Brownian motion limit of random walks in symmetric non-homogeneous media

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) {w_áxyñ} satisfying

w_áxyñ = w_áxyñ (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 w_x,x+1 = 1-w_x,x-1, x Î , being independent and identically distributed (i.i.d.) random variables (note that w_x,x+1 ¹ w_x+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 {w_x,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 t^d with an exponent d < 1 depending on the divergence of the distribution at w_x,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 w_x,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 lim_{t® ¥} (X_i(t)X_j(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 = w_x,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-w_áxyñ /_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 = {w_b}_b Î be an assignment of positive numbers. Each component w_brepresents 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 w_áxyñ = w_áyxñ .

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 {X_{L ,w}(t), t ³ 0} with differential transition matrix W_L = W_L (w) defined by

for all x Î L and u such that u_y = 0 if y Î \ L .

Note that -D_L ,w is a positive matrix,

and, if for all where D_L 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 : L_L: = { x = (x₁,... ,x_d) Î :sup_i|x_i| < L} , and all quantities depending on L will be indexed by L instead.

The probability distribution of {X_L,w(t), t ³ 0} is governed by the semi-group generated by D_L,w. If X_L,w(t) denotes the position of a random walk at time t, then

where (u₀)_x = d_0,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:sup_i|x_i| = 1}, and ¶² = ¶²(k) is given by

The heat kernel T_t(h,z) = e^t ¶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, T_t 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(n_ix) stands for either cos(n_ix) or sin(n_ix) depending on whether n_iis 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)X_L,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 < s_nof positive numbers and any collection f₁,... ,f_n 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 X_L(t) has been scaled as in the central limit theorem: X_L(t) is a sum of about L² independent increments¹ 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 = {w_b}_bÎ be independent and identically distributed random variables such that

with = w_b. 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 w_x,x+1 > 0. The positive condition has been eliminated in our proof. Condition (II.9) is met if w = {w_b} 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 w_b to be bounded away from 0 and ¥.

3. Theorem II.3 holds also if the transition probabilities {w_xy} 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 D_L,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, A_n, ... 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 D_L,wand ¶² are unbounded operators we consider their inverse instead. To formulate the results of this section, we need some definitions.

Let i_L 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 i_L^† :L₀²()® ,

is defined by the equation

á f ,i_Lu ñ = (i_L^† f, u) ,

with the inner product in and L₀²() given, respectively, by

and

We shall denote by

the scaled inverse of D_L,w. The operator kernel of X_L^-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, {X_L^-1;L > L₀} is a sequence of bounded self-adjoint operator in L₀²() which converges

||X_L^-1-(¶²)^-1||® 0 ,

(III.7)

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 X_L^-1, and a corresponding orthogonal projection E^L such that

||E^L-E||® 0 (III.8)

and

|| E^L( X_L^-1-lI)E^L|| ® 0 , (III.9)

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

j(l) = e^t/l (III.10)

and note that j is uniformly continuous at l = 0 with j(0) = 0. We have T_t= j((¶²)^-1) and T_t^L = e^t X_L = j( X_L^-1).

In view of Definition II.1 and the isometry i_L, Theorems II.2 and II.3 can be restated as²

(III.11)

as L ® ¥. We shall prove an equivalent statement:

lim
L® ¥ ||T_t-T_t^L|| = lim
L®¥

||j((¶²)^-1) -j(X_L^-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

E_nf = áe_n,fñ e_n . (III.13)

Because 0 is an accumulation point, we introduce an integer cut-off N < ¥ and let

We have

||j((¶_N²)^-1)-j((¶²)^-1)||² = å
|n| > N j²(l_n) , (III.15)

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 E_n^L is, analogously to E_n, the projector on the invariant subspace: X_L^-1

_n^L = _n^L. Writing

X_L,N^-1 = E^L,NX_L^-1E^L,N (III.16)

and using the fact that E^L,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 ||X_L^-1-X_L,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 (¶²_N)^-1 and X_L,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 l_n^L and eigenvectors e_n^L of L^-2D_L^-1 can be computed explicitly:

and

with n Î L^*: = {1,2,... ,(2L-1)}^d (recall sc( n_ix) stands for cos(n_ix) or sin(n_ix) , depending on whether n_i is an odd or even number). Note that |l_n-l_n^L|, with l_n 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

D_L,w u = f

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 X_L^-1(r,s) .

The procedure starts by looking for two linear independent solutions of the homogeneous equation

( D_L,w u)_x = w_áx-1,xñ(u_x-1-u_x)-w_áx,x+1ñ(u_x-u_x+1) = 0 , (IV.1)

with x Î {-L+1,... ,L-1} and u_-L = u_L = 0. Without loss of generality, we set w_áL-1,Lñ= w_á-L,-L+1ñ= k.

