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Teoriya Veroyatnostei i ee Primeneniya, 1970, Volume 15, Issue 2, Pages 200–215 (Mi tvp1705)  

This article is cited in 13 scientific papers (total in 13 papers)

Phase transitions in random graphs

V. E. Stepanov

Moscow
Abstract: To each subgraph $G$ of a complete graph of $m$ vertices statistical weight $w(G)=x^kh^n$ is assigned, where $k=k(G)$ is the number of components and $n=n(G)$ is the number of edges of graph $G$; $x$ and $h>0$. A random graph $\mathscr G_m(x\mid h)$ is defined by the condition that $\mathbf P(\mathscr G_m(x\mid h)=G)=Z_m^{-1}(x\mid h)w(G)$, where $Z_m(x\mid h)$ is a necessary normalizing coefficient. It is proved that there exists a limit
$$ \lim_{m\to\infty}\frac1m\ln Z_m(x\mid y/m)=\chi(x,y). $$
Limit values of density
$$ \rho(x,y)=\lim_{m\to\infty}\frac1m\mathbf En(\mathscr G_m(x\mid y/m)) $$
and disconnectedness
$$ \varkappa(x,y)=\lim_{m\to\infty}\frac1m\mathbf Ek(\mathscr G_m(x\mid y/m)) $$
of random graph $\mathscr G_m(x\mid y/m)$ are expressed in terms of partial derivatives of $\chi(x,y)$.
An investigation of functions $\rho(x,y)$ and $\varkappa(x,y)$ discovers a surprising analogy of the behaviour of these functions to the behaviour of isotherms of physical systems considered in statistical physics. Connections between some properties of functions $\rho(x,y)$ and $\varkappa(x,y)$ and the structure of random graph $\mathscr G_m(x\mid y/m)$ are under investigation.
Received: 17.03.1969
English version:
Theory of Probability and its Applications, 1970, Volume 15, Issue 2, Pages 187–203
DOI: https://doi.org/10.1137/1115027
Bibliographic databases:
Language: Russian
Citation: V. E. Stepanov, “Phase transitions in random graphs”, Teor. Veroyatnost. i Primenen., 15:2 (1970), 200–215; Theory Probab. Appl., 15:2 (1970), 187–203
Citation in format AMSBIB
\Bibitem{Ste70}
\by V.~E.~Stepanov
\paper Phase transitions in random graphs
\jour Teor. Veroyatnost. i Primenen.
\yr 1970
\vol 15
\issue 2
\pages 200--215
\mathnet{http://mi.mathnet.ru/tvp1705}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=270407}
\zmath{https://zbmath.org/?q=an:0225.90048|0213.45901}
\transl
\jour Theory Probab. Appl.
\yr 1970
\vol 15
\issue 2
\pages 187--203
\crossref{https://doi.org/10.1137/1115027}
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  • https://www.mathnet.ru/eng/tvp/v15/i2/p200
  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
     
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