The Localization Transition in the Ultrametric Ensemble

# The Localization Transition in the Ultrametric Ensemble

author \minisecAbstract We study the hierarchical analogue of power-law random band matrices, a symmetric ensemble of random matrices with independent entries whose variances decay exponentially in the metric induced by the tree topology on . We map out the entirety of the localization regime by proving the localization of eigenfunctions and Poisson statistics of the suitably scaled eigenvalues. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model.

## 1 Introduction

One-dimensional lattice Hamiltonians with random long-range hopping provide a useful and simplified testing ground for the Anderson metal-insulator transition in more complicated systems. Two prominent examples of such models are the random band matrices [4, 13], whose entries are zero outside some band , and the power-law random band matrices (PRBM) [12, 19], whose entries have variances decaying according to some power of the Euclidean distance . Even for these models, the mathematically rigorous understanding is far from complete, although there has been some progress; see [22, 9, 2, 6, 10, 21] and references therein. This article is concerned with a further simplification, the ultrametric ensemble of Fyodorov, Ossipov and Rodriguez [14], which is essentially obtained by replacing the Euclidean distance in the definition of the PRBM with the metric induced by the tree topology.

The index space of the ultrametric ensemble is endowed with the metric

 d(x,y)=min{r≥0|x and y lie in a common member of Pr},

where is the nested sequence of partitions defined by

 Bn={1,...,2r}∪{2r+1,...,2⋅2r}∪...∪{2n−r−12r+1,...,2n}.

The basic building blocks of the ultrametric ensemble are the matrices whose entries are independent (up to the symmetry constraint) centered real Gaussian random variables with variance

 E∣∣⟨δy,Φn,rδx⟩∣∣2=2−r⎧⎨⎩2 if d(x,y)=01 if 1≤d(x,y)≤r0 otherwise. (1.1)

Thus is a direct sum of random matrices drawn independently from the Gaussian Orthogonal Ensemble (GOE) of size . The ultrametric ensemble with parameter refers to the random matrix

 Hn=1Zn,cn∑r=02−(1+c)2rΦn,r (1.2)

where and are independent for . We choose the normalizing constant such that

 ∑y∈BnE∣∣⟨δy,Hnδx⟩∣∣2=1,

which means that grows exponentially in in case and is asymptotically constant in case . The original definition in [14] contains an additional parameter governing the relative strengths of the diagonal and off-diagonal disorder, but this parameter does not significantly alter our analysis and so we omit it altogether. Moreover, the authors of [14] constructed the block matrices from the Gaussian Unitary Ensemble (GUE), and our results apply to both GOE and GUE blocks with only slight changes.

The ultrametric ensemble is a hierarchical analogue of the PRBM in a sense which was first introduced to statistical mechanical models by Dyson [8], and studied rigorously in the context of quantum hopping systems with only diagonal disorder in [3, 15, 16, 17, 20, 24]. Thus, one expects the core features of the PRBM phase transition to be present in the ultrametric ensemble as well. Indeed, the authors of [14] present arguments at a theoretical physics level of rigor as well as numerical evidence for a localization-delocalization transition in the eigenfunctions of as the parameter changes from to . The effect of the Gaussian perturbations on the spectrum of can be described dynamically by Dyson Brownian motion [7] and, in this sense, the critical point is natural because it governs whether the evolution passes the local equilibration time of this system or not.

In this article we establish the localized phase by proving that the eigenfunctions remain localized and the level statistics converge to a Poisson point process if . Both statements are improvements over the rigorous results for band matrices, where the only localization result is due to Schenker [21] and no proof of Poisson statistics is known. For the first main result in this paper, we recall that, by the Wegner estimate [25], the infinite-volume density of states measure defined through

 ν(f)=limn→∞2−n∑λj∈σ(Hn)f(λj) (1.3)

has a bounded Radon-Nikodym derivative, the density of states, whose values we denote by .

###### Theorem 1.1 (Poisson statistics).

Suppose and let be a Lebesgue point of . Then, the random measure

 μn(f)=∑λ∈σ(Hn)f(2n(λ−E))

converges in distribution to a Poisson point process with intensity as .

The proof of Theorem 1.1 is contained in Section 2.

