The resistance between arbitrary nodes of an infinite network of resistors is calculated when the network is perturbed by removing one bond from the perfect lattice. A relation is given between the resistance and the lattice Green’s function of the perturbed resistor network. By solving the Dyson equation, the Green’s function and the resistance of the perturbed lattice are expressed in terms of those of the perfect lattice. Numerical results are presented for a square lattice.

## REFERENCES

1.

B. van der Pol and H. Bremmer,

*Operational Calculus Based on the Two-Sided Laplace Integral*(Cambridge U.P., Cambridge, 1955), 2nd ed., p. 372.2.

P. G. Doyle and J. L. Snell,

*Random Walks and Electric Networks*, The Carus Mathematical Monograph, Series 22 (The Mathematical Association of America, 1984), pp. 83–149.3.

G.

Venezian

, “On the resistance between two points on a grid

,” Am. J. Phys.

62

, 1000

–1004

(1994

).4.

D.

Atkinson

and F. J.

van Steenwijk

, “Infinite resistive lattices

,” Am. J. Phys.

67

, 486

–492

(1999

).5.

J.

Cserti

, “Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors

,” Am. J. Phys.

68

, 896

–906

(2000

).6.

E. N. Economou,

*Green’s Functions in Quantum Physics*(Springer-Verlag, Berlin, 1983), 2nd ed.7.

S.

Katsura

, T.

Morita

, S.

Inawashiro

, T.

Horiguchi

, and Y.

Abe

, “Lattice Green’s Function. Introduction

,” J. Math. Phys.

12

, 892

–895

(1971

).8.

R. E.

Aitchison

, “Resistance between adjacent points of Liebman mesh

,” Am. J. Phys.

32

, 566

(1964

).9.

S.

Kirkpatrick

, “Percolation and Conduction

,” Rev. Mod. Phys.

45

, 574

–588

(1973

).10.

F. Schwabl,

*Quantum Mechanics*(Springer-Verlag, Berlin, 1988).11.

K.

Wu

and R. M.

Bradley

, “Efficient Green’s-function approach to finding the currents in a random resistor network

,” Phys. Rev. E

49

, 1712

–1725

(1994

).12.

M. L.

Glasser

and J.

Boersma

, “Exact values for the cubic lattice Green functions

,” J. Phys. A

33

, 5017

–5023

(2000

).13.

J.

Koplik

, “On the effective medium theory of random linear networks

,” J. Phys. C

14

, 4821

–4837

(1981

).14.

S.

Redner

, “Conductivity of random resistor-diode networks

,” Phys. Rev. B

25

, 5646

–5655

(1982

).15.

P. M.

Duxbury

, P. L.

Leath

, and P. D.

Beale

, “Breakdown properties of quenched random systems: The random-fuse network

,” Phys. Rev. B

36

, 367

–380

(1987

).16.

J.

Boksiner

and P. L.

Leath

, “Dielectric breakdown in media with defects

,” Phys. Rev. E

57

, 3531

–3541

(1998

).17.

P. M. Chaikin and T. C. Lubensky,

*Principles of Condensed Matter Physics*(Cambridge U.P., Cambridge, 1995), Chap. 2;J. M. Ziman,

*Principles of The Theory of Solids*(Cambridge U.P., Cambridge, 1972), Chap. 1;N. W. Ashcroft and N. D. Mermin,

*Solid State Physics*(Saunders College, Philadelphia, PA, 1976), Chap. 4;C. Kittel,

*Introduction to Solid State Physics*(Wiley, New York, 1986), 6th ed., pp. 37–42.18.

M.

Jeng

, “Random walks and effective resistances on toroidal and cylindrical grids

,” Am. J. Phys.

68

, 37

–40

(2000

).
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