Proposition IV.1Let x_LÎ ^2L-1 be a vector valued function of the environment w given by

for all x Î {-L+1,... ,L-1}, where

Then u₁ = x_L and u₂ = 1-x_L are two linear independent solutions of (IV.1).

Proof.u₁ = x_L is a solution of (IV.1) by simple verification and the same can be said of u₂ = 1-x_L. For this, note that

w_{áx-1,x ñ} (Ñx_L)_x = h_L (IV.3)

holds uniformly in x, where (Ñu)_x = u_x - u_x-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),

W = -h_L [ (x_L)_x+1-(x_L)_x] = -h_L , (IV.5)

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

( D_L,wD_L,w^-1) _x,y = 0

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 X_L^-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 z_L: = 2 x_L-1 are introduced into equation (IV.6), the operator kernel (IV.8) can be written as

in view of the fact that (z_L)_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

||K|| £ |||K||| £ 4 sup
-1<r,s<1 |K(r,s)|

and ||X_L^-1-(¶²)^-1||® 0 is implied by the pointwise convergence X_L^-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

| (z_L)_[Lr]-r| < e (IV.10)

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

2 L h_L® k , (IV.11)

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 w_áx-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 r_L(r): = (iz_L)(r) - r and : = 2L h_LX_L^-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{e_n:l_n 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 X_L and E_1/L the projection (not necessarily orthogonal) onto _1/L.

The projection E_1/L is defined by the following set of equations:

(I-E_1/L) X^-1_L E_1/L = 0 (V.1)

(i.e.

_1/L is an invariant subspace) and the two conditions

E_1/LE₀ = E_1/L and E₀E_1/L = E₀ . (V.2)

Note that, under (V.2) E_1/L is a projector

which is consistent with E₀ in the sense that lim_L®¥E_1/L = E₀.

We shall prove that, provided V_L: = X_L^-1 - ¶^-2 is bounded, E_1/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^-1_L, V_L, _1/L and E_1/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

(1-E)¶^-2E = ¶^-2Q - Q ¶^-2 . (V.4)

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

Q = g (Q) , (V.6)

where

g (Q) = X ¶^-2 Q - X (I+ E₀- Q) V (Q-E₀) .

Proposition V.1 The sequence Q_n, n = 0,1,... , of projectors defined by

Q_n = g(Q_n-1) , (V.7)

with initial condition Q₀ = 0 satisfies the conditions (V.2) and converges, Q = lim_n® ¥Q_n, to the unique solution of equation (V.6).

Proof. We have ||Q_n|| £ q < 1 for all n Î provided q is chosen small enough and L is taken so large that if ||Q|| £ q then

||g(Q)|| £ ||X|| ||E₀¶^-2|| q+d^-1 (1+q)²||V_L|| £ q . (V.8)

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)

| g(Q)-g(Q^¢)| £ q| Q-Q^¢|

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 E_1/L such that ||Q_1/L|| = ||E_1/L- E₀|| £ q. Since q can be made arbitrarily small by taking L sufficiently large so that (V.9) holds, we have lim_{L® ¥}||E_1/L- E₀|| = 0 and _1/L® ₀ as L® ¥.

To complete the proof of Theorem III.1, we need to find an orthogonal projector E^L onto ^L º _1/L in order to get (III.8). This is achieved by setting

and noting that the inverse operator (E^†E)^-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,E^†E fñ ³ (1-q)²|f|| .

One can show, in addition, that [X^-1,E^L] = 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, (D_L,w)^-1, with w satisfying the macroscopic homogeneity condition (II.10). This is justified in ref. [AKS] in view of the fact that (D_L,w)^-1, when properly scaled, converge to its expectation for almost all environment w. Thus, the formula ((D_L,w)^-1) ^-1 ~ D_L,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 (D_L,w)^-1.

A simple algebraic manipulation shows

where

is a well defined matrix since, in view of (II.2) and (II.10), -D_L,w is positive and the square root of can be taken.

Choosing = w_b and use (II.1) to write = D_L where D_L is the finite difference Laplacian with 0-Dirichlet boundary condition on L, equation (VI.2) can be written as

D_L,w = ( -D_L) ^-1/2D_L,a( -D_L) ^-1/2 , (VI.3)

where a = {a_b} given by a_b = w_b/-1, are i.i.d. random variables with mean a_b = 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 (D_L,w)^-1 in power of D_L,w due to the small parameter d. The remaining of this section is devoted to the pointwise convergence of the matrix element of [( I -D_L,w) ^-1] ^-1.

