's configuration space.
• The action is 'almost mixing of all orders'—non-mixing configurations are sparse enough to allow the application of the cumulant method.
• For observables with $V_n(f) = O(\lambda^n)$ and $\lambda < p^{-1}$, the $L^\infty$-norm of the projection $\Pi^n f$ decays exponentially as $O(\lambda^n)$, ensuring regularity.
• The Central Limit Theorem holds for ergodic sums over increasing rectangles $D_n$, with convergence in distribution to a normal law under the normalization $|D_n|^{-1/2}$.
• A functional CLT is established for such sums, extending previous results to non-connected, totally disconnected compact abelian groups.
• The CLT is valid under multiple summation schemes, including random walks, due to the decay of cumulants and the regularity of the observable.