The second main result of this paper says that if an eigenfunction of in some mesoscopic energy interval has any mass at some , then actually all but an exponentially small amount of the total mass is carried in an exponentially vanishing fraction of the volume near with high probability. We formulate this result in terms of the eigenfunction correlator

 Qn(x,y;W)=∑λ∈σ(Hn)∩W|ψλ(x)ψλ(y)|,

which in completely delocalized regimes is typically given by

 ∑λ∈σ(Ht)∩W∑y≠x|ψλ(x)ψλ(y)|≈∑λ∈σ(Ht)∩W1≈2n|W|.

Thus, since grows very large for mesoscopic spectral windows, it is a signature of localization if the eigenfunction correlator asymmptotically vanishes for small enough mesoscopic intervals , as is proved in the following theorem.

###### Theorem 1.2 (Eigenfunction localization).

Suppose and let . Then, there exist , , and a sequence with such that for every the -normalized eigenfunctions satisfy

 P⎛⎝∑y∈Bn∖Bmn(x)Qn(x,y;W)>2−μn⎞⎠≤C2−κn

with

 W=[E0−2−(1−w)n,E0+2−(1−w)n].

The proof of Theorem 1.2 may be found in Section 3.

Our results gain additional interest upon noting that, when , the ultrametric ensemble has an essential mean field character and techniques originally developed for Wigner matrices show that the energy levels agree asymptotically with those of the GOE and that the eigenfunctions are completely delocalized. We will now roughly sketch how to apply these results in the present situation and state the corresponding theorems. The key observation is that the normalizing factor , which scales the spectrum to , is given by

 Z2n,c=∑y∈BnE∣∣ ∣∣⟨δy,(n∑r=02−(1+c)2rΦn,r)δx⟩∣∣ ∣∣2=(1−2−(1+c)(n+1))1+O(1)1−2−(1+c),

 Mn\vcentcolon=(maxx,y∈BnE|⟨δy,Hnδx⟩|2)−1={Z2n,c2−o(n)ifc≥−2,2(1+o(1))nifc<−2,

grows like a positive power of the system size when . The results of [11] then show that the semicircle law (i.e. ) is valid on scales of order even for the matrices

 ˜Hn=1Zn,cn−1∑r=02−(1+c)2rΦn,r+1−√TnZn,c2−(1+c)2nΦn,n

with a small part of the final Gaussian component removed. We set with . The validity of the local semicircle law already implies the complete delocalization of the eigenfunctions in mesoscopic windows in the bulk of the spectrum (see [9, Thm. 2.21]).

###### Theorem 1.3 (cf. [9, 11]).

For any compact interval there exist such that for all the -normalized eigenfunctions of in satisfy

 ∥ψλ∥∞=O(M−1/2+ϵn)

with probability .

Random matrix universality of the local statistics may be expressed by saying that the -point correlation functions

 ρ(k)Hn(λ1,..λk)=∫R2n−kρHn(λ1,...,λ2n)dλk+1...dλ2n,

the -th marginals of the symmetrized eigenvalue density , locally agree with the corresponding objects for the GOE asymptotically. For this, we employ the work of Landon, Sosoe and Yau [18, Thm. 2.2] concerning the universality of Gaussian perturbations for

 Hn=˜Hn+√TnZn,c2−(1+c)2nΦn,n.

For the statement of the theorem, let

 Ψ(k)n,E(α1,...,αk) =ρ(k)Hn(E+2−nα1ρsc(E),...,E+2−nαkρsc(E)) −ρ(k)GOE(E+2−nα1ρsc(E),...,E+2−nαkρsc(E)),

where is the -point correlation function of the GOE and is the density of the semicircle law.

###### Theorem 1.4 (cf. [18, 11]).

Suppose , and . Then,

 limn→∞∫RkO(α)Ψ(k)n,E(α)dα=0

for every .

Summing up, these results rigorously prove the existence of a metal-insulator transition in the ensemble of ultrametric random matrices. In particular, our results allow an approach all the way to the critical point from the localized side , which greatly improves upon the best known corresponding result for random band matrices [21]. However, the above arguments do not cover the regime , in which the local eigenvalue statistics are still expected to be of Wigner-Dyson-Mehta type as in the case [14].

