Abstract
We analyse the density of states of the random graph Laplacian in the percolating regime. A symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially localised states due to finite subgraphs. We derive a nonlinear integral equation for the integrated DSPC and solve it with a population dynamics algorithm. We discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Mehta, M.: Random Matrices. Academic Press, Boston (1991)
Guhr, T., Müller-Groeling, A., Weidenmüller, H.A.: Phys. Rep. 299, 190 (1998)
Bray, A.J., Rodgers, G.J.: Phys. Rev. B 38, 11461 (1988)
Fyodorov, Y.V., Mirlin, A.D.: J. Phys. A, Math. Gen. 24, 2219 (1991)
Khorunzhy, O., Shcherbina, M., Vengerovsky, V.: J. Math. Phys. 45, 1648 (1994)
Khorunzhy, O., Kirsch, W., Müller, P.: Ann. Appl. Probab. 16, 295 (2006)
Bauer, M., Golinelli, O.: J. Stat. Phys. 103, 301 (2001)
Biroli, G., Monasson, R.: J. Phys. A 32, L255 (1999)
Kühn, R.: J. Phys. A, Math. Gen. 41, 295002 (2008)
Broderix, K., Aspelmeier, T., Hartmann, A.K., Zippelius, A.: Phys. Rev. E 64, 021404 (2001)
Müller, P., Stollmann, P.: arXiv:1002.5000
Erdős, P., Rényi, A.: Magyar Tud. Akad. Mat. Kut. Int. Kőzl. 5, 17 (1960). Reprinted in: Spencer, J. (ed.): The Art of Counting. MIT Press, Cambridge (1973)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1: Functional Analysis. Academic Press, New York (1972)
Chayes, J.T., Chayes, L., Franz, J.R., Sethna, J.P.: J. Phys. A, Math. Gen. 19, L1173 (1986)
Alexander, S., Orbach, R.: J. Phys. Lett. (Paris) 43, L625 (1982)
Barlow, M.T., Járai, A.A., Kumagai, T., Slade, G.: Commun. Math. Phys. 278, 385 (2008)
Kozma, G., Nachmias, A.: Invent. Math. 178, 635 (2009)
Kesten, H.: Ann. Inst. H. Poincaré Probab. Stat. 22, 425 (1986)
Barlow, M.T., Kumagai, T.: Ill. J. Math. 50, 33 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Aspelmeier, T., Zippelius, A. The Integrated Density of States of the Random Graph Laplacian. J Stat Phys 144, 759–773 (2011). https://doi.org/10.1007/s10955-011-0271-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-011-0271-2