Abstract
The phenomenon of macroscopic homogenization is illustrated with a simple example of diffusion. We examine the conditions under which a d-dimensional simple random walk in a symmetric random media converges to a Brownian motion. For d = 1, both the macroscopic homogeneity condition and the diffusion coeficient can be read from an explicit expression for the Green's function. Except for this case, the two available formulas for the effective diffusion matrix kappa do not explicit show how macroscopic homogenization takes place. Using an electrostatic analogy due to Anshelevich, Khanin and Sinai [AKS], we discuss upper and lower bounds on the diffusion coeficient kappa for d >1.
Brownian motion limit of random walks in symmetric non-homogeneous media
Domingos H. U. Marchetti and Roberto da Silva
Instituto de Física
Universidade de São Paulo
P. O. Box 66 318, 05315-970, São Paulo, SP, Brazil
Received 5 April, 1999
The phenomenon of macroscopic homogenization is illustrated with a simple example of diffusion. We examine the conditions under which a d-dimensional simple random walk in a symmetric random media converges to a Brownian motion. For d = 1, both the macroscopic homogeneity condition and the diffusion coeficient can be read from an explicit expression for the Green's function. Except for this case, the two available formulas for the effective diffusion matrix k do not explicit show how macroscopic homogenization takes place. Using an electrostatic analogy due to Anshelevich, Khanin and Sinai [AKS], we discuss upper and lower bounds on the diffusion coeficient k for d >1.
I Introduction
The long time behavior of random walks on a random environment is reviewed. We focus mainly on the following question:
What are the conditions under which a properly scaled random walk on a non-homogeneous medium converges to a Brownian motion.
This and related phenomena are usually named macroscopic homogenization (and the environmental conditions are called macroscopic homogeneity conditions) because such system looks homogeneous at macroscopic scales. The discussion will be restricted to simple random walks with non-vanishing transition probabilities (or rates) {wáxyñ} satisfying
wáxyñ = wáxyñ (I.1)
for all nearest neighbor sites áxyñ of a d -dimensional lattice . The so-called symmetric medium has been considered by several authors (see e.g. [ABSO, AKS, AV, MFGW, Ku, PV] and references therein).
Except for a basic lemma, the general scheme of our presentation will be dimensional independent. However, the one-dimensional problem plays a central role in this work since, in this case, the macroscopic homogeneity condition can be read from an explicit formula.
The d = 1 case has been mostly investigated. The first mathematical results [KKS, So, Si] were concerned with asymmetric random walks with transition probabilities wx,x+1 = 1-wx,x-1, x Î , being independent and identically distributed (i.i.d.) random variables (note that wx,x+1 ¹ wx+1,x). The trajectories {X(t), t > 0 | X(0) = x0} of asymmetric random walks were shown to behave very anomalously. Symmetric random walks began to be discussed in a series of papers (see e.g. [ABO, BSW, ABSO]) in connection with the problem of disordered chains of harmonic oscillators (see [LM] for an introduction and a selection of reprints) and other problems in physics. Using Dyson's integral equation [D], these authors derived the following asymptotic behavior for the trajectories: if
where denotes the expectation value with respect to the i.i.d. random variables {wx,x+1}, then
as t® ¥, and the convergence of to the Gaussian random variable with zero mean and variance is implied. In addition, if condition (2) is violated, then grows as td with an exponent d < 1 depending on the divergence of the distribution at wx,x+1 = 0.
A mathematical proof of convergence to Brownian motion for d = 1 symmetric random walks was given by Anshelevich and Vologodski [AV]. For d ³ 2 there are at least two different proofs and both require macroscopic homogeneity conditions more stringent than (2). Anshelevich, Khanin and Sinai [AKS] proved the result by developing an expansion for the expected value of the inverse of a non-homogeneous discrete Laplacian. Künnemann [Ku] has proven this result by extending Papanicolaou-Varadhan's approach [PV]. Whether (2) is a necessary (and sufficient) macroscopic homogeneity condition for d ³ 2, remains, to our knowledge, an open problem.
The present paper is inspired by the work of Anshelevich, Khanin and Sinai. We use the logic of this proof in order to simplify the Anshelevich and Vologodski's proofs for d = 1. Our proof, in particular, eliminates the technical hypothesis of wx,x+1 to be strictly positive and illustrates with textbook's mathematical methods the macroscopic homogeneity condition (2).
A Brownian motion is described by the diffusion equation. The random environment induces an effective diffusive matrix k whose elements are given by limt® ¥ (Xi(t)Xj(t)) /t, i, j = 1,¼,d. For the one-dimensional problem, the reciprocal of this constant is given by the macroscopic homogeneity condition: k-1 = wx,x+1-1. For d ³ 2, the two available formulas for the effective diffusion matrix (see [AKS, Ku]) do not explicitly show how macroscopic homogenization takes place and this makes it difficult to obtain estimates. In this review (see also [AKS]) the upper bound
will be shown to hold if |1-wáxyñ /áxyñ | £ d < 1/2 . We also discuss how a lower bound can be obtained using the electrostatic equivalence of the diffusion problem as formulated in [AKS].
