Asymptotic long-range order for the XY-model on random geometric graphs

Abstract

We study the classical XY-model on random geometric graphs 𝒢_{n, ε}, which are obtained by sampling n ∈ ℕ independent points in a finite domain Ω⊂ ℝ^d, d ≥ 2, and connecting two points by and edge if their distance is of order ε > 0. We refer to 𝒢_{n, ε} as the random environment. Letting ε → 0 as n → ∞ at a sufficiently slow rate, these graphs capture the geometry of Ω. Denoting the inverse temperature by β, we show that in the limit β→ ∞ at a rate depending on n and ε, the XY-model on 𝒢_{n, ε} exhibits long range order in the sense that we prove a lower bound away from zero on the two-point function. Our result is quenched in the random environment: long-range order holds with large probability, converging to one as n → ∞. To prove the statement, we show that with high probability the environment is sufficiently regular to apply a convexity argument and the Brascamp--Lieb inequality.

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BibTeX RIS

BibTeX

@misc{disertori2026asymptotic,
  title = {{Asymptotic long-range order for the XY-model on random geometric graphs}},
  author = {Margherita Disertori and Max Mihailescu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02618},
  eprint = {2609.02618},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02618},
}

RIS

TY  - UNPB
TI  - Asymptotic long-range order for the XY-model on random geometric graphs
AU  - Margherita Disertori
AU  - Max Mihailescu
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02618
UR  - https://arxiv.org/abs/2609.02618
AB  - We study the classical XY-model on random geometric graphs 𝒢_n, ε, which are obtained by sampling n ∈ ℕ independent points in a finite domain Ω⊂ ℝ^d, d ≥ 2, and connecting two points by and edge if their distance is of order ε > 0. We refer to 𝒢_n, ε as the random environment. Letting ε → 0 as n → ∞ at a sufficiently slow rate, these graphs capture the geometry of Ω. Denoting the inverse temperature by β, we show that in the limit β→ ∞ at a rate depending on n and ε, the XY-model on 𝒢_n, ε exhibits long range order in the sense that we prove a lower bound away from zero on the two-point function. Our result is quenched in the random environment: long-range order holds with large probability, converging to one as n → ∞. To prove the statement, we show that with high probability the environment is sufficiently regular to apply a convexity argument and the Brascamp--Lieb inequality.
ER  -

Abstract

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