Abstract
We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of “small” eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
L. Arnold, On the asymptotic distribution of the eigenvalues of random matrices, J. Math. Analysis and Appl. 20:262–268 (1967).
M. Bauer and O. Golinelli, On the kernel of tree incidence matrices, Journal of Integer Sequences, Article 00.1.4, Vol. 3 (2000), (www.research.att.com/~njas/sequences/JIS); cond-mat/0003049.
M. Bauer and O. Golinelli, Random incidence matrices: spectral density at zero energy, Saclay preprint T00/087; cond-mat/0006472.
M. Bauer and O. Golinelli, Spectra of random forests, in preparation.
G. Biroli and R. Monasson, A single defect approximation for localized states on random lattices, J. Phys. A: Math. Gen. 32:L255-L261 (1999).
B. Bollobàs, Modern Graph Theory, Graduate text in mathematics 184 (Springer, 1998).
A. Cavagna, I. Giardina, and G. Parisi, Analytic computation of the instantaneous normal modes spectrum in low-density liquids, Phys. Rev. Lett. 83:108–111 (1999).
P. Di Francesco, O. Golinelli, and E. Guitter, Meanders: a direct enumeration approach, Nucl. Phys. B 482:497–535 (1996), or preprint hep-th/9607039.
P. Erdös and A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5:17–61 (1960).
S. N. Evangelou, Quantum percolation and the Anderson transition in dilute systems, Phys. Rev. B 27:1397–1400 (1983).
S. N. Evangelou and E. N. Economou, Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble, Phys. Rev. Lett. 68:361–364 (1992).
S. N. Evangelou, A numerical study of sparse random matrices, J. Stat. Phys. 69:361–383 (1992).
Z. Füredi and J. Komlós, The eigenvalues of random symmetric matrices, Combinatorica 1:233–241 (1981).
F. R. Gantmacher, Matrix Theory (Chelsea, New York, 1964).
A. Brooks Harris, Exact solution of a model of localization, Phys. Rev. Lett. 49:296–299 (1982).
D. Jakobson, S. D. Miller, I. Rivin, and Z. Rudnick, Eigenvalue spacings for regular graphs, in Emerging Applications of Number Theory, D. A. Hejhal et al., eds. (Springer, 1999).
F. Juhász, On the spectrum of a random graph, in Algebraic Methods in Graph Theory, Lovász et al., eds., Coll. Math. Soc. J. Bolyai 25 (North-Holland, 1981), pp. 313–316.
S. Kirkpatrick and T. P. Eggarter, Localized states of a binary alloy, Phys. Rev. B 6:3598–3609 (1972).
M. L. Mehta, Random Matrices, second edition (Academic Press, 1991).
A. D. Mirlin and Y. V. Fyodorov, Universality of level correlation function of sparse random matrices, J. Phys. A: Math. Gen. 24:2273–2286 (1991).
G. J. Rodgers and A. J. Bray, Density of states of a sparse random matrix, Phys. Rev. B 37:3557–3562 (1988).
G. J. Rodgers and C. De Dominicis, Density of states of sparse random matrices, J. Phys. A: Math. Gen. 23:1567–1573 (1990).
N. J. A. Sloane, The encyclopedia of integer sequences, sequence A000296. (http://www.research.att.com/~njas/sequences/index.html)
R.-P. Stanley, Enumerative Combinatorics, Vol. II (Cambridge University Press, Cambridge, 1997).
E. P. Wigner, The Collected Works of Eugene Paul Wigner, Part A, Vol. IV (Springer Verlag, Berlin, 1997), pp. 131–136.
E. P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67:325–327 (1958).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bauer, M., Golinelli, O. Random Incidence Matrices: Moments of the Spectral Density. Journal of Statistical Physics 103, 301–337 (2001). https://doi.org/10.1023/A:1004879905284
Issue Date:
DOI: https://doi.org/10.1023/A:1004879905284