The outline of the present work is as follows. In Section II we formulate the problem and state our results. The proofs will be given in Section III under the assumption that the eigenvalues and eigenvectors of the semi-group generator of the process converge to the eigenvalues and eigenfunctions of a Laplacian. The eigenvalue problem is a consequence of our basic lemma (Lemma III.1) which will be proven in Section for d = 1 by Green's function method. The spectral perturbation theory will be presented in Section V. Finally, the diffusion coefficient will be examined in Section VI.
II Statement of Results
Let denote the set of bonds of the regular lattice and let w = {wb}b Î be an assignment of positive numbers. Each component wbrepresents the transition rate of a random walk to go from the site x to y along the bond b = áxyñ. The assignment w defines an symmetric environment on if the transition rates satisfy wáxyñ = wáyxñ .
Given an environment w and a finite set L Ì , a continuous time random walk on L , with absorbing boundary condition, is a Markov process {XL ,w(t), t ³ 0} with differential transition matrix WL = WL (w) defined by
for all x Î L and u such that uy = 0 if y Î \ L .
Note that -DL ,w is a positive matrix,
and, if for all where DL is the finite difference Laplacian with 0-Dirichlet boundary condition on L . From here on, L is taken to be the hypercube centered at origin with size |L| = (2L-1)d, L Î : LL: = { x = (x1,... ,xd) Î :supi|xi| < L} , and all quantities depending on L will be indexed by L instead.
The probability distribution of {XL,w(t), t ³ 0} is governed by the semi-group generated by DL,w. If XL,w(t) denotes the position of a random walk at time t, then
where (u0)x = d0,x.
The semi-group is the solution of the initial value problem in ,
with initial condition u(0) = u0.
The solution of (II.4) exists for all times t > 0 and all sizes L < ¥ but may depend on the realization of w and on the initial value. We present the sufficient conditions on the environment w by which the solution of (II.4), under suitable scaling of time and space, converges almost everywhere in w to the fundamental solution of the heat equation,
with u(t,¶) = 0. Here, (II.5) is defined in the domain t > 0 and x Î : = (-1,1)d with boundary ¶ = {x:supi|xi| = 1}, and ¶2 = ¶2(k) is given by
The heat kernel Tt(h,z) = et ¶2(h,z) when defined in gives rise to a Wiener process (or Brownian motion) {B(t), t ³ 0} with covariance B(t)B(s) = 4k min(s,t) (see e.g. Simon [S]). In view of the boundary condition, Tt yields a Brownian motion {B0(t), t ³ 0} on the domain with absorbing frontiers. The probability density of B0(s+t)-B0(s) to be equal to x can be obtained by solving equations (II.5) by Fourier method
where and sc(nix) stands for either cos(nix) or sin(nix) depending on whether niis an odd or even number.
This discussion suggests the following definition.
Definition II.1The random walk in a random environment w is said to converge to a Brownian motion if there exist a constant k = k(w), the diffusion coefficient, such that
uniformly in t > 0, x Î and m0 on the space of finite measures with support in . For r Î , [r] Î and has as components the integer part of the components of r; and u0Î L is a vector given by
It is important to note that, by the dominant convergence theorem, this definition implies the convergence in distribution of the random walk process { (1/L)XL,w(L2t), t ³ 0} to the Brownian motion {B0(t), t ³ 0} as it is known in Probability Theory (weak convergence of their distributions): in distribution if, for any collection 0 < s1 < ¼ < snof positive numbers and any collection f1,... ,fn of bounded and continuous functions in , n Î , we have
as N® ¥, where m0 means the expectation of the process starting with the measure m0. Note also that XL(t) has been scaled as in the central limit theorem: XL(t) is a sum of about L2 independent increments1 1 A simple random walk with continuum time jumps according to a Poisson process on time with rate 2 d and there will be 2 dL 2 t jumps in average after a time L 2 t. With the random environment, the Poisson process has a site dependent rate given by å y:|x-y| = 1 w áxyñ . divided up by L.
Theorem II.2(Anshelevich and Vologodski) If d = 1 and the environment w is a stationary process such that the partial sums
converge as x® ¥ to k-1, 0 < k < ¥, almost everywhere in w, then a random walk in a random environment w converges, for almost every w, to a Brownian motion with diffusion coefficient k.
