The spectral edge of some random band matrices

Abstract

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide ($W \gg N^{5/6}$). Otherwise, a different limiting distribution appears.

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      PAGES = {722--725},
      ISSN = {0002-3264},
      MRCLASS = {26.00},
      MRNUMBER = {22 \#8093},
      MRREVIEWER = {S. Mandelbrojt},
      }
  • [W] W. M. Wang, "On localization and density of states for the random Schrödinger operator," PhD Thesis , Princeton University, 1992.
    @phdthesis{W,
      author={Wang, W. M.},
      TITLE={On localization and density of states for the random Schrödinger operator},
      SCHOOL={Princeton University},
      YEAR={1992},
      }
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    @article {W2, MRKEY = {0095527},
      AUTHOR = {Wigner, Eugene P.},
      TITLE = {On the distribution of the roots of certain symmetric matrices},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {67},
      YEAR = {1958},
      PAGES = {325--327},
      ISSN = {0003-486X},
      MRCLASS = {60.00 (15.00)},
      MRNUMBER = {20 \#2029},
      MRREVIEWER = {J. G. Wendel},
      DOI = {10.2307/1970008},
      ZBLNUMBER = {0085.13203},
      }

Authors

Sasha Sodin

School of Mathematical Sciences
Tel Aviv University
Ramav Aviv, Tel Aviv 69978
Israel