Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials

We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials.

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1. Introduction and results

This note is a continuation of our recent paper [3] where we studied the statistics of low-lying eigenvalues of Anderson Hamiltonians in the “homogenization” regime, i.e., under the conditions when a non-trivial continuum limit can be taken. The derivations of [3] were restricted to the class of bounded potentials; here we extend the main conclusions — namely, the convergence of the individual eigenvalues to their continuum (and deterministic) counterparts as well as a proof of Gaussian fluctuations around their mean — to a class of unbounded random potentials satisfying suitable, and essentially sharp, moment conditions.

Our setting is as follows: Let be a bounded open subset of  whose boundary is for some . For any , we define the discretized version of  as

 Dϵ:={x∈Zd:dist∞(ϵx,D\rm c)>ϵ}, (1.1)

where is the -distance in . Given any potential , we now consider the linear operator (a matrix)  acting on the linear space of functions that vanish outside  via

 (HDϵ,ξf)(x):=−ϵ−2(Δ(\rm d)f)(x)+ξ(x)f(x),x∈Zd, (1.2)

where is the lattice Laplacian

 (Δ(\rm d)f)(x):=∑y:|x−y|=1[f(y)−f(x)] (1.3)

with denoting the Euclidean distance. Throughout we will take the potential

random, defined on some probability space

, with an -dependent law satisfying one or both of the following requirements (depending on the context):

Assumption 1.1

For each , are independent with

 ∃K>1∨d/2:supϵ∈(0,1)maxx∈DϵE(|ξ(ϵ)(x)|K)<∞. (1.4)

Moreover, there is such that

 Eξ(ϵ)(x)=U(xϵ),x∈Dϵ. (1.5)
Assumption 1.2

The bound (1.4) holds for some . Moreover, there is such that

 \rm Var(ξ(ϵ)(x))=V(xϵ),x∈Dϵ. (1.6)

To ease our notations, we will often omit marking the -dependence of . We are interested in the behavior of the eigenvalues of in the limit as .

Let denote the continuum Laplacian with Dirichlet boundary conditions outside . As it turns out, the continuum (homogenized) counterpart of is the operator

 HD,U:=−Δ+U(x) (1.7)

acting on the space := closure of in the norm , where  denotes the continuum gradient. The operator is self-adjoint and, thanks to our conditions on  and , of compact resolvent. In particular, its spectrum is real-valued and discrete with no eigenvalue more than finitely degenerate — we will thus write to denote the -th smallest eigenvalue of . Our first conclusion is as follows:

Theorem 1.3

Under Assumption 1.1, for each ,

 λ(k)Dϵ,ξP⟶ϵ↓0λ(k)D. (1.8)
Remark 1.4

As we will show in the Appendix, the moment condition (1.4) is more or less optimal for (1.8) to hold. More precisely, if the negative part of  fails to have -nd moment in , we get as . We expect (although have not addressed mathematically) this to be a result of appearance of localized states.

The formula (1.8) determines the leading-order deterministic behavior of the spectrum of . The control of the subleading orders (or even an expansion in powers of ) is a challenging task which we will not tackle here. We will content ourself with a description of the asymptotic behavior of the leading random correction. For reasons to be explained later, we will do this only for any collection of (asymptotically) simple eigenvalues. In order to state the result, we need to fix and define the truncated potential

 ¯¯¯ξ(x):=ξ(x)1{|ξ(x)|≤ϵ−κ}. (1.9)

Our second main result is then:

Theorem 1.5

Suppose Assumptions 1.11.2 hold, fix and let be distinct indices such that the eigenvalues of are simple. Then, in the limit as

, the law of the random vector

 (λ(k1)Dϵ,ξ−Eλ(k1)Dϵ,¯ξϵd/2,…,λ(kn)Dϵ,ξ−Eλ(kn)Dϵ,¯ξϵd/2) (1.10)

tends weakly to a multivariate normal with mean zero and covariance matrix given by

 (1.11)

where is a collection of

-normalized eigenfunctions of

for indices and  is the function from (1.5).

