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Abstract
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Consider n positive diffusions whose logarithms are Brownian motions whose drift vector at every time point is determined by the order in which the coordinates are arranged as a decreasing sequence. These processes appear naturally in a variety of areas such as modeling market capitalizations in mathematical economics, modeling networks in queueing theory, and particle systems in statistical physics. In a paper in 2005 authors Banner, Fernholz, and Karatzas (BFK) use functions of such processes to model market weights in equity markets. The real-world data of such weights display two remarkable characteristics: stability in shape over eight decades, and a power law decay when arranged in decreasing order. A significant amount of economic theory has been devoted to explain these features. The models in these theories are mostly similar to discrete-time birth and death markov chains. BFK conjectured that rank-based interacting diffusions are a continuous time model which is akin to standard geometric Brownian motion models and lead to a market weight distribution which fits the empirical observation. We will show that under very weak conditions as the size of the market grows to infinity there are only three possible behaviors of the market weights if one assumes a rank-based model. Either they all go to zero, or the maximum grows to one while the rest go to zero, or they stabilize and converge in law to a Poisson-Dirichlet point process. Since the latter (only) non-trivial limit is known to have a power law decay, this settles the conjecture. The proof borrows ideas from Talagrand's analysis of Derrida's Random Energy Model of spin glasses. This is based on separate joint works with Sourav Chatterjee and Jim Pitman. |
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