Theorem II.3 (Anshelevich, Khanin and Sinai) For any d ³ 1 and d < 1/2, let w = {wb}bÎ be independent and identically distributed random variables such that
with = wb. Then a random walk in a random environment w converges, for almost every w, to a Brownian motion with diffusion coefficient matrix k = k(d,d,).
Remark II.4 1. Theorem II.2 was proven in [AV] under the condition (II.9) with wx,x+1 > 0. The positive condition has been eliminated in our proof. Condition (II.9) is met if w = {wb} are i.i d. random variables with 0 < < ¥. Finiteness of first negative moment seems to be, according to arguments presented in [ABSO] (see also [FIN]), a sufficient and necessary homogeneity condition since, otherwise, the walk would spend a extremely large time between jumps leading the process to be sub-diffusive.
2. Whether the homogeneity condition 0 < < ¥ is also sufficient for d > 1 is, to our knowledge, an open problem. It would already be an important achievement to prove Theorem II.3 for any distribution whose support is . It is unfortunate that both proofs (see [AKS, Ku]) require as a technical step wb to be bounded away from 0 and ¥.
3. Theorem II.3 holds also if the transition probabilities {wxy} do no vanish for y-x belonging to a finite set that is able to generate by translations. Under the same sort of condition (II.10), [AKS] have shown convergence to Brownian motion satisfying the diffusion equation (II.5).
4. Theorem II.3 can also be extended to Brownian motion on if one combines the result with both absorbing and periodic boundary conditions on ¶ (see details in [AKS]).
Theorem II.5 Under the conditions of Theorem II.3 and a conjecture formulated in (VI.27), there exist a finite constant C = C(d), such that the bounds
hold with 1 being the d×d identity matrix and a positive matrix such that
in the sense of quadratic forms.
III Basic Lemma
The proof of Theorems II.2 and II.3 presented in this section are based on the uniform convergence of the eigenvalues and eigenvectors of DL,w to the eigenvalues and eigenfunctions of the Laplacian operator ¶2 on . The uniform convergence follows from a classical result in perturbation theory which says the following.
If A1,A2,¼, An, ... is a sequence of bounded operators in a Hilbert space which converge to A in the operator norm, then all isolated pieces of their spectrum and their respective projections converge uniformly, as n® ¥, to those of A.
Because DL,wand ¶2 are unbounded operators we consider their inverse instead. To formulate the results of this section, we need some definitions.
Let iL be an isometry of the vector space into the piecewise constant functions in the vector space L02(), of square-integrable functions f on = (-1,1)d with f(¶) = 0:
given by
with [x] as in Definition II.1
The adjoint operator iL :L02()® ,
is defined by the equation
with the inner product in and L02() given, respectively, by
and
We shall denote by
the scaled inverse of DL,w. The operator kernel of XL-1 is a step function with step-width 1/L which, as we shall see in the next lemma, approximates the kernel (¶2)-1(r,s) in the operator norm induced by L02-norm:
where
LemmaIII.1 (Basic Lemma) Under the conditions of Theorems II.3 and II.3, there exist a finite number L0 = L0(w) such that, {XL-1;L > L0} is a sequence of bounded self-adjoint operator in L02() which converges
as L® ¥, in the operator norm topology.
It thus follows from perturbation theory (see e.g. [F]):
Corollary III.2 If l is an isolated eigenvalue of (¶2)-1 and E the orthogonal projection in its eigenspace , one can find a subspace L Ì L02(), invariant under XL-1, and a corresponding orthogonal projection EL such that
and
as L® ¥
Lemma III.1 will be proven for d = 1 random walks in Section IV. This lemma reduces the Brownian motion limit problem to the convergence of the inverse matrix to the inverse Laplacian (¶2)-1 in the L02-operator norm topology. Corollary III.2 will be proven in Section V.
Proof of Theorems II.2 and II.3 assuming Corollary III.2. (As in Appendix 3 of [AKS]) Let j: be given by
and note that j is uniformly continuous at l = 0 with j(0) = 0. We have Tt= j((¶2)-1) and TtL = et XL = j( XL-1).
In view of Definition II.1 and the isometry iL, Theorems II.2 and II.3 can be restated as2 2 The isometry iLhas been introduced to bring all operators to the same Hilbert space L 0 2( ). Note, however, that X L -1 and TtLremain finite rank operators.
(III.11)
as L ® ¥. We shall prove an equivalent statement:
L® ¥
L®¥
||j((¶2)-1) -j(XL-1) || = 0 .
The inverse Laplacian (¶2)-1 on with 0-Dirichlet boundary condition is a compact operator with spectral decomposition given by (recall equation (II.7))
where and , are eigenvalues and associate eigenfunctions of (¶2)-1 and
Because 0 is an accumulation point, we introduce an integer cut-off N < ¥ and let
We have
|n| > N
which can be made as small as we wish by letting N® ¥. More generally, the uniform continuity of j at 0 means the following: given e > 0 and a non-positive bounded operators A, we can find d > 0 such that if ||A|| < d we have ||j(A)|| < e/3. We shall use this fact often in the sequel.