We note that, for simple eigenvalues, the eigenfunctions are determined up to an overall sign (they can always be chosen real valued). In particular, all choices of the eigenfunctions lead to the same value of the integral (1.11). A deeper, albeit related, reason for excluding degenerate eigenvalues is the fact that we work directly with ordered eigenvalues (and not, e.g., the resolvent or some other symmetric function thereof). We expect that, for degenerate eigenvalues, the individual fluctuations are still Gaussian but the order is decided by combining the fluctuation with the expected value (which we control only to the leading order). We do not find this restriction much of a loss as, for generic  and , all eigenvalues of  will be non-degenerate.

Remark 1.6

Under Assumption 1.1, we will see in (2.1) below that the truncation (1.9) has no effect, with probability tending to 1 as . However, it turns out that the truncation does affect the mean value for small , see again the Appendix. Therefore it is necessary to retain the truncated potential inside the expectations in (1.10).

We refer the reader to our earlier paper [3] for a thorough discussion of the above problem as well as related references. We will only mention to papers where we feel an update is necessary. First, an earlier work of Bal [2] derived very similar homogenization and fluctuation results for the eigenvalues of a continuum Anderson Hamiltonian. However, there are a number of important differences:

1. the weak convergence in [2] is proved around the homogenized eigenvalues rather than mean values,

2. the results hold also for sufficiently fast mixing random potentials,

3. the spatial dimension is assumed to be less than or equal to three, , and

4. stronger moment assumption than ours are required.

In particular, if one applies the method of [2] to discrete independent potentials, it requires boundedness of the fourth moments. We believe this is because we use a completely different, mostly probabilistic approach.

Second, related results concerning the low-lying eigenvalues of a random Laplacian arising from random conductances have recently been obtained by Flegel, Haida and Slowik [8]. Also there homogenization of the individual eigenvalues to those of a continuum (albeit “homogenized”) Laplacian is obtained under more or less optimal moment condition on the random conductances.

Notations

Let us collect the notations that will be needed throughout this work. We write for the canonical -norm of - or -valued functions on . When , we use to denote the associated inner product in . All functions defined a priori only on  will be regarded as extended by zero to . In order to control convergence to the continuum problem, it will sometimes be convenient to work with the scaled -norm,

 (1.12)

For , we will write to denote the inner product associated with . For functions  of a continuum variable, we write the norms as  and the inner product in  as . The discrete gradient is defined as the vector in  whose -th component is , where is the canonical basis of .

Some of our computations in the proofs below will require suitable block averaging. For  and , let and for any , define

 fL(x):=∑y∈Zd1BL(y)(x)∑z∈BL(y)L−df(z). (1.13)

Note that, for each given , exactly one  contributes to the first sum; the resulting function is then constant on square blocks of side  and it equals to the average of  on each of them.

Recall that we assumed  to be a bounded open set in with -boundary for some . This ensures a corresponding level of regularity of the eigenfunction. Indeed, by, e.g., Corollary 8.36 of Gilbarg and Trudinger [6], the eigenfunctions of obey

 φ(k)D∈C1,α(¯¯¯¯¯D), (1.14)

that is, they are continuously differentiable in with the gradient uniformly -Hölder continuous. (In particular, the integral (1.11) is convergent.) Concerning the discrete problem, we denote by an (real-valued) eigenfunction of  normalized in ; this is again determined up to a sign whenever the -th eigenvalue is non-degenerate.

Finally, throughout the paper denotes a constant depending only on and whose value may change from line to line. We write () for a negative (resp. positive) power of for simplicity.

2. Convergence to homogenized eigenvalues

We are now in a position to start the exposition of the proofs. Here we will prove Theorem 1.3 dealing with the convergence of the random eigenvalues to those of the continuum problem.

