Optimal local laws and CLT for the circular Riesz gas

Boursier, Jeanne

doi:10.48550/arxiv.2112.05881

Cited by 4 publications

(8 citation statements)

References 38 publications

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“…Theorem 1.4 (Central Limit Theorem). Suppose that 𝜃 ∈ 𝐶 13 is a compactly supported test function, and let 𝜉 𝑧,𝐿 denote the associated rescaled test function at scale 𝐿 > 𝜔 𝑁 with 𝑧 ∈ 𝖡. Let  𝑁 be the good event of Theorem 4.2.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Local laws and a mesoscopic CLT for β‐ensembles

Peilen

2023

Comm Pure Appl Math

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We study the statistical mechanics of the log‐gas, or β‐ensemble, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on the next order energy that are valid down to microscopic length scales. To our knowledge, this is the first time that this kind of a local quantity has been controlled for the log‐gas. Simultaneously, we exhibit a control on fluctuations of linear statistics that is valid at all mesoscales using Johansson's method and a transport approach. Using these local laws, we are able to exhibit for the first time a CLT at arbitrary mesoscales, improving upon previous results that were true only for power mesoscales.

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Section: Statement Of Resultsmentioning

confidence: 99%

Local laws and a mesoscopic CLT for β‐ensembles

Peilen

2023

Comm Pure Appl Math

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“…Lem. 3.2 that E[q 2 R,0 ] C/R d−s , for any infinite translation-invariant (stationary) point process with finite Jellium energy per unit volume, in the case max(0, d − 2) s < d. One can in fact get the same bound for all 0 < s < d (averaged over [R, 2R]) by integrating(76) against the point process. It is more complicated to deal with non translation-invariant systems (without performing an average over translations, that is, look at the "empirical field").…”

mentioning

confidence: 84%

“…At T = 0, Lemma 25 says that it is uniformly bounded for minimizers, away from the boundary of Ω. The estimate (76) says that it is small in average for r ≫ 1, which is going to be useful later.…”

Section: Local Bounds and Definition Of The Point Processmentioning

confidence: 97%

“…Boursier has recently obtained 76 rigidity results about the fluctuations of the individual points in the case 0 < s < 1 in dimension d = 1, which imply very precise (average) local bounds. In the case of the 1D log gas s = 0, much more is known due to the link with random matrices explained in Section V, see for instance Ref.…”

Section: Local Bounds and Definition Of The Point Processmentioning

confidence: 99%

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Coulomb and Riesz gases: The known and the unknown

Lewin

2022

Journal of Mathematical Physics

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We review what is known, unknown, and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in [Formula: see text] interacting with the Riesz potential ±| x|− s (respectively, −log | x| for s = 0). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g., in random matrix theory). The main question addressed in this Review is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case s < d. For the convenience of the reader, we give the detail of what is known in the short range case s > d. Finally, we discuss phase transitions and mention what is expected on physical grounds.

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“…Boursier has recently obtained 72 rigidity results about the fluctuations of the individual points in the case 0 < s < 1 in dimension d = 1, which imply very precise (average) local bounds. In the case of the 1D log gas s = 0, much more is known due to the link with random matrices explained in Section V, see for instance Ref.…”

Section: Local Bounds and Definition Of The Point Processmentioning

confidence: 99%

Coulomb and Riesz gases: The known and the unknown

Lewin

2022

Preprint

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We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in R d interacting with the Riesz potential ±|x| −s (resp. − log |x| for s = 0). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case s < d. For the convenience of the reader we give the detail of what is known in the short range case s > d. In the last part we discuss phase transitions and mention what is expected on physical grounds.

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Optimal local laws and CLT for the circular Riesz gas

Cited by 4 publications

References 38 publications

Local laws and a mesoscopic CLT for β‐ensembles

Local laws and a mesoscopic CLT for β‐ensembles

Coulomb and Riesz gases: The known and the unknown

Coulomb and Riesz gases: The known and the unknown

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