From Corollary III.2, there exist a projector
where EnL is, analogously to En, the projector on the invariant subspace: XL-1
nL = nL. Writing
and using the fact that EL,N is an orthogonal projector, we have
(here j(A-1) is defined by its power series I + tA + t2A2/2+¼).
We now show that ||XL-1-XL,N-1|| can be made smaller then d = d(e) if L and N are chosen sufficient large. By Lemma III.1, there exist L1³ L0, L1 = L1(d), such that
for all L > L1. From (III.12) and (III.14), there exist N1 = N1(d) such that
if N > N1. By Lemma III.1, there exist L2 > L0, L2 = L2(d), such that
holds for all L > L2 with N fixed.
If L > max(L1, L2) and N > N1, equations (III.18)-(III.20) yield
which implies, due the continuity of j and the orthogonal relation (III.17),
By uniform continuity of j and (III.20), we also have
In addition, using the spectral decomposition of (¶2N)-1 and XL,N-1, and taking into account
and Lemma III.1, we can find L3 > L0, L3 = L3 (e,N), such that
Now, let L > max(L1,L2,L3). It then follows from (III.22), (III.23) and (III.25)
which implies strong convergence of the semi-group and completes the proof of Theorem II.3.
Remark III.3The introduction of the cut-off N in (III.14) is necessary even for homogeneous environment. In this case, the eigenvalues lnL and eigenvectors enL of L-2DL-1 can be computed explicitly:
and
with n Î L*: = {1,2,... ,(2L-1)}d (recall sc( nix) stands for cos(nix) or sin(nix) , depending on whether ni is an odd or even number). Note that |ln-lnL|, with ln given by (III.12), may not be small if |n| = O(L). We always pick N large but fixed and let L® ¥ in order (III.25) to be true.
IV Proof of Lemma III.1 (d = 1)
In this section we prove Lemma III.1 for d = 1. We consider the second-order Sturm-Liouville difference equation
with u (¶) = 0, and use the method of Green to calculate the matrix elements of . This gives, in view of equation (III.5), an explicit formula for the operator kernel XL-1(r,s) .
The procedure starts by looking for two linear independent solutions of the homogeneous equation
with x Î {-L+1,... ,L-1} and u-L = uL = 0. Without loss of generality, we set wáL-1,Lñ= wá-L,-L+1ñ= k.
Proposition IV.1Let xLÎ 2L-1 be a vector valued function of the environment w given by
for all x Î {-L+1,... ,L-1}, where
Then u1 = xL and u2 = 1-xL are two linear independent solutions of (IV.1).
Proof.u1 = xL is a solution of (IV.1) by simple verification and the same can be said of u2 = 1-xL. For this, note that
holds uniformly in x, where (Ñu)x = ux - ux-1. It thus remains to verify that they are linear independent.
Let W = W(u1,u2;x) be the ''Wronskian'' of the two solutions u1 and u2 given by the following determinant
It follows that two solutions u1 and u2 are linear independent if W(u1,u2;x) ¹ 0 for all x Î {-L,...,L}. Plugging u1 and u2 into (IV.4) we have, in view of (IV.3),
which concludes the proof of the proposition.
The inverse matrix can be calculated by the so called Green's function method (see e.g. [J]):
To see this is true, we note ()z,y is the z-component of a vector for each y fixed. So, by definition
holds for all x ¹ y. For x = y we have
by (IV.3), verifying the assertion.
We are now ready to write the operator kernel of XL-1. In view of
valid for any (2L-1)×(2L-1) matrix A and f Î L02(), and definitions (III.5) and (III.1), we have
for any -1 £ r,s £ 1.
If new variables zL: = 2 xL-1 are introduced into equation (IV.6), the operator kernel (IV.8) can be written as
in view of the fact that (zL)x is a monotone increasing function of x.
By Schwarz inequality the operator norm (IV.6) is bounded by the L2-norm of the operator kernel, the Hilbert-Schmidt norm . Since the functions in the Hilbert space has compact support, we have
-1<r,s<1
and ||XL-1-(¶2)-1||® 0 is implied by the pointwise convergence XL-1(r,s)®(¶2)-1(r,s) of the operator kernel. We shall see that the latter convergence sense is consequence of the following result.
Proposition IV.2Given e > 0 and w satisfying the hypothesis of Theorem II.2. Then, there exist an integer number L0 = L0(e,w) such that
holds for all L > L0 and -1 < r < 1.
Proof. Under the hypothesis of Theorem II.2 the strong law of large numbers holds and
as L® ¥, for almost every w (see eq. (IV.2)). Analogously, since [Lr]/L® r as L® ¥,
converges to r as L® ¥ for each r Î (-1,1) and this gives (IV.10).
