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On the concentration of eigenvalues of random symmetric matrices

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Abstract

It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e − t 232s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces.

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Author notes

  1. Van H. Vu

    Present address: Department of Mathematics, UCSD, 92093-0112, La Jolla, CA, USA

Authors and Affiliations

  1. Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Noga Alon & Michael Krivelevich

  2. Microsoft Research, 1 Microsoft Way, 98052, Redmond, WA, USA

    Van H. Vu

Authors
  1. Noga Alon
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  2. Michael Krivelevich
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  3. Van H. Vu
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Corresponding author

Correspondence to Noga Alon.

Additional information

Research supported in part by a USA — Israel BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

Research supported in part by a USA — Israel BSF grant and by a Bergmann Memorial Grant.

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Alon, N., Krivelevich, M. & Vu, V.H. On the concentration of eigenvalues of random symmetric matrices. Isr. J. Math. 131, 259–267 (2002). https://doi.org/10.1007/BF02785860

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  • DOI: https://doi.org/10.1007/BF02785860

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