Title: Characterization of the Fluctuations in a Symmetric Ensemble of Rank-Based Interacting Particles
Description: Within the context of rank-based interacting particle systems, we study the fluctuations in a symmetric ensemble around its stable distribution. This system is inspired by the classic Atlas model but represents its opposite pole since both the highest- and lowest-ranked particles will have non-zero drifts. In the first part of the dissertation, we derive a fine asymptotic analysis that includes a Law of Large Numbers. The lack of monotonicity of the ensemble requires that we develop alternative tools to those traditionally used in the analysis of the Atlas model. In the second part of the dissertation, we characterize the system’s fluctuations and show that, as the number of particles goes to infinity, they converge weakly to the mild solution of the Additive Stochastic Heat Equation on the real line with a symmetric initial condition. To establish this result, we use the technique proposed by Dembo and Tsai, 2017, where the Empirical Measure Process is used as a proxy for the ensemble’s fluctuations.

We expect that a combination of our work, and the available knowledge about the Atlas model, could help draw a full picture of how a finite rank-based interacting particle system with a general drift structure fluctuates around its stationary distribution as the number of particles goes to infinity, a long-standing open question in the field.