1. Introduction and results
This note is a continuation of our recent paper [3] where we studied the statistics of lowlying eigenvalues of Anderson Hamiltonians in the “homogenization” regime, i.e., under the conditions when a nontrivial 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
(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
(1.2) 
where is the lattice Laplacian
(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
(1.4) 
Moreover, there is such that
(1.5) 
Assumption 1.2
The bound (1.4) holds for some . Moreover, there is such that
(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
(1.7) 
acting on the space := closure of in the norm , where denotes the continuum gradient. The operator is selfadjoint and, thanks to our conditions on and , of compact resolvent. In particular, its spectrum is realvalued 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 ,
(1.8) 
Remark 1.4
The formula (1.8) determines the leadingorder 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
(1.9) 
Our second main result is then:
Theorem 1.5
Suppose Assumptions 1.1–1.2 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
(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 nondegenerate.
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:

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

the results hold also for sufficiently fast mixing random potentials,

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

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 lowlying 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
(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
(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 (realvalued) eigenfunction of normalized in ; this is again determined up to a sign whenever the th eigenvalue is nondegenerate.
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
(2.2) 
and assume
(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
(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:
Proof. The proof is based on a number of elementary concentrationofmeasure arguments. Let us fix such that
(2.6) 
Using this sequence, we write
(2.7) 
so that
(2.8) 
First, the AzumaHoeffding inequality shows
(2.9)  
for all sufficiently small . Note that due to our use of the truncated potential, a proper use of AzumaHoeffding 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
(2.11) 
and , a simple application of the Bernstein inequality tells us that the righthand side of (2.10) is bounded by for sufficiently small .
The argument for is almost the same. We write and, using the above sequence,
(2.12) 
so that
(2.13) 
When is sufficiently small, we have
(2.14) 
and we can again appeal to the AzumaHoeffding inequality to get
(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
(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
(2.17) 
and
(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 (2.17–2.18) 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 ,
(2.19) 
holds for all sufficiently small . In particular, under Assumption 1.1, for any ,
(2.20) 
Proof. Consider (a choice of) an ONS of the first eigenfunctions of . Recall that all of these are in . Now define
(2.21) 
Thanks to uniform continuity of the eigenfunctions, we then have
(2.22) 
and so for small the functions are nearly mutually orthogonal. Applying the GramSchmidt orthogonalization procedure, we conclude that there are functions and coefficients , , such that
(2.23) 
with
(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
(2.26) 
The continuity of shows that, also
(2.27) 
Therefore, given any , as soon as is sufficiently small (independent of ) the variational characterization (2.17) yields
(2.28) 
The summands on the righthand side are bounded as
(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 ,
(2.30) 
Proof. For smallenough , 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 squaredomains of sidelength 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 nontrivial 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
(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 blockaveraged function (1.13), define
(2.32) 
Consider the event
(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
(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 signedpower function and for . By equation (40) of [1], there is a constant depending only on such that for any function with finite support
(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
(2.36)  
The key point of using the signedpower function is that and are of the same sign. This permits us to wrap (2.35) as
(2.37)  
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 righthand side is bounded by
(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 righthand inequality dropped in ) we have
(2.40) 
for some constant . The righthand side is a multiple of while, in light of (2.37–2.39), the lefthand side is bounded by a term involving . This turns (2.40) into a recursion relation
(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
(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
Our final regularity lemma addresses approximations of functions by their piecewiseconstant 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,
(2.44) 
Proof. For any , Hölder’s inequality shows
(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:
Proof. Recall that for with as in (2.2). Introducing
(2.47) 
we may write
(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 truncatedfield expectations converge to
), by (2.3) and
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