## 2 Proof of Poisson Statistics

When , the limit exists. Thus, we may drop the normalizing constant from the definition of in the remainder of this paper without any loss of generality. The goal of this section is to prove Theorem 1.1 by approximating with the truncated Hamiltonian

 Hn,m=m∑r=02−1+c2rΦn,r, (2.1)

which has the property that, for any ,

 Hn,m=2n−k⨁j=1H(j)k,m, (2.2)

where each is an independent copy of . Therefore

 μn,m(f)=∑λ∈σ(Hn,m)f(2n(λ−E))

consists of independent components, a fact whose relevance to Theorem 1.1 is contained in the following characterization of Poisson point processes.

###### Proposition 2.1.

Let be a collection of point processes such that:

1. The point processes are independent for all .

2. If is a bounded Borel set, then

 limn→∞supj≤NnP(μn,j(B)≥1)=0.
3. There exists some such that if is a bounded Borel set with , then

 limn→∞Nn∑j=1P(μn,j(B)≥1)=c|B|

and

 limn→∞Nn∑j=1P(μn,j(B)≥2)=0.

Then, converges in distribution to a Poisson point process with intensity .

We recall [1] that a sequence of point processes converges in distribution to whenever

 limn→∞Ee−μn(Pz)=Ee−μ(Pz)

for all , where is the rescaled Poisson kernel

 Pz(λ)=Im1λ−z=Imz(λ−Rez)2+(Imz)2. (2.3)

Hence, Theorem 1.1 follows by furnishing a sequence such that Proposition 2.1 applies to and

 limn→∞Ee−μn,mn(Pz)=limn→∞Ee−μn(Pz) (2.4)

for all .

The difference is a Gaussian perturbation with time parameter . Therefore, Theorem 1.1 of [23] shows that there exists such that for all we have

 12nE∣∣Tr(Hn−zℓ)−1−Tr(Hn,n−1−zℓ)−1∣∣ ≤Cz2−c2n−1(1+23(ℓ−n)) ≤Cz2−c2n23(ℓ−n) (2.5)

with . Our strategy in achieving (2.4) thus consists of applying (2) to the finite-volume density of states measures

 νn(f)=2−nTrf(Hn),νn,m(f)=2−nTrf(Hn,m)

in an iterative fashion.

###### Theorem 2.2.

There exist and such that

 E∣∣νn(Pzℓ)−νn,m(Pzℓ)∣∣≤Cz23(ℓ−(1+δ)m)

for all .

###### Proof.

The estimate (2) proves that

 E∣∣νk(Pzℓ)−νk,k−1(Pzℓ)∣∣≤Cz23(ℓ−(1+δ)k) (2.6)

with when . Since is given by a telescopic sum,

 νn(Pzℓ)−νn,m(Pzℓ)=n∑k=m+1(νn,k(Pzℓ)−νn,k−1(Pzℓ)),

the decomposition (2.2) implies that

 νn,k(Pzℓ)−νn,k−1(Pzℓ)=2−(n−k)2n−k∑j=1(νk(Pzℓ)−νk,k−1(Pzℓ)). (2.7)

Applying (2.6) to each term in (2.7) yields

 E∣∣νn(Pzℓ)−νn,m(Pzℓ)∣∣ ≤n∑k=m+1Cz23(ℓ−(1+δ)k)≤Cz23(ℓ−(1+δ)m).

Theorem 2.2 has two important implications for the measures and which are based on the identities and . The first of these enables us to find a suitable sequence satisfying (2.4).

###### Corollary 2.3.

There exists a sequence with and such that

for all .

###### Proof.

Since , there exists a sequence with , and . By applying Theorem 2.2 with , we obtain

 E|μn(Pz)−μn,mn(Pz)|≤Cz23(n−(1+δ)mn)→0.

To finish the proof of Theorem 1.1, we need to show that satisfies the hypothesis of Proposition 2.1. By (2.2), is a sum of point processes

 μn,mn=2n−mn∑j=1μmn,j

with independent . If is a bounded Borel set, the theorem of Combes-Germinet-Klein [5] asserts that and hence for any :

 P(μmn,j(B)≥ℓ)≤(C|B|2mn−n)ℓℓ!. (2.8)

Since , the first hypothesis of Proposition 2.1 follows. Writing

 X(n,ℓ)=2n−m∑j=1P(μmn,j(B)≥ℓ),

(2.8) implies

 X(n,ℓ)≤2n−mn(C|B|2mn−n)ℓℓ!→0

when . In particular, and the last hypothesis of Proposition 2.1 is satisfied. It remains to prove the remaining hypothesis of Proposition 2.1, which is the second important consequence of Theorem 2.2 and is contained in the following theorem.

