Localization in multiparticle Anderson models with weak interaction

Appendix A.1. Proof of Theorem 1.1

To extend Stollmann’s strategy [16] to a multiparticle system, we use multiparticle multiscale analysis estimates in condition DS\((k,N)\).

For \( \mathbf{x} _0\in \mathbb{Z} ^{Nd}\) and an integer \(k\ge0\), using the notation in Lemma 2.1, we set

and define where the positive parameter \(b\) is chosen later. It is easy to verify that Moreover, if \( \mathbf{x} \in A_{k+1}( \mathbf{x} _0)\), then the cubes \( \mathbf{C} ^{(N)}_{L_k}( \mathbf{x} )\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} _0)\) are separable by Lemma 2.1. Now let where \(\Gamma_k:= \mathbf{x} _0+(L_k/3) \mathbb{Z} ^{Nd}\). Condition DS\((k,N)\) with the cardinality of the set \(A_{k+1}( \mathbf{x} _0)\cap\Gamma_k\) taken into account yields Because \(p\ge(\alpha Nd+1)/2\) (in fact, \(p\ge6Nd\)), we find that \(\sum_{k=0}^{\infty} \mathbb{P} \{\Omega_k( \mathbf{x} _0)\}\) is finite. Setting we obtain \( \mathbb{P} \{\Omega_{\infty}\}=1\) by the Borel–Cantelli lemma and the fact that \( \mathbb{Z} ^{Nd}\) is denumerable. Consequently, it suffices to prove that any eigenfunction \( \boldsymbol{\Psi} \) of \( \mathbf{H} ^{(N)}(\omega)\) for \(\omega\in \Omega_{\infty}\) decays exponentially.

Let \( \boldsymbol{\Psi} \) be a polynomially bounded eigenfunction satisfying the inequality of eigenfunction decay (see Theorem 2.4). Let the norm \(\| \mathbf{1} _{ \mathbf{C} ^{(N)}_1( \mathbf{x} _0)} \boldsymbol{\Psi} \|\ne0\) for \( \mathbf{x} _0\in \mathbb{Z} ^{Nd}\) (if such \( \mathbf{x} _0\) does not exist, then the proof is complete). Then \( \mathbf{C} ^{(N)}_{L_k}( \mathbf{x} _0)\) cannot be an \((E,m,h)\)-NS cube for infinitely many \(k\). Indeed, if \( \mathbf{C} ^{(N)}_{L_k}( \mathbf{x} _0)\) is an \((E,m,h)\)-NS cube for a given \(k\ge0\), then by the inequality of eigenfunction decay and the polynomial estimate for \( \boldsymbol{\Psi} \), we obtain

The right-hand side of this inequality tends to zero as \(L_k\to\infty\), which contradicts the choice of \( \mathbf{x} _0\). Therefore, there exists a finite \(k_1=k_1(\omega,E, \mathbf{x} _0)\) such that \( \mathbf{C} ^{(N)}_{L_k}( \mathbf{x} _0)\) is an \((E,m,h)\)-S cube for all \(k\ge k_1\). At the same time, because \(\omega\in \Omega_{\infty}\), there exists \(k_2=k_2(\omega, \mathbf{x} _0)\) such that \(\Omega_k( \mathbf{x} _0)\) is not realized for \(k\ge k_2\). We therefore conclude that for all \(k\ge\max\{k_1,k_2\}\) and all \( \mathbf{x} \in A_{k+1}( \mathbf{x} _0)\cap\Gamma_k\), \( \mathbf{C} ^{(N)}_{L_k}( \mathbf{x} )\) is an \((E,m,h)\)-NS cube.

