NDLI: A phase transition behavior for Brownian motions interacting through their ranks

Content Provider	Springer Nature Link
Author	Chatterjee, Sourav Pal, Soumik
Copyright Year	2009
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 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson–Dirichlet distribution. The underlying idea of the proof is inspired by 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.
Ending Page	159
Page Count	37
Starting Page	123
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	1
Volume Number	147
Language	English
Publisher	Springer-Verlag
Publisher Date	2009-03-17
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Rank-dependent processes McKean–Vlasov General theory of processes Poisson–Dirichlet Theoretical, Mathematical and Computational Physics Interacting random processes; statistical mechanics type models; percolation theory Probability Theory and Stochastic Processes Interacting diffusions Quantitative Finance Atlas model Point processes Statistics for Business/Economics/Mathematical Finance/Insurance Operations Research/Decision Theory Mathematical Biology in General Phase transition
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

A phase transition behavior for Brownian motions interacting through their ranks

Self-interacting diffusions: a simulated annealing version

Weak convergence of the scaled median of independent Brownian motions

Is the critical percolation probability local?

Percolation in the vacant set of Poisson cylinders

Percolation on the three dot system

Excited Brownian motions as limits of excited random walks

On conditional McKean Lagrangian stochastic models

Percolation on random Johnson–Mehl tessellations and related models

A phase transition behavior for Brownian motions interacting through their ranks

Similar Documents

A phase transition behavior for Brownian motions interacting through their ranks

Self-interacting diffusions: a simulated annealing version

Weak convergence of the scaled median of independent Brownian motions

Is the critical percolation probability local?

Percolation in the vacant set of Poisson cylinders

Percolation on the three dot system

Excited Brownian motions as limits of excited random walks

On conditional McKean Lagrangian stochastic models

Percolation on random Johnson–Mehl tessellations and related models

A phase transition behavior for Brownian motions interacting through their ranks