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November 2017 Equilibrium fluctuation of the Atlas model
Amir Dembo, Li-Cheng Tsai
Ann. Probab. 45(6B): 4529-4560 (November 2017). DOI: 10.1214/16-AOP1171

Abstract

We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite ($\mathbb{Z}_{+}$-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on $\mathbb{R}_{+}$. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-\frac{1}{4}}$, converges as $t\to\infty$ to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on $\mathbb{R}_{+}$ with the Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional Brownian Motion (fBM). In particular, we prove a conjecture of Pal and Pitman [Ann. Appl. Probab. 18 (2008) 2179–2207] about the asymptotic Gaussian fluctuation of the ranked particles.

Citation

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Amir Dembo. Li-Cheng Tsai. "Equilibrium fluctuation of the Atlas model." Ann. Probab. 45 (6B) 4529 - 4560, November 2017. https://doi.org/10.1214/16-AOP1171

Information

Received: 1 March 2015; Revised: 1 April 2016; Published: November 2017
First available in Project Euclid: 12 December 2017

zbMATH: 06838126
MathSciNet: MR3737917
Digital Object Identifier: 10.1214/16-AOP1171

Subjects:
Primary: 60K35
Secondary: 60H15, 82C22

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 6B • November 2017
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