Let \(\rho\in(0,1)\), We choose

and hence

Let \( \mathbf{x} \in \tilde A_{k+1}( \mathbf{x} _0)\). Because \(| \mathbf{x} - \mathbf{x} _0|\ge b_kL_k/(1-\rho)\), we have the estimate

Further, \(| \mathbf{x} - \mathbf{x} _0|\le bb_{k+1}L_{k+1}/(1+\rho)\), and therefore Consequently, We now set \(k_3=\max\{k_1,k_2\}\), and (A.1) then implies that by virtue of Let \(k\ge k_3\). We recall that any cube with a center in \(A_{k+1}( \mathbf{x} _0)\cap \Gamma_k\) and side length \(2L_k\) is an \((E,m,h)\)-NS cube. Therefore, for any \( \mathbf{x} \in \tilde A_{k+1}( \mathbf{x} _0)\), we can choose \( \mathbf{x} _1\in A_{k+1}( \mathbf{x} _0)\) such that \( \mathbf{x} \in \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} _1)\). Consequently, Up to a set of Lebesgue measure zero, we can cover a cube \( \mathbf{C} ^{(N, \mathrm{out} )}_{L_k}( \mathbf{x} _1)\) by at most \(3^{Nd}\) cubes \( \mathbf{C} ^{(N, \mathrm{in} )}_{L_k} ( \tilde { \mathbf{x} })\) with \( \tilde { \mathbf{x} }\in\Gamma_k\) and \(| \tilde { \mathbf{x} }- \mathbf{x} _1|=L_k/3\). Choosing \( \mathbf{x} _2\) that gives a maximum norm, we obtain and hence

By induction, we can thus find \( \mathbf{x} _1, \mathbf{x} _2,\dots, \mathbf{x} _n\) in \(\Gamma_k\cap A_{k+1}( \mathbf{x} _0)\) with the estimate

Because \(| \mathbf{x} _i- \mathbf{x} _{i+1}|=L_k/3\) and \( \operatorname{dist} ( \mathbf{x} , \partial A_{k+1})\ge \rho\cdot| \mathbf{x} - \mathbf{x} _0|\), iterating at most \(\rho\cdot| \mathbf{x} - \mathbf{x} _0|\cdot3/{L_k}\) times, we reach the boundary of the set \(A_{k+1}( \mathbf{x} _0)\).

Using the polynomial estimate for \( \boldsymbol{\Psi} \), we obtain

We conclude that for given \(\rho'\in(0,1)\), we can find \(k_4\ge k_3\) such that for \(k\ge k_4\), if \(| \mathbf{x} - \mathbf{x} _0|\ge b_{k_4}L_{k_4}/(1-\rho)\). This completes the proof of the exponential localization of eigenfunctions in a uniform norm.

Appendix A.2. Proof of Lemma 2.1

We prove the first statement in the lemma. Let \(L>0\), \( \varnothing \ne \mathcal{J} \subset \{1,\dots,n\}\), and \( \mathbf{y} \in \mathbb{Z} ^{nd}\). We call \(\{y_j\}_{j\in \mathcal{J} }\) an \(L\)-cluster if \(\bigcup_{j\in \mathcal{J} } C^{(1)}_L(y_j)\) cannot be decomposed into two nonempty disjoint subsets.

For two given configurations \( \mathbf{x} , \mathbf{y} \in \mathbb{Z} ^{nd}\), we proceed as follows:

1. 1.

   We decompose the vector \( \mathbf{y} \) into maximum \(L\)-clusters \(\Gamma_1,\dots,\Gamma_M\) (the diameter of each cluster does not exceed \(2nL\)), where \(M\le n\).

2. 2.

   Each point \(y_i\) corresponds to exactly one cluster \(\Gamma_j\), \(j=j(i)\in\{1,\dots,M\}\).

3. 3.

