Consider n points, x 1,... , x n , distributed uniformly in [0, 1]d. Form a graph by connecting two points x i and x j if \(\Vert x_i - x_j\Vert \leq r(n)\). This gives a random geometric graph, \(G({\mathcal {X}}_n;r(n))\), which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on \(G({\mathcal {X}}_n; r(n))\) is concentrated, and in fact converges to that of the graph on the deterministic grid.
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References
Bai Z. D. (1993). Convergence rate of expected spectral distributions of large random matrices, part i: Wigner matrices, part ii: sample covariance matrices. Ann. Probab.
Booth L., Bruck J., Franceschetti M., Meester R. (2003). Covering algorithms, continuum percolation and the geometry of wireless networks. Ann. Appl. Probab. 13(2): 722–741
Boyd S., Ghosh A., Prabhakar B., Shah D. (2004). Mixing times of random walk on geometric random graphs. Submitted for publication.
Goel, A., Rai, S., and Krishnamachari, B. (2004). Sharp thresholds for monotone properties in random geometric graphs. In Proceedings of the Symposium on the Theory of Computing, pages 13–23.
Gray, R. M. (1971). Toeplitz and circulant matrices: A review. Revised 2002, URL: http://ee.stanford.edu/ gray/toplitz.pdf.
Guionnet A., Zeituni O. (2000). Concentration of the spectral measure for large matrices. Electron. Commun. Probab. 9, 119–236
Gupta P., Kumar P.R. (1998). Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, chapter Critical power for asymptotic connectivity in wireless networks. Birkhaüser, Boston
Janson S., Luczak T., Ruciński A. (2000). Random Graphs, Wiley Interscience series in discrete mathematics and optimization. Wiley-Interscience, New York
Koltchinskii V.I. (1998). Asymptotics of spectral projections of some random matrices approximating integral operators. In Eberlein, E., Hahn, M., Talagrand, M. (eds.). High Dimensional Probability, volume 43 of Progress in Probability, Birkhäuser, pages 191–227.
Koltchinskii V.I., Giné E. (2000). Random matrix approximation of spectra of integral operator. Bernoulli.
Künnemann R. (1983). The diffusion limit for reversible jump processes on \(\mathbb{Z}^{d}\) with ergodic random bond conductivities. Commun. Math. Phys. 90, 27–68
Leighton F.T., Shor P.W. (1989). Tight bounds for minimax grid matching, with applications to the average case analysis of algorithms. Combinatorica 9, 161–187
Penrose M.D. (2003). Random Geometric Graphs, Volume 5 of Oxford Studies in Probability. Oxford University Press, oxford
Pólya, G., Sze̋go, G. (1998). Problems and Theorems in Analysis I. Springer-Verlag, Berlin (reprint edition).
Shor P.W., Yukich J.E. (1991). Minmax grid matching and empirical measures. Ann. Probab. 19(3), 1338–1348
Trench W.F. (2003). Absolute equal distribution of the spectra of hermitian matrices. Linear Algebra and Its Applications.
Wigner E.P. (1985). On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327
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Rai, S. The Spectrum of a Random Geometric Graph is Concentrated. J Theor Probab 20, 119–132 (2007). https://doi.org/10.1007/s10959-006-0049-7
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DOI: https://doi.org/10.1007/s10959-006-0049-7