The Green's function method can also be used to compute the integral kernel of (¶2)-1 as an operator in the Hilbert space L02(). The two linear independent solutions of the homogeneous equation
with boundary condition f(-1) = f(1) = 0 are f1 = 1+r and f2 = 1-r. Replacing u1(2) and wáx-1,xñ(Ñu1(2)) by f1(2) and k(df1(2)/dr) in (IV.4), gives W = -2 k. Substituting these into (IV.6) following the simplification of (IV.9), yields
Note that |(¶2)-1(r,s)| £ 1/ (2k) and, in view of (IV.9) and Proposition IV.2,
holds uniformly in r, s Î (-1,1) for all L > L0.
Now, let rL(r): = (izL)(r) - r and : = 2L hLXL-1/k. Then, if L > L0, in view of (IV.9), (IV.14) and Proposition IV.2, we have
where
uniformly in r, s Î (-1,1). When combined with (IV.11) and (IV.15) this proves Lemma III.1 for d = 1.
V Perturbation of Spectra
Proof of Corollary III.2. This proof can be found in Appendix B of [AKS] and is essentially based on the perturbation theory of Hermitian bounded operators developed by Friedrichs in [F]. Since it can be described shortly, we repeat the proof's derivation for completeness. Our derivation, however, is more close to [F] in the sense that we perturb an interval of the spectrum. When the interval contains one single eigenvalue this reduces to the derivation of [AKS]. The generalization is however essential in dealing with intervals containing accumulation point of the spectrum. This situation has to be considered in order to show that the spectrum projection in such intervals remains orthogonal when the perturbation is turned on.
We now introduce some notation. Let I0Î be an isolated closed interval of the spectrum s(¶-2) of ¶-2 defined with Dirichlet boundary condition on = (-1,1). There exist 0 < d < ¥ and an interval I Ì I0 such that IÇs(¶-2) = s(¶-2)ÇI0 and
dist ( I0, \ I) > d .
Let
0 denote the eigenspace span{en:ln Î I0} Î L02() associated with I0 and E0 the spectral projection onto 0. Let 1/L denote the subspace of the Hilbert space : = L02() invariant under the action of XL and E1/L the projection (not necessarily orthogonal) onto 1/L.The projection E1/L is defined by the following set of equations:
(i.e.
1/L is an invariant subspace) and the two conditionsNote that, under (V.2) E1/L is a projector
which is consistent with E0 in the sense that limL®¥E1/L = E0.
We shall prove that, provided VL: = XL-1 - ¶-2 is bounded, E1/L depends analytically on 1/L and 1/L tends to 0 as 1/L ® 0. The proof of this statement uses equation (V.1) to write an integral equation. For simplicity, we shall drop the index L of the quantities X-1L, VL, 1/L and E1/L.
Our starting point begins with equation
which comes from the following facts. The operator ¶-2 commutes with the spectral projector E0. Using this and equations (V.2), we have
E¶-2E = E E0¶-2E = E ¶-2E0E = E ¶-2E0 = E E0¶-2 = E ¶-2 ,
and this implies the second line of (V.3). The commutation relation [¶-2, E0] = 0 allows us to replace E in the equation (V.3) by Q : = E-E0
Combining (V.1) with (V.4) and using X-1 = ¶-2 + V, gives
Since the interval I0 is isolate from the rest of the spectrum, ¶-2 is an invertible bounded operator in the subspace (I-E0) . We can solve the left hand side of (V.5) for Q by defining
Note that X ¶-2 = ¶-2X = I-E0 and ||X|| < d-1.
Equation (V.5) can thus be written as
where
Proposition V.1 The sequence Qn, n = 0,1,... , of projectors defined by
with initial condition Q0 = 0 satisfies the conditions (V.2) and converges, Q = limn® ¥Qn, to the unique solution of equation (V.6).
Proof. We have ||Qn|| £ q < 1 for all n Î provided q is chosen small enough and L is taken so large that if ||Q|| £ q then
Note that the smallness of g depends on the smallness of V. Since
equation (V.8) holds provided
Now, for fixed value of q, it can be shown (see ref. [F] for details)
also holds with q < 1 and this implies Proposition V.1 by the Banach fixed point theorem.
We have proven the existence of a unique projector E1/L such that ||Q1/L|| = ||E1/L- E0|| £ q. Since q can be made arbitrarily small by taking L sufficiently large so that (V.9) holds, we have limL® ¥||E1/L- E0|| = 0 and 1/L® 0 as L® ¥.
