Conformal mapping and the random site problem

CONFORMAL

16 May 1988

PHYSICS LETTERS A

Volume 129, number 3

MAPPING

AND THE RANDOM

SITE PROBLEM

E. AHMED a and A. TAWANSI b a Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt b Department of Physics, Faculty of Science, Mansoura University, Mansoura, Egypt

Received 11 January 1988; revised manuscript received 2 1March 1988; accepted for publication 21 March 1988 Communicated by A.A. Maradudin

We prove that for the overlapping figure construction of the two-dimensional random site problem, the percolation threshold of two curves related by a conformal mapping is the same.

In random site percolation problems [ 11, sites are chaotically distributed. Let r, be the position vector of the ith site and define rij by r,j

=

r, - r, .

(1)

Let Q be some function of rip The two sites are considered to be in the same cluster if

mensional cases for which an OLF construction exists. The bonding criterion is that two sites belong to the same cluster if the surfaces d(r) around them intersect. Now we can prove the following theorem: Theorem. If a curve #’ (x’, , x; ) is obtained from the curve @(xl, x2) by a conformal mapping

x;=fk(xI,x*), tij

for some r. The percolation threshold & is the lowest
&r.

(3)

Following ref. [2] there are two formulations of this problem. The first is the inclusive figure construction (IF) in which one draws circles (spheres) with radii r around each site. Two sites belong to the same cluster if one of them is inside the ‘circle (sphere) around the other. The second formulation is the overlapping figure construction (OLF) in which one draws circles (spheres) with radii r/2 around each site. Two sites belong to the same cluster if the two circles (spheres) around them overlap. We will consider the case 6 as a function of rij r,=@(r,,),

k=l,2,

(5)

(2)

Q t

(4)

where $ is a positive homogeneous and increasing function of r. Eq. (4) represents a surface, called the bonding criterion surface. In this Letter we will only consider the two-di-

then the percolation threshold of the two curves is the same. Proof: Consider two sites with coordinates (x,, x2) and (J+, y2 1. Assuming that the surfaces around them are

@(Xl,x2)=4/2,

(6)

(7)

903,.Y2)=~/2,

let x~=fk(xL,x2)~

JJ;=fk(Yl,Y2),

k=1,2

(8)

be a conformal transformation that maps the curves (6) and (7) onto the curves @(I) and tic2). Now, if the two sites belong to the same cluster, the two curves (6) and (7) intersect in points, say, A, B. An important property of conformal mappings is that they map boundaries onto boundaries [ 31. Therefore the image of A and B under ( 8 ) will be on the boundaries of both e(l) and @c2),i.e. @(I) and @(‘) overlap if the original curves (6) and (7) intersect. Since the conformal mappings form a group

0375-9601/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

151

Volume 129. number

3

PHYSICS

the curves @(I) and g(2) overlap if and only if the curves (6) and (7 ) overlap. Therefore we conclude that if two sites belong to the same cluster, they will still belong to the same cluster after being conformally transformed and vice versa, i.e. percolation threshold is invariant under conformal mapping. This completes the proof of the theorem. Now let us recall Riemann’s theorem. It states that every singly connected domain of the complex plane whose boundary consists of more than one point can be conformally mapped onto the interior of the unit circle. Hence we derive the following corollary: Corollary. If a problem admits an overlapping figure construction, then all two-dimensional piecewise smooth closed curves have the same percolation threshold.

This explains and generalizes the results of ref. [ 21 in which it was found that for some curves the shape of the bonding criterion surface (4) has little effect on the percolation threshold. It is also a generalisation in two dimensions of the Sinai theorem [ 4 1. Thus we have proven that, for two systems with bonding surfaces related by a conformal mapping,

152

LETTERS

A

16 May 1988

there is a one to one correspondence between their cluster structure. Hence they same have the same percolation probability P(p). Therefore they have the same critical exponent j? defined by P(P)

N (p-pC)s. P-P,’

(9)

This is a special case of the universality hypothesis [ 5 1. It states that given the dimensionality, the critical exponents are independent of the lattice structure and the type of the problem. This hypothesis has been confirmed experimentally [6] and by computer simulation [4]. Yet it has never been mathematically proven. We have presented here a mathematical proof for a special case of this hypothesis.

References [ I] D. Stauffer, Introduction

I

to percolation theory (Taylor and Francis, London, 1985). G. Pike and C. Seager, Phys. Rev. B 10 ( 1974) 142 1. A. Sveshnikov and A. Tiknonov, The theory of functions of a complex variable (Mir, Moscow, 1982). B. Shklovskii and A. Efros, Electronic properties of doped semiconductors (Springer, Berlin, 1984). D. Amit, Field theory, renormalisation group and critical phenomena (McGraw-Hill, New York, 1978). T. Noh, S. Lee and J. Gaines, Phys. Lett. A 114 ( 1986) 207.