2.1. Truncation

As is common whenever unbounded random variables get involved, we will deal with large values of the potential via a suitable truncation. We begin by noting:

Lemma 2.1

Under Assumption 1.1, for each we have

 (2.1)

Proof. This follows from a union bound, Chebyshev’s inequality, the bound (1.4) and the fact that definition (1.1) implies that is order . ∎

We henceforth fix a so that (2.1) holds, pick  satisfying

 1∨d/2

and assume

 maxx∈Dϵ|ξ(x)|≤ϵ−κ. (2.3)

This is tantamount to working with the truncated potential in place of , which we will however ignore notationally; thanks Lemma 2.1, it suffices to prove Theorem 1.3 under this additional assumption.

Given any choice of the normalized eigenfunctions of the operator (1.7), for each and each  define the event

 Ek,ϵ,γ:=⎧⎪⎨⎪⎩ξ:max1≤j≤k∣∣⟨ξ−U(ϵ⋅),φ(j)D(ϵ⋅)2⟩ϵ,2∣∣<γ∥ξ∥ϵ,r<4|D|maxx∈DϵE[|ξ(x)|r]⎫⎪⎬⎪⎭. (2.4)
Remark 2.2

The constant 4 above plays no special role in the proof. Any larger constant would work as well. We will make use of this observation (only) in the proof of Lemma 3.4 below.

Then we observe:

Lemma 2.3

Under Assumption 1.1 and (2.3), for all and all , and all  sufficiently small,

 P(E\rm ck,ϵ,γ)≤exp{−ϵ0−}. (2.5)

Proof. The proof is based on a number of elementary concentration-of-measure arguments. Let us fix such that

 01K,n=1,…,N. (2.6)

Using this sequence, we write

 ξ(x)−U(ϵx)=(ξ(x)−U(ϵx))1{|ξ(x)|<ϵ−a0}+N∑n=1(ξ(x)−U(ϵx))1{ϵ−an−1≤|ξ(x)|<ϵ−an}=:η(x)+N∑n=1ζn(x) (2.7)

so that

 P(∣∣⟨ξ−U(ϵ⋅),φ(j)D(ϵ⋅)2⟩ϵ,2∣∣≥γ)≤P(∣∣∣∑x∈Dϵϵdη(x)φ(j)D(ϵx)2∣∣∣≥γ2)+N∑n=1P(∑x∈Dϵϵd|ζn(x)|φ(j)D(ϵx)2≥γ2N). (2.8)

First, the Azuma-Hoeffding inequality shows

 P(∣∣ ∣∣∑x∈Dϵϵdη(x)φ(j)D(ϵx)2∣∣ ∣∣≥γ2) ≤2exp{−cϵ−d+2a0} (2.9) ≤exp{−ϵ0−}

for all sufficiently small . Note that due to our use of the truncated potential, a proper use of Azuma-Hoeffding requires an additional intermediate step reflecting on the fact that may not be zero. This is handled by replacing  above with and noting that the difference converges to zero uniformly in . Our implicit truncation (2.3) also sometimes requires this type of considerations and they will be done implicitly in what follows.

Next, we deal with the second term in (2.8). When is sufficiently small, we can bound each summand by

 (2.10)

Since are stochastically dominated by independent Bernoulli variables with success probability

 P(ζn(x)≠0)≤P(|ξ(x)|>ϵ−an−1)≤ϵan−1Ksupϵ∈(0,1)supx∈DϵE[|ξ(x)|K] (2.11)

and , a simple application of the Bernstein inequality tells us that the right-hand side of (2.10) is bounded by for sufficiently small .

The argument for is almost the same. We write and, using the above sequence,

 |ξ(x)|r=|ξ(x)|r1{|ξ(x)|<ϵ−a0}+N∑n=1|ξ(x)|r1{ϵ−an−1≤|ξ(x)|<ϵ−an}=:η(x)+N∑n=1ζn(x) (2.12)

so that

 P(∑x∈Dϵϵd|ξ(x)|r≥4M)≤P(∑x∈Dϵϵdη(x)≥3M)+N∑n=1P(∑x∈Dϵϵdζn(x)≥MN). (2.13)

When is sufficiently small, we have

 ∑x∈DϵϵdE[η(x)]≤2M (2.14)

and we can again appeal to the Azuma-Hoeffding inequality to get

 P(∑x∈Dϵϵdη(x)≥3M)≤P(∑x∈Dϵϵd(η(x)−E[η(x)])≥M)≤2exp{−cϵ−d+2a0}. (2.15)

