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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Asymptotics in random $ (O,\,1)$-matrices


Author: Patrick Eugene O’Neil
Journal: Proc. Amer. Math. Soc. 25 (1970), 290-296
MSC: Primary 05.25
DOI: https://doi.org/10.1090/S0002-9939-1970-0255430-9
MathSciNet review: 0255430
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Abstract: Let $ {M^n}(i)$ be the class of $ n \times n(0,1)$-matrices with $ i$ ones. We wish to find the first and second moments of Perm $ B$, the permanent of the matrix $ B$, as $ B$ ranges over the class $ {M^n}(i)$. We succeed for $ i > {n^{3/2 + \varepsilon }}$ in finding an asymptotic estimate of these quantities. It turns out that the square of the first moment is asymptotic to the second moment, so we may conclude that almost all matrices in $ {M^n}(i)$ have asymptotically the same permanent. It is suggested that the technique employed will also enable us to evaluate asymptotically the number of hamiltonian circuits in a random graph with $ i$ links on $ n$ vertices.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0255430-9
Keywords: Permanent, $ (0,1)$-matrices, asymptotic enumeration, direct graph, hamiltonian circuits, matchings, bipartite graph, probabilistic
Article copyright: © Copyright 1970 American Mathematical Society