(Non)-hyperuniformity of perturbed lattices

This paper investigates how perturbations affect the hyperuniformity of stationary lattices, demonstrating that such properties are preserved in dimensions one and two under finite moment conditions but can be destroyed by arbitrarily small perturbations in dimensions three and higher.

The Gist

Imagine you have a perfect, infinite grid of dots, like a sheet of graph paper stretching forever in every direction. In the world of physics and mathematics, this is called a lattice. It is the ultimate example of order.

Now, imagine you take a pair of scissors and gently nudge every single dot on that grid just a tiny bit. Maybe you push them left, maybe right, maybe up or down. This is what the paper calls a "perturbed lattice."

The big question the authors ask is: If you nudge these dots, does the grid lose its special "super-orderly" nature?

The Concept: "Hyperuniformity" (The Perfect Crowd)

To understand the answer, we need to understand a concept called hyperuniformity.

Think of a crowd of people at a concert.

• A Random Crowd (Poisson Process): If people arrive randomly, the number of people in a specific section (say, a 10-foot square) will fluctuate wildly. Sometimes there are 50 people, sometimes 100. The "noise" (variance) in the count is high.
• A Hyperuniform Crowd: Imagine a crowd that is so perfectly organized that if you count the people in any large square, the number is almost exactly the same every time. The "noise" is incredibly low.

In physics, this is called hyperuniformity. It's a state where the system looks disordered up close (you can't predict exactly where a specific dot is), but it is incredibly ordered from a distance (the density is perfectly uniform). The paper studies whether our "nudging" destroys this perfect balance.

The Main Findings: How Much Can You Nudge?

The authors discovered that the answer depends entirely on how big the dimension is (1D, 2D, or 3D) and how hard you push the dots.

1. The "One-Dimensional" and "Two-Dimensional" Worlds (Lines and Sheets)

Imagine a line of dots (1D) or a flat sheet of dots (2D).

• The Good News: If you nudge the dots, but the nudges are generally small (mathematically, if the "average size" of the push is finite), the grid stays hyperuniform. It keeps its super-orderly nature.
• The Sharp Limit: The paper proves this limit is "sharp." If you start pushing the dots so hard that the "average size" of the push becomes infinite (meaning you occasionally give a dot a massive, chaotic shove), the order breaks. The grid becomes messy, and the number of dots in a given area starts fluctuating wildly again.
• Analogy: Think of a line of dominoes. If you gently tap them, they stay in a neat line. If you occasionally kick one with the force of a truck, the whole line becomes chaotic.

2. The "Three-Dimensional" World (Space)

Now imagine the dots are floating in 3D space.

• The Bad News: In 3D, the rules change. The authors found that you can nudge the dots by arbitrarily tiny amounts—so small you can't even see them—and still break the hyperuniformity.
• The Catch: You have to nudge them in a very specific, coordinated way (they can't be random; they have to depend on each other).
• Analogy: In a 1D line, a tiny nudge is harmless. But in a 3D room, if you coordinate tiny nudges perfectly, you can create a "ripple" that ruins the perfect density of the crowd, even if no one moved far.

The "Slow Decay" Surprise

The paper also discovered something weird about the "noise" in these systems.
Usually, when a system is hyperuniform, the noise (fluctuations) dies down quickly as you look at larger and larger areas.

• The Discovery: The authors constructed examples where the system is still hyperuniform (the noise eventually goes to zero), but it does so painfully slowly.
• Analogy: Imagine a noisy room that is supposed to get quiet. Usually, it gets quiet in a minute. These new examples are like a room that gets quieter, but only by a tiny, almost unnoticeable whisper every hour. It does get quiet, but it takes an eternity.

Summary of the "Rules"

• In 1D and 2D: As long as your nudges aren't "too crazy" (finite average size), the grid stays perfectly ordered.
• In 3D: You can break the order with nudges so small they are almost invisible, provided the nudges are coordinated correctly.
• The "Slow" Case: You can have a grid that is ordered, but the "order" is so fragile that it takes forever to show up when you look at large scales.

Why This Matters (According to the Paper)

The authors don't claim this fixes a specific engineering problem or predicts a new material. Instead, they are mapping the mathematical boundaries of order. They are showing us exactly how much chaos a system can tolerate before it loses its "super-ordered" status, and how that tolerance changes depending on the dimension of the universe we are looking at. They are essentially drawing the map of the edge between order and chaos.

Technical Summary

Technical Summary: (Non)-hyperuniformity of perturbed lattices

1. Problem Statement and Setting

The paper investigates the stability of hyperuniformity under random perturbations of stationary lattices in $\mathbb{R}^d$. A point process $X$ is defined as hyperuniform if its rescaled number variance vanishes asymptotically:
$$\sigma(r) := \frac{\text{Var}[|X \cap B_r|]}{|B_r|} \to 0 \quad \text{as } r \to \infty,$$
where $B_r$ is a ball of radius $r$. A stronger condition, class-I hyperuniformity, requires the variance to grow at most as the surface area ($O(r^{d-1})$).

The authors consider a perturbed lattice $P_L = \{x + U + p_x : x \in L\}$, where:

• $L$ is a lattice in $\mathbb{R}^d$ (e.g., $\mathbb{Z}^d$).
• $U$ is a random shift uniformly distributed in a fundamental domain of $L$.
• $(p_x)_{x \in L}$ is a family of random perturbations that are identically distributed and $L$-invariant (jointly stationary), but potentially dependent.