Using (VI.3), we have

To write (VI.5) in a more convenient form, let Ñ_L : ® be the finite difference operator:

(Ñ_L u)_áxyñ = -(Ñ_L u)_áyxñ = s_áxyñ( u_y-u_x) .

where the sign , according to whether áxyñ is positively ( = 1) or negatively (= -1) oriented. Ñ_L maps a 0-form u into a 1-forms Ñ_Lu. Let Ñ_L^*:® be its adjoint (w,Ñ_L u) = ( Ñ_L^*w,u) , i.e. the finite divergent operator which maps a 1-form w into a 0-form Ñ_L^*w given by

and let M_a : ® be the multiplication operator by a: (M_aw)_b: = a_b w_b.

With these notations, we have

D_L,a = Ñ_L^*M_aÑ_L , (VI.6)

and its bilinear form reads

recovering expression (II.2) for the quadratic form.

Define

and note that, since (-D_L)_x,y^-1 is the Coulomb potential between two unit charges located at x and y, F_b,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,... ,b_n),

and

with J and n being 1-forms given by Ñ(-D_L) ^-1/2v and Ñ(-D_L)^-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 F_b,b'^¢for L >> dist (b,b') >> 1 can be estimated by its spectrum decomposition³,

where

and e_n^L 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 z_l^(k) = z_l+d_k,l, and make a change of variables, j_i = (p/2L)n_i, i = 1,... ,d, we have

where Ñ^kf(z) = f(z^(k)) - f(z) is the difference operator in the k-th direction. The |x-y| >> 1 behavior of F_b,b^¢ is given by restricting the integral (VI.12) around a e-neighborhood of 0 with e |x-y| = O(1):

As a consequence, F_b,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× _L^nk, with n_i ³ 1. Note that W_G = W_G1¼W_Gk.

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,... ,v_s} is the vertex set and E = {e₁,... ,e_s^¢} 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,... ,b_n} as a the vertex set of a graph G whose connectivity is determined by the presence of interactions F_b,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 {v_i0,e_i1,... ,e_in,v_in} of alternately vertices and edges of G such that e_ik = (v_ik-1v_ik) holds for each k; the edges {e_i1,... ,e_in} are the steps of the path and the vertices {v_i0,... ,v_in} 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,... ,b_n} as a set of vertices visited by a path. In view of the fact that a_b has zero mean, we have

if there exist at least one bond b_i which are not repeated in the list G = {b₁,... ,b_n}.

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 = {b_i1,... b_is} and edges E = {( b₁b₂) ,... ,( b_n-1b_n) } 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 (b_i-1b_i) of G exactly once which starts at b₁ and ends at b_n. 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,... ,G_s), each of which capable of generating admissible graphs G_i. Here, with the notation of (VI.10) and footnote in Remark VI.1,

for n > 1 with F_b1,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 A_G = 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 (b_i b_i+1) be the only edge of a cut-set and let G = (G₁,... ,G_s) be a decomposition of a path in G. Either both b_i and b_i+1 belongs to some G_j 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 G_j into two elements G_j⁽¹⁾ and G_j⁽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 ( b_i b_i+1) bridges the two paths G_j⁽¹⁾ and G_j⁽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 A_G = 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,... ,b_n} in a given graph G = (V,E) of size |E| = n-1 with a partition P = (P ₁, ... ,P_r) 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 P_jcorresponds to one vertex b_i1 = b_i2 = ¼ = b_ip of G visited p = |P_j| 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, ... , G_s) into paths G_i'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 £ i_n-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 F_b,b¢ in this formula can be useful. We propose that C_[G] = C_n,r depends on the number of vertices r = |V| and edges n = |E| as follows.

Conjecture VI.6 Let (n,r) = C_n,rP(n,r). There exist a geometric constant C < ¥ such that

holds for n,r Î , n ³ r with

(r,r) = C^r.

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 a_b'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(-D_{L, w}) ()^-1/2 converges to [ (I-D_{L, w})^-1]^-1 with probability 1.

Proof of the upper bound of II.11. Let us recall some facts about the matrix -D_{L, w}. By equation (II.2) it is a positive definite matrix and its square root is well defined. We also have (-D_{L, w}) = = -D_L and, by Lemma III.1, i(-D_{L, w})^-1i^†/L² converges with probability 1 to (-¶² (k))^-1 exactly as i(-Ñ_L · kÑ_L)^-1i^†/L² does.

