[Paper Review] Truchet Tilings and Renormalization

W. Patrick Hooper|arXiv (Cornell University)|Mar 30, 2011

Mathematical Dynamics and Fractals13 references3 citations

TL;DR

This paper investigates the prevalence of closed curves in Truchet tilings generated by product measures derived from two-sided Markov chains on {±1}. Using renormalization techniques from dynamical systems, it proves that for symmetric Markov chains (where ±1 occur with equal probability), the curve through any fixed edge is almost surely closed. The key result establishes almost sure closure of curves under symmetric, stationary measures, contrasting with the Drift Theorem which shows bi-infinite curves are common under asymmetric measures.

ABSTRACT

The Truchet tiles are a pair of square tiles decorated by arcs. When the tiles are pieced together to form a Truchet tiling, these arcs join to form a family of simple curves in the plane. We consider a family of probability measures on the space of Truchet tilings. Renormalization methods are used to investigate the probability that a curve in a Truchet tiling is closed.

Motivation & Objective

• To investigate the probability that a curve in a Truchet tiling is closed, particularly under structured random measures.
• To apply renormalization techniques from dynamical systems to analyze curve behavior in Truchet tilings.
• To determine conditions under which closed curves are prevalent, especially when the underlying measures are stationary and symmetric.
• To compare results under symmetric vs. asymmetric measures, showing a sharp contrast in curve behavior.
• To extend understanding of Truchet tilings through connections to rectangle exchange maps and corner percolation models.

Proposed method

• The paper models Truchet tilings using functions τω,ω′(m,n) = ωmω′n, where ω and ω′ are sequences in {±1}Z.
• It defines probability measures μ and μ′ on the space of sequences as stationary measures of Markov chains with transition probabilities p and p′.
• The analysis uses renormalization: the tiling is iteratively rescaled and analyzed via recursive structure in curve length and configuration space.
• A key component is the use of a renormalization map that tracks the length L(α) of the longest initial run of a’s in a configuration α.
• The proof relies on bounding the expected value of a function f_k(α) over configuration spaces Ak, using inductive estimates on β_k and γ_k.
• The convergence of β_k to zero is established via induction, showing that the measure of configurations with long curves vanishes as k→∞.

Experimental results

Research questions

• RQ1 Under what conditions on the underlying Markov chains is the curve through a fixed edge almost surely closed?
• RQ2 How does the asymmetry of the measure (i.e., p ≠ 1/2) affect the likelihood of closed vs. bi-infinite curves?
• RQ3 Can renormalization techniques be used to derive stronger results on the distribution of curve lengths?
• RQ4 What is the role of shift-invariance and stationarity in determining curve behavior in Truchet tilings?
• RQ5 How do the results compare to known models like corner percolation or rectangle exchange maps?

Key findings

• For symmetric Markov chains (p = p′ = 1/2), the curve through any fixed edge is almost surely closed, as shown in Theorem 2.
• Under asymmetric measures (p ≠ 1/2 or p′ ≠ 1/2), the Drift Theorem guarantees that the probability of a bi-infinite curve is at least max{|p|, |q|}, so closed curves are not full measure.
• The probability that a curve has length L = n is given by P(L(α) = n) = 2(n+3)/[(k+3)(k+4)] for n > 0 and 12/[(k+3)(k+4)] for n = 0, in finite approximations of size k.
• The expected value of the function f_k(α) over configuration space Ak tends to zero as k → ∞, which implies the main result via convergence of β_k to zero.
• The inductive proof shows that β_k → 0 by demonstrating that for any m > 0, there exists K_m such that γ_k ≤ 2(8/9)^m for all k ≥ K_m.
• The renormalization process reveals that the contribution of long curves diminishes exponentially, supporting the almost sure closure of curves under symmetric measures.