Rigidity of one-dimensional point processes via optimal transport

TL;DR

We address rigidity properties of one-dimensional stationary point processes and show that the existence of an $L^1$ transport map from a lattice or from Lebesgue measure to a process implies Number-Rigidity and Cyclic-Factor. The main mechanism combines optimal transport, well-ordered matchings, and ergodic averaging to deduce rigidity, then applies the framework to one-dimensional non-singular Riesz gases with $-2<s\le -1$, proving they are $L^2$-perturbed lattices, Hyperuniform of type I, Number-Rigid, and possessing Cyclic-Factor; the Coulomb case $s=-1$ yields crystallization. The approach provides a rigorous link between transport costs and rigidity phenomena, with finite-volume to infinite-volume entropy methods enabling construction of infinite-volume Riesz gases. These results illuminate crystallization tendencies in 1D long-range-interacting systems and suggest open questions for the regime $-1<s<0$.

Abstract

We investigate rigidity phenomena in one-dimensional point processes. We show that the existence of an $L^1$ transport map from a stationary lattice or the Lebesgue measure to a point process is sufficient to guarantee the properties of Number-Rigidity and Cyclic-Factor. We then apply this result to non-singular Riesz gases with parameter $s\in(-2,-1]$, defined in infinite volume as accumulation points of stationarized finite-volume Riesz gases. This includes, for $s=-1$, the well-known one-dimensional Coulomb gas (also called Jellium plasma, or the one-component 1D plasma).

Figures (1)

• Figure 1: Conjectured phase diagram for the one-dimensional Riesz gas with $-2<s\leq 0$LewinSurvey2022lelotte2023phasetransitionsonedimensionalriesz. The known and studied cases are as follows. The case $s=-1$ corresponds to the 1D Coulomb gas. The case $s=0$ corresponds to the Sine$_\beta$ gas. At $s=0$ and $\beta =2$, this is the Dyson sine process, a determinantal point process. It is expected that crystallization occurs at any temperature for $-2<s\leq -1$ (no phase transition), and that a double phase transition in temperature takes place for $-1<s<0$: from crystal, to quasisolid, and then to fluid as temperature increases. Theorem \ref{['thm2']} indicates crystallization for $s\leq -1$. Other conjectures remain open.

Theorems & Definitions (19)

• Lemma 1.5
• Lemma 1.6
• Theorem 2.2: Rigidities in one dimension
• Remark 2.3: False contrapositive
• Remark 2.4: No equivalent in dimension $d\geq 2$
• Remark 2.5: Alternate proof
• Theorem 2.7: Existence of Riesz gas
• Theorem 2.8: Rigidities of Riesz gases
• Conjecture 2.9
• Lemma 3.3