The rest of the argument is very similar to above and we omit further details. ∎

2.2. Upper bound by homogenized eigenvalue

We will now prove the upper bound in Theorem 1.3. Instead of individual eigenvalues, we will work with their sums

 Λϵk(ξ):=k∑i=1λ(i)Dϵ,ξandΛk:=k∑i=1λ(i)D. (2.16)

These quantities are better suited for dealing with degeneracy because they admit a variational characterization (a.k.a. the Ky Fan Maximum Principle KyFan) of the form

 Λϵk(ξ)=infh1,…,hkONSk∑i=1(ϵ−2∥∇(\rm d)hi∥22+⟨ξ,h2i⟩) (2.17)

and

 Λk=infψ1,…,ψkONSk∑i=1(∥∇ψi∥2L2(Rd)+⟨U,ψ2i⟩L2(Rd)), (2.18)

where the acronym “ONS” imposes that the -tuple of functions (all assumed in the domain of the gradient in the latter case) forms an orthonormal system in the subspace corresponding to Dirichlet boundary conditions.

The infima in (2.172.18) are both achieved by a collection of lowest- eigenfunctions of operators , resp., . This offers a strategy for comparing the two quantities: Take the eigenfunctions of one problem and use them, after discretizing/undiscretizing, as trial functions in the other variational problem. Starting from the continuum problem, this strategy is relatively easy to implement as attested by:

Proposition 2.4

For any and any ,

 Ek,ϵ,γ⊆{Λϵk(ξ)≤Λk+3γ} (2.19)

holds for all sufficiently small . In particular, under Assumption 1.1, for any ,

 limϵ↓0P(Λϵk(ξ)≤Λk+δ)=1. (2.20)

Proof. Consider (a choice of) an ONS of the first  eigenfunctions of . Recall that all of these are in . Now define

 fi(x):={φ(i)D(xϵ),if x∈Dϵ,0,otherwise. (2.21)

Thanks to uniform continuity of the eigenfunctions, we then have

 ⟨fi,fj⟩ϵ,2⟶ϵ↓0⟨φ(i)D,φ(j)D⟩L2(D)=δij (2.22)

and so for  small the functions are nearly mutually orthogonal. Applying the Gram-Schmidt orthogonalization procedure, we conclude that there are functions and coefficients , , such that

 hϵi=k∑j=1(δij+aij(ϵ))fj,i=1,…,k, (2.23)

with

 ⟨hϵi,hϵj⟩ϵ,2=δijandmaxi,j|aij(ϵ)|⟶ϵ↓00. (2.24)

Moreover, the definition of  and the -regularity of the eigenfunctions imply

 (2.25)

and the same applies to instead of as well. Since and are also bounded, we thus get

 ϵ−1∥∇(\rm d)hϵi∥ϵ,2⟶ϵ↓0∥∇φ(i)D∥L2(Rd). (2.26)

The continuity of  shows that, also

 ⟨U(ϵ⋅),(hϵi)2⟩ϵ,2⟶ϵ↓0⟨U,(φ(i)D)2⟩L2(Rd). (2.27)

Therefore, given any , as soon as is sufficiently small (independent of ) the variational characterization (2.17) yields

 Λϵk(ξ)≤Λk+γ+k∑i=1⟨ξ−U(ϵ⋅),(hϵi)2⟩ϵ,2. (2.28)

The summands on the right-hand side are bounded as

 ∣∣⟨ξ−U(ϵ⋅),(hϵi)2⟩ϵ,2∣∣ (2.29)

Noting that the first term is at most  and is bounded on , this will be less than  as soon as  is sufficiently small (again, independent of ). ∎

Corollary 2.5

For each  and each  there is such that for all ,

 Ek,ϵ,γ⊆{Λϵk(ξ)≤ck,γ} (2.30)