###### Theorem 2.4.

Let be a bounded Borel set. Then,

 limn→∞X(n,1)=ν(E)|B|.
###### Proof.

By (2.2) we have for any , and so we conclude from Theorem 2.2 with that

 limn→∞|E[νn(Pzn)−ν(Pzn)]| =limn→∞limp→∞|E[νn(Pzn)−νp(Pzn)]| =limn→∞limp→∞|E[νp,n(Pzn)−νp(Pzn)]| ≤limn→∞Cz2−3δn=0.

This shows that the measures satisfy

 limn→∞(Eμn(Pz)−λn(Pz))=0

and Corollary 2.3 implies that also

 limn→∞(Eμmn(Pz)−λn(Pz))=0. (2.9)

For any bounded Borel set , the indicator is in the -closure of the finite linear combinations from the set and the measures are absolutely continuous with uniformly bounded densities by the Wegner estimate. Together, these two observations yield that (2.9) is valid for any bounded Borel set . Moreover, since is a Lebesgue point of ,

 limn→∞λn(B)=limn→∞2nν(2−nB+E)=ν(E)|B|,

and hence we have shown that

 limn→∞Eμmn(B)=ν(E)|B|. (2.10)

Since takes values in the non-negative integers

 limn→∞X(n,1)=limn→∞2n−m∑j=1Eμnm,j(B)−limn→∞∑ℓ≥2X(n,ℓ)

so  (2.8), (2.10) and the dominated convergence theorem give

 limn→∞X(n,1)=limn→∞2n−m∑j=1Eμnm,j(B)=ν(E)|B|.

## 3 Proof of Eigenfunction Localization

In this section, we prove Theorem 1.2 by comparing the eigenfunctions of with the obviously localized eigenfunctions of . As in Section 2, we drop the normalizing constant from the definition of . The core of this argument again consists of resolvent bounds for Gaussian perturbations, and thus we consider the Green functions

 Gn(x,y;z)=⟨δy,(Hn−z)−1δx⟩,Gn,m(x,y;z)=⟨δy,(Hn,m−z)−1δx⟩.

If for some , Theorem 1.2 of [23] proves that there exists such that

 2−k∑y∈Bk(x)E|Gk(x,y;E+iη)−Gk,k−1(x,y;E+iη)| ≤C2−c2k(1+23((1+ℓ)n−k)) =C23(1+ℓ)n−3(1+δ)k

with whenever . Iterating this result, we see that

 2−n∑y∈BnE|Gn(x,y;E+iη)−Gn,m(x,y;E+iη)| ≤2−nn∑k=m+1∑y∈BnE|Gn,k(x,y;E+iη)−Gn,k−1(x,y;E+iη)| =2−nn∑k=m+1∑y∈Bk(x)E|Gk(x,y;E+iη)−Gk,k−1(x,y;E+iη)| ≤2−nn∑k=m+12kC23(1+ℓ)n−3(1+δ)k≤C2(3(1+ℓ)−1)n2−(3(1+δ)−1)m.

Since , we can choose , , and such that

 2μ\vcentcolon=(1−ϵ)(3(1+δ)−1)−(3(1+ℓ)−1)−w>0.

Thus, setting and

 W=[E−2−(1−w)n,E+2−(1−w)n],

and using that if show that

 ∑y∈Bn∖Bmn(x)E∫W|ImGn(x,y;E+iη)|dE≤C2−2μn.

Applying Markov’s inequality, we arrive at

 P⎛⎝∑y∈Bn∖Bmn(x)∫W|ImGn(x,y;E+iη)|dE>2−μn⎞⎠≤C2−μn,

so Theorem 1.2 follows from Theorem 5.1 of [23], which says that

 ∑y∈Bn∖Bmn(x)Qn(x,y;W)≤∑y∈Bn∖Bmn(x)∫W|ImGn(x,y;E+iη|dE+log2n2wn

with probability .

