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

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Abstract

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.

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References

  1. Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  2. Beatson, R., Greengard, L.: A short course on fast multipole methods. In: Wavelets, Multilevel Methods and Elliptic PDEs, Leicester, 1996. Numer. Math. Sci. Comput., pp. 1–37. Oxford Univ. Press, New York (1997)

    Google Scholar 

  3. Bessis, D., Itzykson, C., Zuber, J.B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1(2), 109–157 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150(1), 185–266 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bornemann, F.: On the numerical evaluation of distributions in random matrix theory: a review. Markov Process. Relat. Fields 16(4), 803–866 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79(270), 871–915 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3. New York University Courant Institute of Mathematical Sciences, New York (1999)

    Google Scholar 

  8. Deift, P., Gioev, D.: Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics, vol. 18. Courant Institute of Mathematical Sciences, New York (2009)

    Google Scholar 

  9. Duchon, P., Flajolet, P., Louchard, G., Schaeffer, G.: Boltzmann samplers for the random generation of combinatorial structures. Comb. Probab. Comput. 13(4–5), 577–625 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43(11), 5830–5847 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Erdős, L., Yau, H.-T.: Universality of local spectral statistics of random matrices. Bull. Am. Math. Soc. 49(3), 377–414 (2012)

    Article  Google Scholar 

  13. Gautschi, W.: Orthogonal polynomials: applications and computation. Acta Numer. 5, 45–119 (1996)

    Article  MathSciNet  Google Scholar 

  14. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001) (electronic)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series, vol. 51. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  16. Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc., Math. Phys. Eng. Sci. 467(2130), 1563–1576 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22(4), 1611–1641 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  19. Mehta, M.L.: Random Matrices, 2nd edn. Academic Press, Boston (1991)

    MATH  Google Scholar 

  20. Mezzadri, F.: How to generate random matrices from the classical compact groups. Not. Am. Math. Soc. 54(5), 592–604 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Pfrang, C.W., Deift, P., Menon, G.: How long does it take to compute the eigenvalues of a random symmetric matrix? arXiv:1203.4635 (2012)

  22. Rao, N.R., Olver, S., Trogdon, T.: Sampling invariant ensembles (2013, in preparation)

  23. Stewart, G.W.: The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numer. Anal. 17(3), 403–409 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Yang, X., Choi, M., Lin, G., Karniadakis, G.E.: Adaptive ANOVA decomposition of stochastic incompressible and compressible flows. J. Comput. Phys. 231(4), 1587–1614 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Acknowledgements

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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  1. Brown University, Providence, RI, USA

    Xingjie Helen Li & Govind Menon

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  1. Xingjie Helen Li
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  2. Govind Menon
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Correspondence to Govind Menon.

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Li, X.H., Menon, G. Numerical Solution of Dyson Brownian Motion and a Sampling Scheme for Invariant Matrix Ensembles. J Stat Phys 153, 801–812 (2013). https://doi.org/10.1007/s10955-013-0858-x

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