Hyperuniform Delone Realizations and Rigidity

Abstract: We prove a measurable realization theorem for hyperuniform Delone point processes. In dimensions (d\geq2), for every prescribed (q\geq1), every essentially free ergodic p.m.p.\ action of (\mathbb R^d) admits, at every sufficiently large prescribed intensity, a generating Delone realization whose return-time point process (η) is measurably isomorphic to the original action and whose Bartlett spectrum (ση) satisfies [ ση(B_\varepsilon)=o(\varepsilon^{2q}) \qquad(\varepsilon\downarrow0). ] Thus arbitrarily high finite-order low-frequency suppression can be imposed without changing the prescribed measurable dynamics. The same realizations can be chosen with surface-order ball variance and linear rigidity to any prescribed finite order, while also being maximally rigid and almost surely bounded-displacement equivalent to a lattice. For essentially free Euclidean-motion actions whose translation subaction is ergodic, the construction can be made isotropic and (V)-ergodic, and hence (V)-weakly mixing. In dimension one, every essentially free ergodic flow admits generating Delone realizations with logarithmic interval discrepancy, maximal rigidity, and near-quadratic decay of the Bartlett spectrum at the origin.

• The paper proves that, for dimensions d≥2 and sufficiently large intensity, every essentially free ergodic action admits generating Delone realizations with arbitrarily high finite-order Bartlett suppression, maximal rigidity, and bounded-displacement equivalence to a lattice.
• In dimension one, the construction achieves logarithmic interval discrepancy, O(log² R) number variance, and hyperuniform Bartlett decay, while showing that uniformly bounded discrepancy would force a Koopman eigenvalue and therefore fails for weakly mixing flows.
• The paper combines one-ended measured forests, exact moment cancellation, and recoverable marker configurations to preserve measurable dynamics, while establishing optimal surface-order variance O(R^{d−1}) despite arbitrarily strong finite-order spectral decay.

This paper by Michael Björklund establishes measurable realization theorems for hyperuniform Delone point processes. The central question is whether an arbitrary prescribed measure-preserving (p.m.p.) action of $\mathbb{R}^d$ can be represented — in the strong sense of being measurably isomorphic, not merely factored — by a Delone point process with prescribed low-frequency suppression of its Bartlett spectrum, together with strong rigidity and lattice-like geometry. The answer is affirmative and remarkably flexible in dimensions $d\geq 2$, while dimension one exhibits genuine spectral obstructions.

Problem statement and framework

Let $\eta$ be a translation-invariant point process on $V\cong\mathbb{R}^d$ with positive intensity and local second moments. Its Bartlett spectrum $\sigma_\eta$ is the unique positive translation-bounded Radon measure satisfying

$$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$

and $\eta$ is hyperuniform if $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$, equivalently $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$. The paper works with the stronger notion of Bartlett suppression of order $p$: $d\geq 2$0.

Given a p.m.p. Borel action $d\geq 2$1, a Borel set $d\geq 2$2 is a translation cross-section whose return sets $d\geq 2$3 define a point process $d\geq 2$4. The section is generating if $d\geq 2$5 is injective on an invariant conull set, so that $d\geq 2$6 and $d\geq 2$7 are measurably isomorphic as $d\geq 2$8-spaces. This information-preserving requirement distinguishes the paper from prior transport constructions (Klatt et al., 6 Jun 2025, Lotz et al., 21 May 2026), where a spatial mechanism is applied to an existing random measure without regard to recovering an underlying dynamical state.

Main results

The two realization theorems are as follows.

• Dimension $d\geq 2$9: for every essentially free p.m.p. action of $\eta$0 ($\eta$1 closed) whose restricted $\eta$2-action is ergodic, every integer $\eta$3, and every sufficiently large intensity $\eta$4, there exists a $\eta$5-invariant generating Delone translation cross-section of intensity exactly $\eta$6 whose return-time process satisfies $\eta$7, is maximally rigid, and — when $\eta$8 — has surface-order ball variance $\eta$9.
• Dimension one: every essentially free ergodic flow admits generating Delone realizations at all large intensities that are maximally rigid, have interval discrepancy bounded by $V\cong\mathbb{R}^d$0, number variance $V\cong\mathbb{R}^d$1, and Bartlett decay $V\cong\mathbb{R}^d$2; hence they are hyperuniform.