   If there exists \(j\in\{1,\dots,M\}\) such that \(\Gamma_j\cap \Pi \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} )= \varnothing \), then the cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{y} )\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} )\) are separable. Otherwise, \(\Gamma_k\cap\Pi \mathbf{C} ^{(n)}_L( \mathbf{x} )\ne \varnothing \) for all \(k=1,\dots,M\). For each \(k=1,\dots,M\), there then exists \(i\in\{1,\dots,n\}\) such that \(\Gamma_k\cap C^{(1)}_L(x_i)\ne \varnothing \). Further, for each \(j=1,\dots,n\), there exists \(k\in\{1,\dots,M\}\) such that \(y_j\in\Gamma_k\). For this \(k\), there then exists \(i\in\{1,\dots,n\}\) such that \(\gamma_k\cap C^{(1)}_L(x_i)\ne \varnothing \). Now let \(z\in\Gamma_k\cap C^{(1)}_L(x_i)\), and hence \(|z-x_i|\le L\). By virtue of \(y_j\in\Gamma_k\), we have

   $$|y_j-x_i|\le |y_j-z|+|z-x_i|\le 2nL-L+L=2nL.$$

We note that we have the estimate \(|y_j-z|\le nL-L\) because \(y_j\) is the center of the \(L\)-cluster \(\Gamma_k\). For each \(j=1,\dots,n\), the coordinate \(y_j\) in the \(n\)-dimensional configuration \((y_1,\dots,y_n)\) must then belong to one of the cubes \(C^{(1)}_{2nL}(x_i)\). We set \(\kappa(n)=n^n\). Whatever the choice from at most \(\kappa(n)\) possibilities for the element \( \mathbf{y} =(y_1,\dots,y_n)\), this element must belong to the Cartesian product of \(n\) cubes of side length \(2L\), i.e., an \(nd\)-dimensional cube of side length \(2nL\), whence follows the first statement in the lemma.

We prove the second statement. We set

We consider a cube \( \mathbf{C} ^{(n)}_L( \mathbf{x} )\) for which \(| \mathbf{y} - \mathbf{x} |\ge R( \mathbf{y} )\). There then exists \(i_0\in\{1,\dots,n\}\) such that \(|y_{i_0}-x_{i_0}|\ge R( \mathbf{y} )\). We consider the maximum connected component \(\Lambda_{ \mathbf{x} }:= \bigcup_{i\in \mathcal{J} }C^{(1)}_L(x_i)\) containing \(x_{i_0}\) in the union \(\bigcup_iC^{(1)}_L(x_i)\). The diameter of \(\Lambda_{ \mathbf{x} }\) does not exceed \(2nL\). We have Because we obtain We recall that \( \operatorname{diam} \Lambda_{ \mathbf{x} }\le2nL\) and for some \(j\in\{1,\dots,n\}\) such that \(v\in C^{(1)}_L(y_j)\). Finally, we obtain the inequality and the right-hand side is strictly positive. It hence follows that \( \mathbf{C} ^{(n)}_L( \mathbf{x} )\) is \( \mathcal{J} \) separable from \( \mathbf{C} ^{(n)}_L( \mathbf{y} )\).

Appendix A.3. Proof of Lemma 4.1

We introduce the notation \(R:=2L+r_0\) and assume that

If the union of cubes \(C^{(1)}_{R/2}(u_i)\), \(i=1,\dots,n\), is not decomposable into two (or more) disjoint groups, then it is a connected set, its diameter is hence bounded by \(n(2(R/2))=nR\), and then \( \operatorname{diam} \Pi \mathbf{u} \le nR\), which contradicts the assumption made. Therefore, there exists an index subset \( \mathcal{J} \subset\{1,\dots,n\}\) such that \(|u_{j_1}-u_{j_2}|\ge2(R/2)\) for all \(j_1\in \mathcal{J} \) and \(j_2\in \mathcal{J} ^{ \mathrm{c} }\). We hence have

Appendix A.4. Proof of Lemma 4.2

If

for some \(R>0\), then there exists \(j_0\in\{1,\dots,n\}\) such that \(|x_{j_0}-y_{j_0}|\ge R\). By hypothesis, \( \mathbf{C} ^{(n)}_L( \mathbf{x} )\) and \( \mathbf{C} ^{(n)}_L( \mathbf{y} )\) are FI cubes, and therefore By the triangle inequality, for any \(i,j\in\{1,\dots,n\}\) and \(R\ge7nL\ge 6nL+2nr_0\), we have Therefore, which proves the statement in the lemma.

