A phase transition behavior for Brownian motions interacting through their ranks

TL;DR

This paper investigates the asymptotic behavior of a point process derived from interacting Brownian motions with rank-based drifts, revealing a phase transition to zero, a degenerate maximum, or a Poisson-Dirichlet distribution as the number of points grows.

Contribution

It establishes a universality result for the BFK models, connecting their large-scale behavior to well-known distributions like Poisson-Dirichlet.

Findings

Points converge to zero under certain drifts.

Maximum point approaches one in some regimes.

Processes converge to Poisson-Dirichlet distribution asymptotically.

Abstract

Consider a time-varying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as $n$ tends to infinity. Under a certain `continuity at the edge' condition on the drifts, we show that one of the following must happen: either (i) all points converge to zero, or (ii) the maximum goes to one and the rest go to zero, or (iii) the processes converge in law to a non-trivial Poisson-Dirichlet distribution. The proof employs, among other things, techniques from Talagrand's analysis of the low temperature phase of Derrida's Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson-Dirichlet law.