Journal of Statistical Physics

, Volume 153, Issue 5, pp 801–812 | Cite as

Numerical Solution of Dyson Brownian Motion and a Sampling Scheme for Invariant Matrix Ensembles

  • Xingjie Helen Li
  • Govind Menon


The Dyson Brownian Motion (DBM) describes the stochastic evolution of N points on the line driven by an applied potential, a Coulombic repulsion and identical, independent Brownian forcing at each point. We use an explicit tamed Euler scheme to numerically solve the Dyson Brownian motion and sample the equilibrium measure for non-quadratic potentials. The Coulomb repulsion is too singular for the SDE to satisfy the hypotheses of rigorous convergence proofs for tamed Euler schemes (Hutzenthaler et al. in Ann. Appl. Probab. 22(4):1611–1641, 2012). Nevertheless, in practice the scheme is observed to be stable for time steps of O(1/N 2) and to relax exponentially fast to the equilibrium measure with a rate constant of O(1) independent of N. Further, this convergence rate appears to improve with N in accordance with O(1/N) relaxation of local statistics of the Dyson Brownian motion. This allows us to use the Dyson Brownian motion to sample N×N Hermitian matrices from the invariant ensembles. The computational cost of generating M independent samples is O(MN 4) with a naive scheme, and O(MN 3logN) when a fast multipole method is used to evaluate the Coulomb interaction.


Random matrix theory Invariant matrix ensembles Dyson Brownian motion (DBM) Explicit tamed Euler scheme 



We thank Percy Deift for several stimulating discussions on numerical computations in random matrix theory. We thank Xiu Yang for assistance with the methods of [24]. This work has been supported in part by NSF grant DMS 07-48482.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Brown UniversityProvidenceUSA

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