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On the norm of a random jointly exchangeable matrix

Author

Listed:
  • Konstantin Tikhomirov

    (Princeton University)

  • Pierre Youssef

    (Université Paris Diderot)

Abstract

In this note, we show that the norm of an $$n\times n$$ n × n random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $$\lfloor n/2\rfloor \times \lfloor n/2\rfloor $$ ⌊ n / 2 ⌋ × ⌊ n / 2 ⌋ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and its top right $$\lfloor n/2\rfloor \times \lfloor n/2\rfloor $$ ⌊ n / 2 ⌋ × ⌊ n / 2 ⌋ submatrix. The result has an application to estimating the spectral gap of random undirected d-regular graphs in terms of the second singular value of directed random graphs with predefined degree sequences.

Suggested Citation

  • Konstantin Tikhomirov & Pierre Youssef, 2019. "On the norm of a random jointly exchangeable matrix," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1990-2005, December.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:4:d:10.1007_s10959-018-0844-y
    DOI: 10.1007/s10959-018-0844-y
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    References listed on IDEAS

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    1. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
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