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
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
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