Geodesic
Eigenvalue Distribution of a Large Weighted Bipartite Random Graph
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2014) no. 2, pp. 240-255 
V. Vengerovsky. Eigenvalue Distribution of a Large Weighted Bipartite Random Graph. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2014) no. 2, pp. 240-255. http://geodesic.mathdoc.fr/item/JMAG_2014_10_2_a4/
@article{JMAG_2014_10_2_a4,
     author = {V. Vengerovsky},
     title = {Eigenvalue {Distribution} of a {Large} {Weighted} {Bipartite} {Random} {Graph}},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {240--255},
     year = {2014},
     volume = {10},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2014_10_2_a4/}
}
TY  - JOUR
AU  - V. Vengerovsky
TI  - Eigenvalue Distribution of a Large Weighted Bipartite Random Graph
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2014
SP  - 240
EP  - 255
VL  - 10
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/JMAG_2014_10_2_a4/
LA  - en
ID  - JMAG_2014_10_2_a4
ER  - 
%0 Journal Article
%A V. Vengerovsky
%T Eigenvalue Distribution of a Large Weighted Bipartite Random Graph
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2014
%P 240-255
%V 10
%N 2
%U http://geodesic.mathdoc.fr/item/JMAG_2014_10_2_a4/
%G en
%F JMAG_2014_10_2_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

We study an eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of the weighted random bipartite graph $\Gamma= \Gamma_{N,p}$. We assume that the graph has $N$ vertices, the ratio of parts is $\displaystyle\frac{\alpha}{1-\alpha}$, and the average number of the edges attached to one vertex is $\alpha p$ or $(1-\alpha) p$. To every edge of the graph $e_{ij}$, we assign the weight given by a random variable $a_{ij}$ with all moments finite. We consider the moments of the normalized eigenvalue counting measure $\sigma_{N,p, \alpha}$ of $A^{(N,p, \alpha)}$. The weak convergence in probability of the normalized eigenvalue counting measures is proved.

[1] M. Bauer, O. Golinelli, Random Incidence Matrices: Spectral Density at Zero Energy, Saclay preprint T00/087, arXiv: cond-mat/0006472

[2] B. Bollobas, Random Graphs, Acad. Press, 1985 | MR | Zbl

[3] M. Bauer, O. Golinelli, “Random Incidence Matrices: Moments and Spectral Density”, J. Stat. Phys., 103 (2001), 301–336 | DOI | MR

[4] Fan R. K. Chung, Spectral Graph Theory, AMS, 1997 | MR

[5] D. M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Acad. Press, 1980 | MR | Zbl

[6] S. N. Evangelou, “Quantum Percolation and the Anderson Transition in Dilute Systems”, Phys. Rev. B, 27 (1983), 1397–1400 | DOI

[7] S. N. Evangelou, E. N. Economou, “Spectral Density Singularities, Level Statistics, and Localization in Sparse Random Matrices”, Phys. Rev. Lett., 68 (1992), 361–364 | DOI | MR

[8] S. N. Evangelou, “A Numerical Study of Sparse Random Matrices”, J. Stat. Phys., 69 (1992), 361–383 | DOI | MR | Zbl

[9] Y. V. Fyodorov, A. D. Mirlin, Strong Eigenfunction Correlations Near the Anderson Localization Transition, arXiv: cond-mat/9612218v1

[10] Ch. Godzil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001 | MR

[11] S. Janson, T. Łuczak, A. Rucinski, Random Graphs, John Wiley Sons, Inc., New York, 2000 | MR

[12] D. Jacobson, S. D. Miller, I. Rivin, Z. Rudnick, “Eigenvalue Spacing for Regular Graphs”, Emerging Applications of Number Theory, eds. D. A. Hejhal et al., Springer-Verlag, 1999 | MR | Zbl

[13] O. Khorunzhy, M. Shcherbina, V. Vengerovsky, “Eigenvalue Distribution of Large Weighted Random Graphs”, J. Math. Phys., 45:4 (2004), 1648–1672 | DOI | MR | Zbl

[14] O. Khorunzhy, B. Khoruzhenko, L. Pastur, M. Shcherbina, The Large-$n$ Limit in Statistical Mechanics and Spectral Theory of Disordered Systems, Phase Transition and Critical Phenomena, 15, Academic Press, 1992

[15] M. L. Mehta, Random Matrices, Academic Press, New York, 1991 | MR | Zbl

[16] A. D. Mirlin, Y. V. Fyodorov, “Universality of the Level Correlation Function of Sparce Random Matrices”, J. Phys. A Math. Jen., 24 (1991), 2273–2286 | DOI | MR | Zbl

[17] G. J. Rodgers, A. J. Bray, “Density of States of a Sparse Random Matrix”, Phys. Rev. B, 37 (1988), 3557–3562 | DOI | MR

[18] G. J. Rodgers, C. De Dominicis, “Density of States of Sparse Random Matrices”, J. Phys. A Math. Jen., 23 (1990), 1567–1566 | DOI | MR

[19] E. P. Wigner, “On the Distribution of the Roots of Certain Symmetric Matrices”, Ann. Math., 67 (1958), 325–327 | DOI | MR | Zbl