About topological invariants and statistical mechanics

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Physica A 358 (2005) 22–29 www.elsevier.com/locate/physa

About topological invariants and statistical mechanics D. Gandolfo PhyMat, Math Department, UTV, BP 132, F-83957 La Garde Cedex & CPT-CNRS, Luminy case 907, F-13288 Marseille, France Available online 29 June 2005

Abstract A short introduction on topological properties of (regular and random) geometrical sets is presented along with some recent results concerning the behaviour of the Euler–Poincare´ characteristic with respect to the (Fortuin–Kasteleyn) random cluster measure. r 2005 Elsevier B.V. All rights reserved. Keywords: Topological invariants; Euler–Poincare´ characteristic; Fortuin–Kasteleyn representation; Alexander duality; Phase transitions

1. Introduction Recently, in the study of the critical properties of clusters in percolation theory new insights have emerged based on ideas from mathematical morphology [1] and integral geometry [2,3]. These mathematical theories provide a set of geometrical and topological measures enabling to quantify the morphological properties of random systems. In particular these tools have been applied to the study of random cluster configurations in percolation theory and statistical physics [4–7]. Euler’s formula for planar graphs and convex polyhedra V
E þ F ¼ 2 relating the number of vertices, edges and faces of such objects has been generalized by Poincare´ to more complex sets.

E-mail address: [email protected]. 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.06.003

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The Euler–Poincare´ characteristic is a well-known descriptor of the topological features of geometric patterns. It belongs to the finite set of Minkowski functionals whose origin lies in the mathematical study of convex bodies and integral geometry (see [2,3]). The application of these description tools for the study of random systems in statistical mechanics has already provided interesting results. In [4,5], the computation of the Euler–Poincare´ characteristic w for a system of penetrable disks in several models of continuum percolation has led to conjecture of new bounds for the critical value of the continuum percolation density. In [4], an exact calculation shows that a close relation exists between the zero of the Euler–Poincare´ characteristic and the critical threshold for continuum percolation in 2 and 3 dimensions. Of similar interest in the same domain, we recall that for the problem of bond percolation on regular lattices, Sykes and Essam [8] were able to show, using standard planar duality arguments, that for the case of self-dual matching lattices (e.g. Z2 ), the mean value of the Euler–Poincare´ characteristic changes sign at the critical point (this even led them to announce a proof for the value of the critical probability of bond percolation on Z2 ). More recently, Wagner [6] was able to compute the Euler–Poincare´ characteristic of all plane regular mosaics (the 11 Archimedean lattices) as a function of the site occupancy probability p 2 ½0; 1 and showed that a close connection exists between the threshold for site percolation on these lattices and the point where the Euler–Poincare´ characteristic (expressed as a function of p) changes sign. In the case of the Fortuin–Kasteleyn representation of the q-state Potts model on Zd the Euler–Poincare´ characteristic shows a nontrivial behaviour at the transition point. (The details of proofs may be found in [9].)

2. Euler characteristic on Zd 2.1. The 2-dimensional case Let us consider the square lattice Z2 ¼ fx ¼ ðx1 ; x2 Þ : xi 2 Z; i ¼ 1; 2g whose elements are called sites. Two sites x and y are nearest neighbours if jx1
y1 j þ jx2
y2 j ¼ 1. We call bonds b ¼ hxyi the subsets of R2 which are the straight line segments with the nearest-neighbours sites x and y as endpoints. We call plaquettes p ¼ ½x; y; z; t the subsets of R2 which are unit squares whose corners are the sites x; y; z; t. The boundary qb of the bond b ¼ hxyi is the set fx; yg and the boundary qp of the plaquette p ¼ ½x; y; z; t is the set of bonds fhxyi; hyzi; hzti; htxig. With this structure the lattice becomes a cell complex L ¼ fL0 ; L1 ; L2 g, where L0 ¼ Z2 is the set of sites, L1 is the set of bonds and L2 is the set of plaquettes (see e.g. [10]). More generally, the elements of Lp will be called p-cells. We also introduce the coboundary dx ¼ fhxyi 2 L1 : y 2 L0 g.