### Acknowledgments

We thank Y. V. Fyodorov for introducing us to the ultrametric ensemble. This work was supported by the DFG (WA 1699/2-1).

Per von Soosten Zentrum Mathematik, TU München Boltzmannstraße 3, 85747 Garching, Germany vonsoost@ma.tum.de Simone Warzel Zentrum Mathematik, TU München Boltzmannstraße 3, 85747 Garching, Germany warzel@ma.tum.de

### References

1. M. Aizenman and S. Warzel. Random operators: Disorder effects on quantum spectra and dynamics, volume 168 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015.
2. P. Bourgade, L. Erdős, H.-T. Yau, and J. Yin. Universality for a class of random band matrices. arXiv:1602.02312, 2016.
3. A. Bovier. The density of states in the Anderson model at weak disorder: a renormalization group analysis of the hierarchical model. J. Statist. Phys., 59(3-4):745–779, 1990.
4. G. Casati, L. Molinari, and F. Izrailev. Scaling properties of band random matrices. Phys. Rev. Lett., 64:1851–1854, Apr 1990.
5. J.-M. Combes, F. Germinet, and A. Klein. Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys., 135(2):201–216, 2009.
6. M. Disertori, H. Pinson, and T. Spencer. Density of states for random band matrices. Comm. Math. Phys., 232(1):83–124, 2002.
7. F. J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Mathematical Phys., 3:1191–1198, 1962.
8. F. J. Dyson. Existence of a phase-transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys., 12(2):91–107, 1969.
9. L. Erdős. Universality of Wigner random matrices: a survey of recent results. Russian Mathematical Surveys, 66(3):507, 2011.
10. L. Erdős, A. Knowles, H.-T. Yau, and J. Yin. Delocalization and diffusion profile for random band matrices. Comm. Math. Phys., 323(1):367–416, 2013.
11. L. Erdős, A. Knowles, H.-T. Yau, and J. Yin. The local semicircle law for a general class of random matrices. Electron. J. Probab., 18:no. 59, 58, 2013.
12. F. Evers and A. D. Mirlin. Anderson transitions. Rev. Mod. Phys., 80:1355–1417, Oct 2008.
13. Y. V. Fyodorov and A. D. Mirlin. Scaling properties of localization in random band matrices: A -model approach. Phys. Rev. Lett., 67:2405–2409, Oct 1991.
14. Y. V. Fyodorov, A. Ossipov, and A. Rodriguez. The Anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices. Journal of Statistical Mechanics: Theory and Experiment, 2009(12):L12001, 2009.
15. E. Kritchevski. Hierarchical Anderson model. In Probability and mathematical physics, volume 42 of CRM Proc. Lecture Notes, pages 309–322. Amer. Math. Soc., Providence, RI, 2007.
16. E. Kritchevski. Poisson statistics of eigenvalues in the hierarchical Anderson model. Ann. Henri Poincaré, 9(4):685–709, 2008.
17. S. Kuttruf and P. Müller. Lifshits tails in the hierarchical Anderson model. Ann. Henri Poincaré, 13(3):525–541, 2012.
18. B. Landon, P. Sosoe, and H.-T. Yau. Fixed energy universality for Dyson Brownian motion. Preprint available at arXiv:1609.09011, 2016.
19. A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman. Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices. Phys. Rev. E, 54:3221–3230, Oct 1996.
20. S. Molchanov. Hierarchical random matrices and operators. Application to Anderson model. In Multidimensional statistical analysis and theory of random matrices (Bowling Green, OH, 1996), pages 179–194. VSP, Utrecht, 1996.
21. J. Schenker. Eigenvector localization for random band matrices with power law band width. Comm. Math. Phys., 290(3):1065–1097, 2009.
22. S. Sodin. The spectral edge of some random band matrices. Ann. of Math. (2), 172(3):2223–2251, 2010.
23. P. von Soosten and S. Warzel. Local stability of the resolvent flow under Dyson Brownian motion. Preprint available at arXiv:1705.00923, 2017.
24. P. von Soosten and S. Warzel. Renormalization group analysis of the hierarchical Anderson model. Ann. Henri Poincaré, 2017 (in press).
25. F. Wegner. Bounds on the density of states in disordered systems. Z. Phys. B, 44(1-2):9–15, 1981.
264062