The point-process corollaries are recoding statements: any translation-ergodic essentially free point process $V\cong\mathbb{R}^d$3 admits a $V\cong\mathbb{R}^d$4-equivariant measurable isomorphism onto a Delone-supported process combining arbitrarily high finite-order Bartlett decay, surface-order variance, maximal rigidity, linear $V\cong\mathbb{R}^d$5-rigidity for any prescribed finite $V\cong\mathbb{R}^d$6, and almost sure bounded-displacement equivalence to the lattice $V\cong\mathbb{R}^d$7 via Laczkovich's criterion. For $V\cong\mathbb{R}^d$8 or $V\cong\mathbb{R}^d$9, translation ergodicity implies translation weak mixing (the Koopman eigenvalue orbit argument), so the isotropic realizations are $\sigma_\eta$0-weakly mixing. Applicable examples include restrictions of ergodic $\sigma_\eta$1-actions by Howe–Moore, Gaussian fields with radial covariance, determinantal processes with radial kernels, isotropic STIT tessellations, and the pinwheel tiling hull.

A notable quantitative feature: the surface-order bound is optimal for these processes. The averaged Beck-type lower bound yields

$\sigma_\eta$2

so the variance cannot be improved to $\sigma_\eta$3 despite arbitrarily high finite-order spectral decay. Thus finite-order Bartlett suppression does not force class-I* behavior.

The one-dimensional construction

The flow is modeled as a suspension over a separated cocompact cross-section with roof function bounded between $\sigma_\eta$4 and $\sigma_\eta$5 (via Slutsky's regular cross-section theorem). A nested hierarchy of return sets $\sigma_\eta$6 is built from a two-height Rokhlin partition (towers of heights 2 and 3 with base measures $\sigma_\eta$7 each), so each level-$\sigma_\eta$8 block is a union of two or three level-$\sigma_\eta$9 blocks, with $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$0. Hierarchical rounding assigns integers $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$1 such that each return block carries discrepancy at most 1 against its real mass. Since any finite orbit interval decomposes into at most $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$2 maximal return blocks, the orbit discrepancy is logarithmic, and integrating gives the exact mean $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$3, hence intensity exactly $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$4.

Each roof interval receives $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$5 points: a recognizable three-point marker encoding the label $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$6 via its edge lengths, plus equally spaced filler points. Uniform Delone bounds hold with separation $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$7 and covering radius $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$8, $$\operatorname{Var}_\eta(Sf)=\int_V |\widehat f(\xi)|^2\,d\sigma_\eta(\xi),$$9. Injectivity of the configuration map follows from intrinsic decoding of markers; since complete markers occur arbitrarily far out, the process is maximally rigid. Testing indicator functions against the Bartlett identity converts the $\eta$0 variance into near-quadratic spectral decay.

Crucially, the logarithmic discrepancy cannot generally be improved: the paper proves that uniformly bounded interval discrepancy forces the intensity $\eta$1 to be a nonzero Koopman eigenvalue of the flow via a bounded-cocycle coboundary argument. Consequently no weakly mixing flow admits such a realization. This is complemented by the companion result cited from (Björklund, 28 Jul 2026): the condition $\eta$2 forces lattice induction. Strong low-frequency suppression therefore constrains the measurable dynamics in dimension one.

The higher-dimensional construction

The proof solves two structural problems: integerizing local point counts while retaining recoverable finite information.

Framed scaffolds and integerization. An auxiliary cocompact cross-section $\eta$3 for the full $\eta$4-action yields, via positions, frames, and labels of returns, uniformly Delone scaffolds $\eta$5 in $\eta$6 with equivariant placement isometries $\eta$7. Labelled Voronoi cells $\eta$8 form an exact Borel partition after tie-breaking by a Borel linear order on labels. The proximity graph on $\eta$9 (edges at $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$0-distance $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$1) is locally finite, p.m.p., aperiodic, and — using $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$2 here, the only place dimension enters geometrically — has one-ended components. The measured one-ended spanning forest theorem of Conley–Gaboriau–Marks–Tucker-Drob then supplies a Borel parent map with uniformly bounded displacement, bounded child counts, and finite descendant sets. Integer masses $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$3 are defined by a divergence formula along this forest; each cell sheds a thin shell of $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$4-mass equal to its fractional part, which is transferred to the parent, producing packets $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$5 with $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$6, uniform core radius, and symmetric-difference mass $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$7.

