Uniform displacement bounds and Gibbs limits for periodic one-dimensional Riesz gases
Abstract
For the neutral periodic one-dimensional Riesz gas with pair potential locally −|x|a, 0<a<1, we prove a particle-displacement variance bound of order β−1, uniformly in the number of particles. Log-concavity also gives exponential displacement tails. Every stationary periodic thermodynamic limit is simple, has intensity one, and admits a stationary ordered matching to the unit lattice with the same bounds. Each limit satisfies the canonical Gibbs equations for the full-line interaction, with an ordinary symmetric spatial principal value for the exterior potential. The matching implies uniformly bounded interval number variance and a positive limiting second moment of the reciprocal-lattice Fourier average at sufficiently low temperature. The main estimate compares the inverse random Hessian, through deterministic electrical flows, to a transient long-range network.
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BibTeX
@misc{pei2026uniform,
title = {{Uniform displacement bounds and Gibbs limits for periodic one-dimensional Riesz gases}},
author = {Yan Ru Pei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28503},
eprint = {2609.28503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28503},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform displacement bounds and Gibbs limits for periodic one-dimensional Riesz gases
AU - Yan Ru Pei
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.28503
UR - https://arxiv.org/abs/2609.28503
AB - For the neutral periodic one-dimensional Riesz gas with pair potential locally −|x|^a, 0<a<1, we prove a particle-displacement variance bound of order β^−1, uniformly in the number of particles. Log-concavity also gives exponential displacement tails. Every stationary periodic thermodynamic limit is simple, has intensity one, and admits a stationary ordered matching to the unit lattice with the same bounds. Each limit satisfies the canonical Gibbs equations for the full-line interaction, with an ordinary symmetric spatial principal value for the exterior potential. The matching implies uniformly bounded interval number variance and a positive limiting second moment of the reciprocal-lattice Fourier average at sufficiently low temperature. The main estimate compares the inverse random Hessian, through deterministic electrical flows, to a transient long-range network.
ER -