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To a set of bonds X  L1 we associate the subcomplex LðX Þ  L defined as the maximal closed subcomplex that contains X: this subcomplex LðX Þ is the union L0 ðX Þ [ L1 ðX Þ [ L2 ðX Þ where L1 ðX Þ ¼ X , is the set of bonds itself, L0 ðX Þ ¼ fx 2 L0 : x \ X a;g , is the set of sites which are endpoints of bonds of X, and L2 ðX Þ ¼ fp ¼ ½x; y; z; t 2 L2 : hxyi; hyzi; hzti; htxi 2 X g , is the set of plaquettes whose all four bounds in their boundary belong to X. For a set of bonds X, we define the boundary BðX Þ of X as the set of bonds of X that belong to a plaquette of L2 nL2 ðX Þ. Fig. 1 shows a cell complex LðX Þ associated with a given set of bonds X. We also associate to the set of bonds X the occupation number ( 1 if b 2 X ; nb ¼ (2.1) 0 otherwise ; and denote by n  fnb gb2L1 the associated configuration. We then use N 0 ðnÞ  jL0 ðX Þj to denote the number of sites of L0 ðX Þ, N 1 ðnÞ  jL1 ðX Þj to denote the number of bonds of L1 ðX Þ, and N 2 ðnÞ  jL2 ðX Þj to denote the number of plaquettes of L2 ðX Þ. Hereafter, we will use jEj to denote the cardinality of the set E. The Euler characteristic of LðX Þ is defined by vðnÞ ¼ N 0 ðnÞ
N 1 ðnÞ þ N 2 ðnÞ ,

Fig. 1. A set of bonds X and its associated cell complex LðX Þ.

(2.2)

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It satisfies the Euler–Poincare´ formula vðnÞ ¼ p0 ðnÞ
p1 ðnÞ þ p2 ðnÞ , 0

(2.3)

1

where p ðnÞ and p ðnÞ are, respectively, the number of connected components and the maximal number of independent cycles of LðX Þ. Here p2 ðnÞ ¼ 0 because LðX Þ is a two-dimensional closed complex. Our aim is to study the mean value of the Euler characteristic with respect to the Fortuin–Kasteleyn measure. To introduce this measure we define, for a ‘‘volume’’ V  L1 , the partition functions with boundary conditions X 1 0 0 ðeb
1ÞN ðnÞ qjL0 ðV Þj
N ðnÞþp ðnÞ kbc ðnÞ , (2.4) Z bc ðV ; b; qÞ ¼ n2f0;1gV

and the corresponding finite volume expectations of local functions f of the bonds variable fnb g are X 1 0 1 0 hf ibc ðV ; b; qÞ ¼ bc f ðeb
1ÞN ðnÞ qjL0 ðV Þj
N ðnÞþp ðnÞ kbc ðnÞ , (2.5) Z ðV ; b; qÞ n2f0;1gV where kbc ðnÞ refers to the boundary condition bc. In particular, we will be interested f in the free boundary Q condition k ðnÞ ¼ 1 and in the ordered or wired boundary w condition k ðnÞ ¼ b2BðV Þ nb . Notice that for a configuration n 2 f0; 1gjV j , one has Y X N 0 ðnÞ ¼ L0 ðV Þ
ð1
nb Þ , x2L0 ðV Þ b2dx\V

N 1 ðnÞ ¼

X 1 X nb , 2 x2L ðV Þ b2dx\V 0

N 2 ðnÞ ¼

X 1 X Y nb , 4 x2L ðV Þ p2PðxÞ b2qp\V 0

where PðxÞ is the set of the four plaquettes that intersect x. This representation motivates the following definition of the local Euler characteristic: Y 1X 1 X Y wx ðnÞ  1
ð1
nb Þ
nb þ nb . (2.6) 2 b2dx 4 p2PðxÞ b2qp b2dx 2.1.1. Results Our first result relies on existence properties. Theorem 1. (a) The following limits exist and coincide:  f v lim ðV ; b; qÞ ¼ lim hw0 if ðV ; b; qÞ  wf ðb; qÞ V !L1 jL0 ðV Þj V !L1

(2.7)

and define the mean local Euler characteristic with free boundary condition wf ðb; qÞ.

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(b) Analogously, the following limits exist and coincide:  w v lim ðV ; b; qÞ ¼ lim hw0 iw ðV ; b; qÞ  ww ðb; qÞ V !L1 jL0 ðV Þj V !L1

(2.8)

and define the mean local Euler characteristic with wired boundary condition ww ðb; qÞ. Proof. The proofs of these results are standard consequences of FKG inequalities [11]. & Theorem 2. (a) The mean local Euler characteristics satisfy the duality relation wf ðb; qÞ ¼
ww ðb ; qÞ ,

(2.9)

where the dual inverse temperature b is given by 

ðeb
1Þðeb
1Þ ¼ q .