The central question is: Under what conditions on the moments and dependence structure of $(p_x)$ does $P_L$ retain hyperuniformity? While the case of independent perturbations is well-understood (finite first moment suffices for hyperuniformity), the behavior under dependent perturbations was largely open.

2. Methodology

The paper employs a combination of spectral analysis, optimal transport theory, and probabilistic constructions.

Spectral Analysis and Structure Factors

The authors utilize the connection between the number variance and the structure factor $S(k)$ (the Fourier transform of the pair correlation function). Specifically, hyperuniformity corresponds to $S(0)=0$ (or more precisely, $S(B_\epsilon) = o(\epsilon^d)$).

• Dimension 2 ($d=2$): The proof relies on a Taylor expansion of the characteristic function of the perturbation difference $p_x - p_0$. By analyzing the spectral measure of the perturbation field, the authors show that if the perturbations have a finite second moment, the "spectral hyperuniformity" is preserved. This involves bounding the difference between the reduced second factorial moment measures of the perturbed and unperturbed lattices using Poisson's summation formula and careful estimates of the error terms involving the decay of the characteristic function.
• Dimension 1 ($d=1$): A similar but simpler analysis is provided in the appendix, utilizing the specific form of the Fourier transform of the indicator function of a ball (Bessel functions) to relate the number variance directly to the moments of the perturbations.

Optimal Transport Perspective

The results are rephrased in terms of the Wasserstein distance ($Wass_p$) between stationary point processes. The paper establishes that in dimensions $d=1, 2$, a finite $Wass_d$ distance to a stationary lattice implies hyperuniformity. This provides a geometric interpretation of the moment conditions.

Probabilistic Constructions

To demonstrate the sharpness of the positive results and the existence of counter-examples, the authors construct specific perturbation fields:

• Infinite Variance Constructions: They define perturbations based on random variables with infinite moments, showing that these can lead to infinite number variance in finite balls.
• Arbitrarily Small Perturbations: For $d \ge 3$, they construct perturbations bounded by an arbitrarily small $\epsilon$ that break hyperuniformity. This is achieved by creating a "displacement field" that pushes points out of large balls with a specific probability, leveraging the geometry of high-dimensional space.
• Slow Decay Examples: They construct hyperuniform processes where the rescaled number variance decays arbitrarily slowly, challenging standard classifications (Class I, II, III).

3. Key Results

Theorem 1: (Non)-hyperuniformity of Perturbed Lattices

• Finite Moments: For any $d \ge 1$, if $E[|p_0|^d] < \infty$, the number variance in finite balls is finite.
• Dimensions 1 and 2: If $E[|p_0|^d] < \infty$, the perturbed lattice is hyperuniform.
  • This condition is sharp: If $E[|p_0|^d] = \infty$, there exist perturbations (bounded by the random variable) such that the resulting process has infinite number variance in finite balls.
• Dimensions $d \ge 3$: Hyperuniformity is fragile. For any $\epsilon > 0$, there exist perturbations with $|p_0| \le \epsilon$ almost surely such that the resulting process is not hyperuniform (the rescaled number variance diverges to $+\infty$).

Theorem 2: Class-I Hyperuniformity

• Dimension 1: If $E[|p_0|^2] < \infty$, the process is class-I hyperuniform. This condition is sharp; if $E[|p_0|^2] = \infty$, one can construct a non-class-I process.
• Dimension 2: There exist perturbations bounded by an arbitrarily small constant that destroy class-I hyperuniformity. This suggests that in $d=2$, preserving class-I hyperuniformity likely requires constraints on the correlation structure of the perturbations, not just their magnitude.

Theorem 3: Arbitrarily Slow Decay

The authors construct hyperuniform processes in any dimension where the rescaled number variance $\sigma(r)$ decays slower than any prescribed rate $\tilde{\sigma}(r) \to 0$. This demonstrates that the standard classification of hyperuniformity (Classes I, II, III) does not cover the full spectrum of possible behaviors.

4. Significance and Claims

The paper claims to provide the first rigorous mathematical constructions of non-hyperuniform perturbed lattices in dimensions $d=2$ (with infinite variance perturbations) and $d \ge 3$ (with arbitrarily small perturbations).

• Sharpness of Conditions: The work establishes that the condition $E[|p_0|^d] < \infty$ is the precise threshold for retaining hyperuniformity in low dimensions ($d=1, 2$).
• Dimensional Dependence: It highlights a fundamental difference between low dimensions ($d \le 2$) and high dimensions ($d \ge 3$). In low dimensions, finite moments of the perturbations are sufficient to preserve order; in high dimensions, even infinitesimal perturbations can destroy hyperuniformity if they are sufficiently correlated.
• Optimal Transport Connection: The paper links the stability of hyperuniformity to the finiteness of the Wasserstein distance to a lattice, offering a new geometric perspective on the problem.
• Limitations of Classification: By exhibiting hyperuniform processes with arbitrarily slow variance decay, the paper suggests that existing phenomenological classifications of hyperuniformity are incomplete.

The authors note that while their constructions for slow decay and non-hyperuniformity are "ad hoc" and may lack direct physical reality, they serve to delineate the mathematical boundaries of the concept. The paper does not propose new experimental setups but rather clarifies the theoretical landscape of perturbed lattices, filling gaps left by previous physics literature which had hinted at these phenomena without explicit construction.