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We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $$n$$ vertices, where $$n\geq1$$ is a fixed positive integer. The method uses a bijection between mappings $$f\colon\{1,\ldots,n\}\to\{1,\ldots,n\}$$ and doubly rooted trees on $$n$$ vertices. The main application is a concentration inequality for the number of vertices connected to an independent set in a uniformly random tree, which is then used to prove partial unimodality of its independent set sequence. So, we give probabilistic arguments for inequalities that often use combinatorial arguments.
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On the number of Hadamard matrices via anti-concentration
Abstract Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$ , where the coordinates of the vector x are restricted to take values in some small subset (e.g. $$\{\pm 1\}$$ ) of the underlying field. The classical ways of bounding this quantity are to use either a rank bound observation due to Odlyzko or a vector anti-concentration inequality due to Halász. The former gives a stronger conclusion except when the number of equations is significantly smaller than the number of variables; even in such situations, the hypotheses of Halász’s inequality are quite hard to verify in practice. In this paper, using a novel approach to the anti-concentration problem for vector sums, we obtain new Halász-type inequalities that beat the Odlyzko bound even in settings where the number of equations is comparable to the number of variables. In addition to being stronger, our inequalities have hypotheses that are considerably easier to verify. We present two applications of our inequalities to combinatorial (random) matrix theory: (i) we obtain the first non-trivial upper bound on the number of $$n\times n$$ Hadamard matrices and (ii) we improve a recent bound of Deneanu and Vu on the probability of normality of a random $$\{\pm 1\}$$ matrix.
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- Award ID(s):
- 1764176
- PAR ID:
- 10407725
- Date Published:
- Journal Name:
- Combinatorics, Probability and Computing
- Volume:
- 31
- Issue:
- 3
- ISSN:
- 0963-5483
- Page Range / eLocation ID:
- 455 to 477
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0963548321000377
