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The singular Gaussian scaling limit for a tagged particle in the infinite-dimensional Dyson model at β=2
Abstract
We prove a logarithmic central limit theorem for a tagged particle in infinite-dimensional Dyson Brownian motion at inverse temperature β=2, starting from the sine2 point process. The fluctuation is of order √(log t), which is subdiffusive and substantially slower than the polynomial fluctuations of classical short-range interacting particle systems, reflecting the rigidity induced by long-range logarithmic repulsion. We further identify the scaling limit as the singular Gaussian field corresponding to the zero-Hurst parameter, which provides a realisation of the singular Gaussian field as a scaling limit of a tagged particle raised by Dürr, Goldstein and Lebowitz (Comm. Pure Appl. Math. 38 (1985)).
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BibTeX
@misc{osada2026singular,
title = {{The singular Gaussian scaling limit for a tagged particle in the infinite-dimensional Dyson model at $β=2$}},
author = {Hirofumi Osada and Kohei Suzuki},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08365},
eprint = {2611.08365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08365},
}
RIS
TY - UNPB
TI - The singular Gaussian scaling limit for a tagged particle in the infinite-dimensional Dyson model at β=2
AU - Hirofumi Osada
AU - Kohei Suzuki
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08365
UR - https://arxiv.org/abs/2611.08365
AB - We prove a logarithmic central limit theorem for a tagged particle in infinite-dimensional Dyson Brownian motion at inverse temperature β=2, starting from the sine_2 point process. The fluctuation is of order √(log t), which is subdiffusive and substantially slower than the polynomial fluctuations of classical short-range interacting particle systems, reflecting the rigidity induced by long-range logarithmic repulsion. We further identify the scaling limit as the singular Gaussian field corresponding to the zero-Hurst parameter, which provides a realisation of the singular Gaussian field as a scaling limit of a tagged particle raised by Dürr, Goldstein and Lebowitz (Comm. Pure Appl. Math. 38 (1985)).
ER -