Asymptotics in random $(0, 1)$-matrices
Author:
Patrick Eugene O’Neil
Journal:
Proc. Amer. Math. Soc. 25 (1970), 39-45
MSC:
Primary 05.25
DOI:
https://doi.org/10.1090/S0002-9939-1970-0258657-5
MathSciNet review:
0258657
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Abstract | References | Similar Articles | Additional Information
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 $i > {n^{3/2 + \epsilon }}$ 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.
- P. Erdős and A. Rényi, On random matrices, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1964), 455–461 (1964) (English, with Russian summary). MR 167496
- P. Erdős and A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61 (English, with Russian summary). MR 125031
- Patrick Eugene O’Neil, Asymptotics and random matrices with row-sum and column-sum restrictions, Bull. Amer. Math. Soc. 75 (1969), 1276–1282. MR 257116, DOI https://doi.org/10.1090/S0002-9904-1969-12393-1
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Additional Information
Keywords:
Permanent,
<IMG WIDTH="50" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$(0,1)$">-matrices,
asymptotic enumeration,
direct graph,
hamiltonian circuits,
matchings,
bipartite graph,
probabilistic
Article copyright:
© Copyright 1970
American Mathematical Society