Effective resistance of random trees

Effective resistance of random trees

Louigi Addario-Berry, Nicolas Broutin, and Gábor Lugosi

Source: Ann. Appl. Probab. Volume 19, Number 3 (2009), 1092-1107.

Abstract

We investigate the effective resistance R_n and conductance C_n between the root and leaves of a binary tree of height n. In this electrical network, the resistance of each edge e at distance d from the root is defined by r_e=2^dX_e where the X_e are i.i.d. positive random variables bounded away from zero and infinity. It is shown that ER_n=nEX_e−(Var (X_e)/EX_e)ln n+O(1) and Var (R_n)=O(1). Moreover, we establish sub-Gaussian tail bounds for R_n. We also discuss some possible extensions to supercritical Galton–Watson trees.

Primary Subjects: 60J45

Secondary Subjects: 31C20

Keywords: Random trees; electrical networks; Efron–Stein inequality

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1245071020
Digital Object Identifier: doi:10.1214/08-AAP572
Zentralblatt MATH identifier: 05580234

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References

[1] Addario-Berry, L. and Reed, B. A. (2009). Minima in branching random walks. Ann. Probab. To appear.

[2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR373040

[3] Barndorff-Nielsen, O. E. (1994). A note on electrical networks and the inverse Gaussian distribution. Adv. in Appl. Probab. 26 63–67.

Mathematical Reviews (MathSciNet): MR1260303

Digital Object Identifier: doi:10.2307/1427579

JSTOR: links.jstor.org

[4] Barndorff-Nielsen, O. E. and Koudou, A. E. (1998). Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. in Appl. Probab. 30 409–424.

Mathematical Reviews (MathSciNet): MR1642846

Digital Object Identifier: doi:10.1239/aap/1035228076

Project Euclid: euclid.aap/1035228076

[5] Benjamini, I. and Rossignol, R. (2008). Submean variance bound for effective resistance on random electric networks. Comm. Math. Phys. 280 445–462.

Mathematical Reviews (MathSciNet): MR2395478

Digital Object Identifier: doi:10.1007/s00220-008-0476-7

[6] Benjamini, I., Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance. Ann. Probab. 31 1970–1978.

Mathematical Reviews (MathSciNet): MR2016607

Digital Object Identifier: doi:10.1214/aop/1068646373

Project Euclid: euclid.aop/1068646373

[7] Boucheron, S., Bousquet, O., Lugosi, G. and Massart, P. (2005). Moment inequalities for functions of independent random variables. Ann. Probab. 33 514–560.

Mathematical Reviews (MathSciNet): MR2123200

Digital Object Identifier: doi:10.1214/009117904000000856

Project Euclid: euclid.aop/1109868590

[8] de Bruijn, N. G. (1981). Asymptotic Methods in Analysis, 3rd ed. Dover Publications, New York.

Mathematical Reviews (MathSciNet): MR671583

[9] Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22. Mathematical Association of America, Washington, DC.

Mathematical Reviews (MathSciNet): MR920811

[10] Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596.

Mathematical Reviews (MathSciNet): MR615434

Digital Object Identifier: doi:10.1214/aos/1176345462

Project Euclid: euclid.aos/1176345462

[11] Falik, D. and Samorodnitsky, A. (2007). Edge-isoperimetric inequalities and influences. Combin. Probab. Comput. 16 693–712.

Mathematical Reviews (MathSciNet): MR2346808

Digital Object Identifier: doi:10.1017/S0963548306008340

[12] Flajolet, P. and Odlyzko, A. (1982). The average height of binary trees and other simple trees. J. Comput. System Sci. 25 171–213.

Mathematical Reviews (MathSciNet): MR680517

Digital Object Identifier: doi:10.1016/0022-0000(82)90004-6

[13] Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931–958.

Mathematical Reviews (MathSciNet): MR1062053

Digital Object Identifier: doi:10.1214/aop/1176990730

Project Euclid: euclid.aop/1176990730

[14] Lyons, R. and Pemantle, R. (1992). Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 125–136.

Mathematical Reviews (MathSciNet): MR1188053

Digital Object Identifier: doi:10.1214/aop/1176989540

Project Euclid: euclid.aop/1176989540

[15] Lyons, R. and Peres, Y. (2006). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge.

[16] Lyons, R., Pemantle, R. and Peres, Y. (1997). Unsolved problems concerning random walks on trees. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). The IMA Volumes in Mathematics and Its Applications 84 223–237. Springer, New York.

Mathematical Reviews (MathSciNet): MR1601753

[17] Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 1229–1241.

Mathematical Reviews (MathSciNet): MR942765

Digital Object Identifier: doi:10.1214/aop/1176991687

Project Euclid: euclid.aop/1176991687

[18] Peres, Y. (1999). Probability on trees: An introductory climb. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Mathematics 1717 193–280. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1746299

[19] Soardi, P. M. (1994). Potential Theory on Infinite Networks. Lecture Notes in Mathematics 1590. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1324344

[20] Steele, J. M. (1986). An Efron–Stein inequality for nonsymmetric statistics. Ann. Statist. 14 753–758.

Mathematical Reviews (MathSciNet): MR840528

Digital Object Identifier: doi:10.1214/aos/1176349952

Project Euclid: euclid.aos/1176349952

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