Discretization with exact moments. Each packet is replaced by exactly $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$8 points via a parameterized equal-mass rectangular partition and bi-Lipschitz radial parametrization of convex cells. A $\operatorname{Var}_\eta(N_{B_R})=o(R^d)$9-point marker $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$0 with distinct pairwise distances — allowing Borel recovery of position, frame, and label from any unlabelled placed copy — is inserted together with auxiliary reservoir points preserving cardinality. A Newton/Picard correction along gradient flows of a basis of $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$1, using positive definiteness of the Gram matrix $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$2, enforces exact moment identities through degree $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$3 with perturbations of size $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$4. Global assembly gives a Borel, injective, $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$5-equivariant configuration map, hence a generating cross-section; markers occurring arbitrarily far out give maximal rigidity.

Spectral consequences. Moment cancellation yields distributional derivative representations $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$6 with translation-covariant primitives of uniformly bounded local variation. The covariance spectra $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$7 exist by a Schwartz-kernel/Bochner–Schwartz argument, and $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$8-ergodicity removes their atoms at zero. Testing against low-frequency cutoffs gives $\sigma_\eta(B_\varepsilon)=o(\varepsilon^d)$9, sharpened to $p$0 by continuity from above. Surface-order variance follows when $p$1 via Bessel decay of $p$2, and polynomial cutoffs give linear $p$3-rigidity whenever $p$4 — a slightly stronger sufficient condition than the borderline $p$5 of Lachièze-Rey for absolutely continuous spectra, valid here for arbitrary translation-bounded spectral measures.

Relation to existing constructions

The cellwise moment-cancellation mechanism descends from Gabrielli–Joyce–Torquato's tiling constructions and the transport frameworks of Klatt–Last–Lotz–Yogeshwaran and Lotz–Klatt. The new constraint is the realization problem: the measurable action is prescribed first, and the construction must produce equivariant, generating coordinates. The forest integerization and the finite recovery sets are what make established cancellation mechanisms compatible with generation. The results also run opposite to the Haynes–Kelly–Weiss program: rather than studying BD/BL classes of nets arising from structured actions, arbitrary essentially free ergodic actions are shown to admit generating sections whose configurations are BD-equivalent to lattices — though the BD matching is pointwise and not asserted to be measurable or equivariant, so it does not identify the dynamics with lattice dynamics. The distinction between the Bartlett spectrum (a Dworkin-subspace observable depending on coordinates) and the full Koopman spectrum underlies why Osada's isomorphisms between determinantal processes and Poisson processes do not conflict with these spatial statistics; as a concrete instance, the homogeneous Poisson action in $p$6 admits generating Delone coordinates with all the listed properties simultaneously.

Limitations and open questions

Several caveats are stated explicitly. The threshold $p$7 and local cardinalities are not effectively tracked in $p$8: although the correction space has dimension $p$9, conditioning and derivative constants are uncontrolled, so no useful explicit bound on $d\geq 2$00 is obtained. The logarithmic discrepancy bound in dimension one is not proved optimal; intermediate sublogarithmic rates remain open. Whether genuinely stronger low-frequency conditions (beyond any fixed finite order, e.g., stealthiness) impose dynamical constraints in dimensions $d\geq 2$01 remains open — the companion paper resolves the stealthy regime only for lattice-induced actions. Finally, the optimality of surface-order variance applies to the constructed processes; it does not preclude other realizations of the same dynamics with smaller variance exponent.

Conclusion

The paper shows that in dimensions $d\geq 2$02, the combination of arbitrarily high finite-order Bartlett suppression, optimal surface-order number variance, maximal and finite-order linear rigidity, and bounded-displacement lattice geometry imposes no constraint whatsoever on the underlying measurable dynamics: every essentially free ergodic action admits generating Delone coordinates with all these properties at any sufficiently large intensity. Dimension one is sharply different, with uniformly bounded discrepancy forcing Koopman eigenvalues. The technical contribution lies in coupling the classical moment-cancellation mechanism to a Borel equivariant generating encoding, via one-ended measured forests for integerization and isolated finite marker sets for state recovery.