In view of this, we can apply Schwarz inequality to the following identity⁴:

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 R_L: is a matrix whose elements, in view of (VI.21) and (VI.29), are bounded by

with K_L £ d^¢C/(1-d^¢) . Note from equations (VI.30) and (VI.31) that Q_L 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 ((-D_{L, w})^-1)^-1i^† = ( L iÑ^*i^† ) (iR_Li^† ) ( L iÑi^† ) with L iÑ^*i^† and L iÑi^† converging in L₀²( ) to the operator ¶ = (¶/¶x₁,¼,¶/¶x_d) . In addition, we claim that the kernel of i( I-R_L) i^† converges in distribution, as L® ¥, to the delta function d(h,x) times a d × d matrix r, since the matrix elements of R_L decay faster than 1/dist(b,b^¢)^d as dist(b,b^¢)® ¥. For this, note that i(I-R_L) i^† (h,x) = ( L^-d) if h ¹ x and = ( L^d) 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:

• [ABO] S. Alexander J. Bernasconi and R. Orbach. ''Spectral diffusion in a one-dimensional percolation model'', Phys. Rev. B 17, 4311-4314 (1978)
• [ABSO] S. Alexander, J. Bernasconi, W. R. Schneider and R. Orbach. ''Excitation dynamics in random one-dimensional systems'', Rev. Mod. Phys. 53, 175-198 (1981)
• [AKS] V. V. Anshelevich, K. M. Khanin and Ya. G. Sinai. ''Symmetric random walks in random environments'', Commun. Math. Phys. 85, 449-470 (1982)
• [AV] V. V. Anshelevich and A. V. Vologodskii. ''Laplace operator and random walk on one-dimensional nonhomogeneous lattice'', Journ. Stat Phys. 25, 419-430 (1981)
• [BSW] J. Bernasconi, W. R. Schneider and W. Wyss. ''Diffusion and hopping conductivity in disordered one-dimensional lattice systems'', Z. Physik B 37, 175-184 (1980)
• [D] F. J. Dyson. ''The dynamics of a disordered chain of harmonic oscilators'', Phys. Rev. 92, 1331-1338 (1953)
• [F] K. O. Friedrichs. ''Perturbation of spectra in Hilbert space''. AMS Lect. in Appl. Math., Providence, Rhode Island (1965)
• [FIN] L. R. G. Fontes, M. Isopi and C. M. Newman. ''Chaotic time dependence in a disordered spin system'', Preprint (1998)
• [J] Alouf Jirari. ''Second-order Sturm-Liouville difference equations and orthogonal polynomials'' Memoirs of AMS 542 (1995)
• [K] P. W. Kasteleyn. ''Graph theory and crystal physics'', in Graph theory and theoretical physics, ed. F. Harary, AP New York (1967)
• [KKS] H. Kesten, M. W. Koslow and F. Spitzer. ''A limit law for random walk in random environment'', Compos. Math. 30, 145-168 (1975)
• [Ku] Rolf Künnemann. ''The diffusion limit for reversible jump processes on Z^d with ergodic random bond conductivities'', Commun. Math. Phys. 90, 27-68 (1983)
• [LM] Elliot H. Lieb and Daniel C. Mattis. ''Mathematical physics in one-dimension: Exactly soluble models of interacting particles'', Academic Press (1966)
• [MFGW] A. de Masi, P. A. Ferrari, S. Goldstein an W. D. Wick. ''An invariant principle for reversible Markov processes. Aplications to random motions in random environments'', Journ. Stat. Phys. 55, 787-855 (1989)
• [PPNM] Aldo Procacci, Emmanuel Pereira, Armando G. M. Neves and Domingos H. U. Marchetti. ''Coulomb interaction symmetries and the Mayer series in two dimensional dipole gas'', Journ. Stat. Phys. 87, 877-889 (1997)
• [PV] G. Papanicolaou and S. R. S. Varadhan. ''Diffusion with random coefficients'', in Statistics and Probability: Essays in Honor of C. R. Rao'', 547-552, North-Holland (1982)
• [S] Barry Simon, ''Functional integration and quantum physics'', Academic Press (1979)
• [Si] Ya. G. Sinai. ''The limiting behavior of a one-dimensional random walk in a random medium'', Theo. of Prob. and its Appl. 27, 256-268 (1982)
• [So] F. Solomon. ''Random walks in a random environment'', Ann. Prob. 3, 1-31 (1975)
• [W] Herbert S. Wilf. ''Generatingfunctionology'', second edition, Academic Press (1994)

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 .

²

The isometry

i_Lhas been introduced to bring all operators to the same Hilbert space L

₀

²(

). Note, however, that X

_L

^-1 and

T_t^Lremain finite rank operators.

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}

u_x v_x =

L^d (

u,v):

. The spectrum resolution of the identity is written in terms of this base.

⁴

In the following, for any two matrices

A and

B,

A £

B means (

u,

A

u) £ (

u,

B

u) for all vectors

u.

Publication Dates

• Publication in this collection
  07 Aug 2000
• Date of issue
  Sept 1999

History

• Received
  05 Apr 1999