Appendix A.5. Proof of Lemma 4.5

We assume that \(M^{ \mathrm{sep} }( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} ),E)\) does not exceed 2 (i.e., there do not exist two separable cubes of radius \(L_k\) in \( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\)) but \(M( \mathbf{C} ^{(n)}( \mathbf{u} ),E)\ge\kappa(n)+2\). Then \( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\) must contain at least \(\kappa(n){+}2\) cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{v} _i)\), \(0\le i\le\kappa(n)+1\) that are separable, but in this case, \(| \mathbf{v} _i- \mathbf{v} _{i'}|\ge7NL_k\) for all \(i\ne i'\).

On the other hand, by Lemma 2.1, there exist at most \(\kappa(n)\) cubes \( \mathbf{C} ^{(n)}_{2nL_k}( \mathbf{y} _i)\) such that each cube \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{x} )\) with a center \( \mathbf{x} \notin\bigcup_j \mathbf{C} ^{(n)}_{2nL_k}( \mathbf{y} _j)\) is separable from \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{v} _0)\). Hence, \( \mathbf{v} _i\in\bigcup_j \mathbf{C} ^{(n)}_{2nL_k}( \mathbf{y} _j)\) for all \(i=1,\dots,\kappa(n)+1\). But \(| \mathbf{v} _i- \mathbf{v} _{i'}|\ge7NL_k\) for \(i\ne i'\), and each cube \( \mathbf{C} ^{(n)}_{2nL_k}(y_j)\), \(j=1,\dots,\kappa(n)\), must contain at least one center \( \mathbf{v} _i\). This leads to the contradiction \(\kappa(n)+1\le\kappa(n)\).

The reasoning holds if we consider only PI cubes.

Appendix A.6. Proof of Lemma 4.6

Let \(M_{ \mathrm{PI} }( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} ),I)\ge\kappa(n)+2\). Then \(M^{ \mathrm{sep} }_{ \mathrm{PI} }( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} ),I)\ge2\) by Lemma 4.5, i.e., there are at least two separable \((E,m,h)\)-S PI cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j_1)}\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j_2)})\) in \( \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\). The number of possible pairs of centers \(\{ \mathbf{u} ^{(j_1)}, \mathbf{u} ^{(j_2)}\}\) such that

is bounded above by \((3^{2nd}/2)L_{k+1}^{2nd}\). Setting we then obtain where \( \mathbb{P} \{ \mathrm{B} _k\}\le L_k^{-4^Np}+L_k^{-4p4^{N-n}}\).

Appendix A.7. Proof of Lemma 4.8

We assume that there exist \(2\ell\) pairwise separable FI cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j)})\subset \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\), \(j=1,\dots,2\ell\). Then by Lemma 4.2, for any pair \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(2i-1)})\) and \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(2i)})\), the corresponding random Hamiltonians \( \mathbf{H} ^{(n)}_{ \mathbf{C} ^{(n)}_{L_k} ( \mathbf{u} ^{(2i-1)})}\) and \( \mathbf{H} ^{(n)}_{ \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(2i)})}\) are independent, as also are their spectra and their Green’s functions. We consider the events

By condition DS\((k,n)\), we have for \(i=1,\dots,\ell\), and because the events \(A_1,\dots,A_{\ell}\) are independent, To complete the proof, we note that the total number of different families of \(2\ell\) cubes \( \mathbf{C} ^{(n)}_{L_k}( \mathbf{u} ^{(j)})\subset \mathbf{C} ^{(n)}_{L_{k+1}}( \mathbf{u} )\), \(j=1,\dots,2\ell\), is bounded by