Proof. For small-enough , this follows from (2.19) and the fact that  is deterministic. In the complementary range of , we note that (2.3) gives for each . This reduces the problem to bounding the sum of the first  eigenvalues of -times the (negative) Dirichlet Laplacian in square-domains of side-length proportional to , for which the spectrum is explicitly computable (and the eigenvalues are bounded uniformly in ). ∎

2.3. Elliptic regularity for eigenfunctions

For the corresponding lower bound of  by , we will start with the collection of the eigenfunctions of  and turn these into functions over the continuum domain . The main technical obstacle is that the discrete eigenfunctions are random

and so the derivation of the needed regularity estimates (which for the upper bound were supplied by the fact that the eigenfunctions of

are ) require a non-trivial use of elliptic regularity theory. As usual, a starting point for these is a suitable functional inequality:

Lemma 2.6 (Sobolev inequality)

Let  obey in . Then there is such that

 ϵ−2∥∥∇(\rm d)f∥∥2ϵ,2+∥∥f∥∥2ϵ,2≥c(D,q)∥∥f∥∥2ϵ,q (2.31)

holds for all and all  with .

Although this is quite standard, we provide a (short) proof in the Appendix (this will also make it clear that our normalizations are legitimate). A considerably deeper use of elliptic regularity theory is required to control the individual eigenfunctions of . In order to state our first such estimate, pick , where  is as in (2.2), set and, recalling the definition of block-averaged function (1.13), define

 ¯¯¯ξL(x):=(U(ϵ⋅)−ξ(⋅))L(x) (2.32)

Consider the event

 Fϵ,γ:={ξ:∥¯¯¯ξL∥ϵ,r<γ}. (2.33)

Then we have:

Proposition 2.7

Suppose Assumption 1.1. For all , all , and any choice of the -th eigenfunction of , we have

 sup0<ϵ<1supξ∈Ek,ϵ,γ∩Fϵ,γ∥∥ϵ−d/2g(k)Dϵ,ξ∥∥ϵ,p<∞ (2.34)

uniformly in sufficiently small .

Remark 2.8

In Lemma 2.3 we showed that  will occur with overwhelming probability for small enough  and , and a similar statement will be shown for in Lemma 2.11. The reason why event needs to be included in the statement above is that it ensures, via Proposition 2.12 with  below, a lower bound on the principal eigenvalue (uniform in ). Combining with Corollary 2.5 we then get an upper bound on the individual eigenvalues for each , which then feeds into the proof of (2.34) for . Since, for , Corollary 2.5 bounds the principal eigenvalue directly, the inclusion of event in (2.34) is redundant and no logical conflict arises.

Proof of Proposition 2.7. The proof is based on the Moser iteration scheme for solutions of elliptic PDEs. This technique needs to be adapted to the discrete setting which has fortunately already been done in a recent paper of Andres, Deuschel and Slowik [1] on homogenization of the random conductance model with general ergodic random conductances subject (only) to suitable moment conditions. We cite both notation and conclusions at liberty from there.

Given , let us write for the signed-power function and for . By equation (40) of [1], there is a constant depending only on  such that for any function with finite support

 ∑x∈Zd∣∣∇(\rm d)ϕ[s](x)∣∣2≤c(s)∑x∈Zdd∑i=1(ϕ[s−1](x)+ϕ[s−1](x+^\rm ei))2∣∣∇(\rm d)iϕ(x)∣∣2, (2.35)

where is the -th component of the discrete gradient. We further use equation (42) of [1] — with the specific choices and — to get

 (ϕ[s−1](x)+ϕ[s−1](x+^\rm ei))2∣∣∇(\rm d)iϕ(x)∣∣ ≤2(|ϕ(x)|2s−2+|ϕ(x+^\rm ei)|2s−2)|∇(\rm d)iϕ(x)| (2.36)

The key point of using the signed-power function is that and are of the same sign. This permits us to wrap (2.35) as

 ∑x∈Zd∣∣∇(\rm d)ϕ[s](x)∣∣2 ≤2c(s)∑x∈Dϵ∑i|∇(\rm d)iϕ[2s−1](x)||∇(\rm d)iϕ(x)| (2.37) =2c(s)⟨∇(\rm d)ϕ[2s−1],∇(\rm d)ϕ⟩.

where we recall that the brackets stand for the usual inner product in .