To complete the proof of Theorem III.1, we need to find an orthogonal projector EL onto L º 1/L in order to get (III.8). This is achieved by setting
and noting that the inverse operator (EE)-1 exist because ||Q|| £ q implies
||E0f|| £ ||E f|| + q ||f|| = ||E f|| + q ||E0f|| ,
for any f Î such that E0f = f. As a consequence ||E f|| ³ (1-q) ||f|| and
áf,EE fñ ³ (1-q)2|f|| .
One can show, in addition, that [X-1,EL] = 0 for all L. Therefore, for any l Î I0Çs(¶-2),
Since the right hand side goes to zero as L® ¥ this concludes the proof of Corollary III.2.
VI Diffusion Coefficient
This section is devoted to the proof of Theorem II.5. The diffusion coefficient will be estimated throughout an expansion for the expectation of the inverse matrix, (DL,w)-1, with w satisfying the macroscopic homogeneity condition (II.10). This is justified in ref. [AKS] in view of the fact that (DL,w)-1, when properly scaled, converge to its expectation for almost all environment w. Thus, the formula ((DL,w)-1) -1 ~ DL,k is expect to hold in the limit as L® ¥. We will see that very important cancellations take place by inverting the series expansion of (DL,w)-1.
A simple algebraic manipulation shows
where
is a well defined matrix since, in view of (II.2) and (II.10), -DL,w is positive and the square root of can be taken.
Choosing = wb and use (II.1) to write = DL where DL is the finite difference Laplacian with 0-Dirichlet boundary condition on L, equation (VI.2) can be written as
where a = {ab} given by ab = wb/-1, are i.i.d. random variables with mean ab = 0, such that
holds in view of (II.10).
Equation (VI.1) suggests us the use of Neumann series to develop a formal expansion of (DL,w)-1 in power of DL,w due to the small parameter d. The remaining of this section is devoted to the pointwise convergence of the matrix element of [( I -DL,w) -1] -1.
Using (VI.3), we have
To write (VI.5) in a more convenient form, let ÑL : ® be the finite difference operator:
(ÑL u)áxyñ = -(ÑL u)áyxñ = sáxyñ( uy-ux) .
where the sign , according to whether áxyñ is positively ( = 1) or negatively (= -1) oriented. ÑL maps a 0-form u into a 1-forms ÑLu. Let ÑL*:® be its adjoint (w,ÑL u) = ( ÑL*w,u) , i.e. the finite divergent operator which maps a 1-form w into a 0-form ÑL*w given by
and let Ma : ® be the multiplication operator by a: (Maw)b: = ab wb.
With these notations, we have
and its bilinear form reads
recovering expression (II.2) for the quadratic form.
Define
and note that, since (-DL)x,y-1 is the Coulomb potential between two unit charges located at x and y, Fb,b¢ is the dipole interaction potential between two unit dipoles located at b and b¢. Note that F maps 1-form into 1-form.
In view of (VI.6) and (VI.7), equation (VI.5) can be rewritten as
where, for G = (b1,b2,... ,bn),
and
with J and n being 1-forms given by Ñ(-DL) -1/2v and Ñ(-DL)-1/2u, respectively.
Concern the convergence, as L
, of a generic term of the expansion (VI.8), the following remark is now in order.Remark VI.1The asymptotic behavior of the dipole potential Fb,b'¢for L >> dist (b,b') >> 1 can be estimated by its spectrum decomposition3 3 Since we have not rescaled the space L Ì , it is convenient to introduce a base n Î L*, L * = {1,... ,2 L-1} d, normalized with respect to the scalar product (( u,v)): = å x Î L ux vx = Ld ( u,v): . The spectrum resolution of the identity is written in terms of this base. ,
where
and enL as in (III.27). If we take b = áxx(i)ñ and b' = áyy(j)ñ, where z(k) is a nearest site of z whose components are given by zl(k) = zl+dk,l, and make a change of variables, ji = (p/2L)ni, i = 1,... ,d, we have
where Ñkf(z) = f(z(k)) - f(z) is the difference operator in the k-th direction. The |x-y| >> 1 behavior of Fb,b¢ is given by restricting the integral (VI.12) around a e-neighborhood of 0 with e |x-y| = O(1):
As a consequence, Fb,b¢ is not summable in absolute value,
and the uniform convergence with respect to L of the G-summation in (VI.8) requires cancellations due to the dipole orientations (see ref. [PPNM]).
We shall exhibit in the following another kind of cancellation due to the inversion of the expected value of (VI.8).
Inverting the expectation of (VI.8) gives
where
and G Î ×¼× Lnk, with ni ³ 1. Note that WG = WG1¼WGk.
To see how the log-divergent terms in (VI.14) cancel out, it is convenient to use graph-theoretical language. A graph G consists of two sets (V,E): V = {v1,... ,vs} is the vertex set and E = {e1,... ,es¢} the connecting set of edges. To each edge e its assigned an ordered pair of vertices ( vv¢) (its extremities) which are called adjacent if v ¹ v¢; otherwise e is said to be a ''loop''. To the problem at our hand, we shall identify the bonds {b1,... ,bn} as a the vertex set of a graph G whose connectivity is determined by the presence of interactions Fb,b¢.
