On the spectral norm of Gaussian random matrices

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Let X be a d × d symmetric random matrix with independent but nonidentically distributed Gaussian entries. It has been conjectured by Lata̷la that the spectral norm of X is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order √log log d. Moreover, dimension-free bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes.

Original languageEnglish (US)
Pages (from-to)8161-8178
Number of pages18
JournalTransactions of the American Mathematical Society
Issue number11
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


  • Nonasymptotic bounds
  • Random matrices
  • Spectral norm


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