Now let us assume that solves the equation in  and vanishes outside . Then we have

 (2.38)

Since and  have the same sign, the right-hand side is bounded by

 ϵ2+d⟨|ϕ|2s,(λ+−ξ)⟩≤ϵ2(∑x∈Dϵϵd|λ+−ξ(x)|r)1/r(∑x∈Dϵϵd|ϕ(x)|2sr′)1/r′=ϵ2∥λ+−ξ∥ϵ,r∥ϕ∥2sϵ,2sr′, (2.39)

where stands for the positive part of and is the Hölder conjugate of . On the other hand, by Lemma 2.6, for any  satisfying (with the right-hand inequality dropped in ) we have

 (2.40)

for some constant . The right-hand side is a multiple of  while, in light of (2.372.39), the left-hand side is bounded by a term involving . This turns (2.40) into a recursion relation

 ∥ϕ∥ϵ,sq≤^c∥ϕ∥ϵ,2sr′ (2.41)

for . For  as in (2.2) we get in  and so, in all , we can find with  and get an improvement in regularity.

Now pick  and let and  and invoke the argument alluded to in Remark 2.8: For , both and are bounded on uniformly in  by definition and Corollary 2.5, and so  is bounded by an absolute constant. Moreover, by definition and, since , for  such that , Hölder’s inequality yields

 ∥ϕ∥ϵ,2sr′≤∥ϕ∥~αϵ,2∥ϕ∥1−~αϵ,sq≤^c1−~α∥ϕ∥~αϵ,2∥ϕ∥1−~αϵ,2sr′, (2.42)

where the second inequality follows from (2.41). This bounds  by ; an iterative use of (2.41) then yields (2.34), as desired.

For , we first use the conclusion for to complete the proof of Proposition 2.12, which shows that is bounded from below on . Then combining with Corollary 2.5, we obtain the boundedness of on and the rest of the computation is the same as before. ∎

As a corollary, we get a regularity result for gradients of eigenfunctions as well:

Corollary 2.9

Under Assumption 1.1, for all , and any choice of the -th eigenfunction of ,

 sup0<ϵ<1supξ∈Ek,ϵ,γ∩Fϵ,γϵ−2∥∥∇(\rm d)g(k)Dϵ,ξ∥∥22<∞, (2.43)

uniformly in  .

Proof. Just plug (2.34) in (2.372.39) with . ∎

Our final regularity lemma addresses approximations of functions by their piecewise-constant counterparts. Recall the definition of  from (1.13). Then we have:

Lemma 2.10

There is such that, for any , any and any with finite support,

 ∥f2−f2L∥p

Proof. For any , Hölder’s inequality shows

 ∥f2−f2L∥p≤∥f−fL∥2∥f+fL∥2p2−p. (2.45)

The first term on the right is bounded by due to the Poincaré inequality and our definition of , while the second terms is at most since is a contraction. ∎

2.4. Lower bound by homogenized eigenvalue

We are now ready to tackle the lower bound in Theorem 1.3. We start by showing that the event from (2.33) occurs with overwhelming probability when  is sufficiently small:

Lemma 2.11

Under Assumption 1.1 and (2.3), for any and all  sufficiently small,

 P(F\rm cϵ,γ)≤exp{−ϵ0−}. (2.46)

Proof. Recall that  for with  as in (2.2). Introducing

 ΞL(y):=∑y∈LZd(ϵL)d∣∣ ∣∣∑z∈BL(y)L−d(U(zϵ)−ξ(z))∣∣ ∣∣r (2.47)

we may write

 ∥¯¯¯ξL∥rϵ,r=∑y∈LZd(ϵL)dΞL(y). (2.48)

Note that is the reciprocal of the number of ’s with up to a multiplicative constant. In addition, note also that in probability for each

(by the Law of Large Numbers and the fact that the truncated-field expectations converge to

), by (2.3) and

 supϵ∈(0,1)supy∈Z