Two graphs G and G¢ are isomorphic (denoted G ~ G¢) if there is a one-to-one correspondence between their elements which preserves the incidence relation. A path G on G is an ordered sequence {vi0,ei1,... ,ein,vin} of alternately vertices and edges of G such that eik = (vik-1vik) holds for each k; the edges {ei1,... ,ein} are the steps of the path and the vertices {vi0,... ,vin} are the points visited by the path. G may be identified with one of these ordered sets since it can be uniquely determined by each of them. Two vertices v,v¢Î V may be connected by more than one path. A graph G is said to be connected if any two vertices v,v¢ can be joined by at least one path G on G. The components of a non-connected graph are its maximum connected subgraphs. Given two vertices v,v¢, the disconnecting set of edges is a set whose removal from the graph G destroys all paths between v,v¢. A cut-set is a minimal set of edges the removal of which from a connected graph G causes it to fall into two components G1,G2.
Turning back to equation (VI.14), one may interpret G = {b1,... ,bn} as a set of vertices visited by a path. In view of the fact that ab has zero mean, we have
if there exist at least one bond bi which are not repeated in the list G = {b1,... ,bn}.
The condition (VI.16) says that the path G must visit each vertex at least twice otherwise its contribution to (VI.14) vanishes. The set of distinct bonds V = {bi1,... bis} and edges E = {( b1b2) ,... ,( bn-1bn) } form a connected graph G with even valency (b) ³ 4 for each vertex b Î G. Graphs with this property will be called admissible graphs. Note that each path G yields only one graph G but there are possibly many n-step paths covering each edge (bi-1bi) of G exactly once which starts at b1 and ends at bn. If we denote by [G]G the set of all paths G satisfying these conditions for a given admissible graph G, we have
Proposition VI.2 Equation (VI.15) can be written as
with
where we sum over all sizes n Î , n ³ 1, all admissible graphs G of size |E| = n, over all paths G in [G]G and over all decompositions of G into s,s ³ 1, successive paths (G1,... ,Gs), each of which capable of generating admissible graphs Gi. Here, with the notation of (VI.10) and footnote in Remark VI.1,
for n > 1 with Fb1,b1 = 1 for n = 1 (the case that G is the trivial graph ({b1 }, Æ)).
We shall in the sequel state two lemmas and prove Theorem II.5 under an extra assumption.
Lemma VI.3 If G is an admissible graph with at least one cut-set contained one edge (i.e. G falls into two components by cutting a single edge), then AG = 0.
Lemma VI.4There exist a constant C[G] < ¥, depending on the equivalence class [G] of isomorphic graphs G, such that
holds uniformly in L for all admissible graph G with |E| = n and cut-sets with no less than two elements.
Remark VI.5 The proof of Lemmas VI.3 and VI.4 are essentially given in [AKS] (see Assertions I and II of Section 4). Note that our estimate (VI.20) have not included the logarithmic corrections which appears in that reference. To get rid of these one has to control the loop subgraphs of G carefully as it is done in the ref. [PPNM]. The uniform upper bound (VI.20) results from the hypothesis that G remains connected by cutting one single edge.
Graphs with single edge cut-sets do not contribute to (VI.17) due to the following cancellation in Lemma VI.3.
Proof of Lemma VI.3. Let (bi bi+1) be the only edge of a cut-set and let G = (G1,... ,Gs) be a decomposition of a path in G. Either both bi and bi+1 belongs to some Gj or they belong to two successive ones. We call the latter decomposition type A and the former type B. It turns out that there is an one-to-one correspondence between type A and type B decompositions differing only by the splitting of Gj into two elements Gj(1) and Gj(2). Lemma VI.3 follows from the fact that the contribution to (VI.18) of a pair of decompositions established by this correspondence have the same absolute value with opposite signals. Note if the edge ( bi bi+1) bridges the two paths Gj(1) and Gj(2).
In view of Proposition VI.2 and Lemmas VI.3 and VI.4, equation (VI.15) can be estimated as
where
with the sum running over the equivalence classes [G] of isomorphic admissible graphs G of size |E| = n and C[G] as in Lemma VI4. Note that AG = A[G].
