Convergence of the eigenvalue density for Laguerre beta ensembles on short scales

Philippe Sosoe (Princeton University)
Percy Wong (D.E. Shaw & Co.)

Abstract


In this note, we prove that the normalized trace of the resolvent of the beta-Laguerre ensemble eigenvalues is close to the Stieltjes transform of the Marchenko-Pastur (MP) distribution with very high probability, for values of the imaginary part greater than $m^{1+\varepsilon}$. As an immediate corollary, we obtain convergence of the one-point density to the MP law on short scales. The proof serves to illustrate some simplifications of the method introduced in our previous work to prove a local semi-circle law for Gaussian beta-ensembles.

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Pages: 1-18

Publication Date: March 15, 2014

DOI: 10.1214/EJP.v19-2638

References

  • Aizenman, Michael. Localization at weak disorder: some elementary bounds. Special issue dedicated to Elliott H. Lieb. Rev. Math. Phys. 6 (1994), no. 5A, 1163--1182. MR1301371
  • Abraomowitz, M., Segun, I., eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, 1964.
  • Beals, Richard; Wong, Roderick. Special functions. A graduate text. Cambridge Studies in Advanced Mathematics, 126. Cambridge University Press, Cambridge, 2010. x+456 pp. ISBN: 978-0-521-19797-7 MR2683157
  • Cacciapuoti, C., Maltsev, A., Schlein, B., Local Marchenko-Pastur Law at the Hard Edge of Sample Covariance Matrices, preprint, arXiv:1206.1730.
  • Deift, P.; Kriecherbauer, T.; McLaughlin, K. T-R; Venakides, S.; Zhou, X. Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (1999), no. 12, 1491--1552. MR1711036
  • Dumitriu, Ioana. Eigenvalue statistics for beta-ensembles. Thesis (Ph.D.)–Massachusetts Institute of Technology. ProQuest LLC, Ann Arbor, MI, 2003. (no paging). MR2717094
  • Dumitriu, Ioana; Edelman, Alan. Matrix models for beta ensembles. J. Math. Phys. 43 (2002), no. 11, 5830--5847. MR1936554
  • Dumitriu, Ioana; Edelman, Alan. Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models. J. Math. Phys. 47 (2006), no. 6, 063302, 36 pp. MR2239975
  • Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 (2009), no. 3, 815--852. MR2537522
  • Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 (2009), no. 2, 641--655. MR2481753
  • Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN 2010, no. 3, 436--479. MR2587574
  • Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer; Yin, Jun. The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 1, 1--46. MR2919197
  • Erdős, László; Ramírez, José A.; Schlein, Benjamin; Yau, Horng-Tzer. Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15 (2010), no. 18, 526--603. MR2639734
  • Erdös, Schlein, B., Yau, H.-T. and Yin, J. : phThe Local Semi-circle Law for a General Class of Random Matrices, preprint, arXiv:1212.0164.
  • Krasikov, Ilia. Nonnegative quadratic forms and bounds on orthogonal polynomials. J. Approx. Theory 111 (2001), no. 1, 31--49. MR1840019
  • Krasikov, Ilia. Inequalities for orthonormal Laguerre polynomials. J. Approx. Theory 144 (2007), no. 1, 1--26. MR2287374
  • Krasovsky, I. V. Asymptotic distribution of zeros of polynomials satisfying difference equations. J. Comput. Appl. Math. 150 (2003), no. 1, 56--70. MR1946882
  • Marčenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices. (Russian) Mat. Sb. (N.S.) 72 (114) 1967 507--536. MR0208649
  • Popescu, Ionel. General tridiagonal random matrix models, limiting distributions and fluctuations. Probab. Theory Related Fields 144 (2009), no. 1-2, 179--220. MR2480789
  • Sosoe, Philippe; Wong, Percy. Local semicircle law in the bulk for Gaussian $\beta$-ensemble. J. Stat. Phys. 148 (2012), no. 2, 204--232. MR2966359
  • Tao, Terence; Vu, Van. Random covariance matrices: universality of local statistics of eigenvalues. Ann. Probab. 40 (2012), no. 3, 1285--1315. MR2962092
  • Saff, E. B.; Ullman, J. L.; Varga, R. S. Incomplete polynomials: an electrostatics approach. Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), pp. 769--782, Academic Press, New York-London, 1980. MR0602801
  • Wishart, J., The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A, 1928.
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