(2.10) pffiffiffi  (b) At the self-dual point bt ¼ logð1 þ qÞ solution of (2.10) with b ¼ b we have wf ðbt ; qÞ ¼
ww ðbt ; qÞ .

(2.11)

Proof. The proof makes use of Alexander duality properties allowing to establish one-to-one correspondence between FK measures with free and wired bc. & Remark 3. Let us mention that for every babt wf ðb; qÞ ¼ ww ðb; qÞ ,

(2.12)

for q ¼ 2 and qX4. This is a consequence of differentiability of free energy with respect to b for the corresponding values of q combined with FKG inequality (see [12]) and [13–15]. Let us also mention that for the percolation model (q ¼ 1) the free energy is obviously differentiable for any (inverse) temperature b. Therefore, in that case wf ðb; q ¼ 1Þ ¼ ww ðb; q ¼ 1Þ ¼ wbc ðb; q ¼ 1Þ. This implies, by Theorem 2 that at the transition point bt ¼ ln 2, one has wbc ðbt ; q ¼ 1Þ ¼ 0 (for any bc). Theorem 4. Assume q is large enough. Then, the mean local Euler characteristic is discontinuous at bt : wf ðbt ; qÞ ¼
ww ðbt ; qÞ ¼ 2q
1=2 þ Oðq
7=12 Þ .

(2.13)

Proof. The proof is mainly based on a contour representation of the model derived in [16]. & 2.2. Euler characteristic on Zd ; dX3 It turns out that the results concerning the behaviour of the mean Euler–Poincare´ characteristic per site at the transition point remain valid in higher dimensions

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(dX3). Contour representation together with Pirogov–Sinaı¨ theory must be used as in dimension 2 to prove discontinuities of the Euler–Poincare´ characteristic. However, in dimensions larger than 2 Alexander duality relates the model to gauge or hypergauge models (see [9]).

3. Conclusion and outlooks The mean Euler–Poincare´ characteristic per site of the FK measure exhibits a nontrivial behaviour at the transition point. It satisfies, in dimension two, a duality property implying that at the transition point it is either zero or it is discontinuous. For the percolation model (q ¼ 1) it actually takes the value zero at the transition point. Moreover, for large q, using an analysis of the order–disorder transition, a jump of order Oðq
1=2 Þ occurs. See Fig. 2 for q ¼ 5000. In higher dimensions, the magnitude of the jump is of order Oðq
1=d Þ for even space dimensions and Oðq
ðd
1Þ=d Þ for odd space dimensions. For small values of q (to which the above-mentioned analysis does not apply) it would be, in particular, interesting to see whether the mean local Euler–Poincare´ characteristic is continuous and takes the value 0 when the transition is second order, as it is the case for the percolation model. Numerical simulations performed in [17] indicate that this is actually the case. We illustrate in Fig. 3, when d ¼ 2 and q ¼ 2, the behaviour of the mean local Euler–Poincare´ characteristic as a function of the temperature that confirms this prediction. In the numerics we have directly sampled the FK distribution through the heat bath Monte Carlo method (Glauber dynamics). As we needed the number of connected components to compute the weights of this distribution, the Hoshen– Kopelman [18] cluster algorithm has been used. We think that the fact that the mean local Euler–Poincare´ characteristic disappears at the transition gives new nontrivial information on the topology of typical configurations at this point.

0.03

1

0.02

0.8

0.01 0.6 0 0.4 -0.01 0.2

-0.02 -0.03

2

3

4

5

6

0

2

3

4

5

6

Fig. 2. Mean Euler characteristic per site and mean bond occupation number as functions of b, for q ¼ 5000 in dimension d ¼ 2.

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0.2

0.1

0

-0.1

-0.2 0.4

0.6

0.8

1

1.2

1.4

Fig. 3. Mean Euler characteristic per site as a function of b, for q ¼ 2 in dimension d ¼ 2.

Acknowledgements This work results from collaborations with Ph. Blanchard and S. Fortunato from the Theoretical Physics Department of Bielefeld University, Germany [17] and J. Ruiz and S. Shlosman from CTP-CNRS Marseille, France [9].

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10]

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