Now we show that, if one uses, as in refs. [AKS] and [PPNM], the upper bound
for some geometric constant C < ¥ where r = |V| is the number of vertices in G, the equation (VI.22) cannot be bounded uniformly in L. Taking into account property (VI.4),
holds uniformly in G and equation (VI.22) can be bounded by
Here, we have identified each path G = {b1,... ,bn} in a given graph G = (V,E) of size |E| = n-1 with a partition P = (P 1, ... ,Pr) of {1, 2, ... ,n } into r = |V| pairwise disjoint subsets. This association is one-to-one since G is an ordered set of elements. Note that each component Pjcorresponds to one vertex bi1 = bi2 = ¼ = bip of G visited p = |Pj| times by the path G. One can thus replace the sum over all equivalent classes of graphs [G] and over all paths G in [G] by the sum over all partitions P. The factor P(n, r) counts the number of partitions of {1, 2, ... , n} into r subsets. The bound (VI.25) disregards the fact that G is an admissible graph. Also, the consistency of each decomposition G = (G1, ... , Gs) into paths Gi's which gives rise to admissible graphs has been not considered. The binomial factor ([(2n -1 ) || n ] ) £ 4n counts the decomposition of G with n steps into any number of paths with the number of steps £ n (the cardinality of the set {1 £ i1£ i2£ ¼ £ in-1 £ n-1}). In addition, for the upper limit in the second sum we note that r = |V| £ (2 L-1)d (the number of vertices of G cannot be larger than the number of sites in L).
Equation (VI.25) cannot be uniformly bounded since, from the recursion relation P(n, r) = P(n-1, r-1) + r P(n-1, r) (see [W]), we have
P(n, r) ³ rP(n-1, r) ³ r n-r P(r, r) = r n-r,
which gives a factorial growth
after replacing the sum by the term with r = min (n, |L|)/2.
A sharper upper bound for (VI.23) may be assumed if one think of C[G] as being given by
As one varies the partition P of {1, ... , n}, the graph G, and the path G over it, varies accordingly and the decay of Fb,b¢ in this formula can be useful. We propose that C[G] = Cn,r depends on the number of vertices r = |V| and edges n = |E| as follows.
Conjecture VI.6 Let (n,r) = Cn,rP(n,r). There exist a geometric constant C < ¥ such that
holds for n,r Î , n ³ r with
(r,r) = Cr.Note that P(n,r) satisfies (VI.27) with C replaced by r. Assuming (VI.27) and using that (1,1) = C and (k, l) = 0 if k < l, we have
which leads (VI.25) to be bounded by
where d¢ = 4 (1+C)d.
This concludes the preliminaries and we are now ready to prove Theorem II.5. We observe at this point that no restrictions about the random variables ab's has been made beside (VI.16) and (VI.24) with d small enough. Has Conjecture VI.6 been proved one could work along similar expansions to show that ()-1/2(-DL, w) ()-1/2 converges to [ (I-DL, w)-1]-1 with probability 1.
Proof of the upper bound of II.11. Let us recall some facts about the matrix -DL, w. By equation (II.2) it is a positive definite matrix and its square root is well defined. We also have (-DL, w) = = -DL and, by Lemma III.1, i(-DL, w)-1i/L2 converges with probability 1 to (-¶2 (k))-1 exactly as i(-ÑL · kÑL)-1i/L2 does.
In view of this, we can apply Schwarz inequality to the following identity4 4 In the following, for any two matrices A and B, A £ B means ( u, A u) £ ( u, B u) for all vectors u. :
in order to get
which implies k £ and concludes our assertion.
Proof of the lower bound of II.11. From equations (VI.1), (VI.3) and (VI.14), we have
where RL: is a matrix whose elements, in view of (VI.21) and (VI.29), are bounded by
with KL £ d¢C/(1-d¢) . Note from equations (VI.30) and (VI.31) that QL is a positive matrix.
Using the isometry operators (III.1) and (III.2) and the fact that i i is the identity matrix in |, we have L2 i ((-DL, w)-1)-1i = ( L iÑ*i ) (iRLi ) ( L iÑi ) with L iÑ*i and L iÑi converging in L02( ) to the operator ¶ = (¶/¶x1,¼,¶/¶xd) . In addition, we claim that the kernel of i( I-RL) i converges in distribution, as L® ¥, to the delta function d(h,x) times a d × d matrix r, since the matrix elements of RL decay faster than 1/dist(b,b¢)d as dist(b,b¢)® ¥. For this, note that i(I-RL) i (h,x) = ( L-d) if h ¹ x and = ( Ld) if h = x.
Whether r is a diagonal matrix cannot be decided by our estimates. The results from this section leads to and this implies
where 1 is the d×d identity and r is a positive matrix satisfying £ d¢C/( 1-d¢) .
Acknowledgment
We wish to thank Luiz R. Fontes for helpful discussions. R. da Silva was supported by CNPq under the PIBIC project and D.H.U. Marchetti was partially supported by FAPESP and CNPq.
Footnotes:
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Publication Dates
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Publication in this collection
07 Aug 2000 -
Date of issue
Sept 1999
History
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Received
05 Apr 1999