Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

Introduction

The general formulation of the universality conjecture for disordered systems states that there are two distinctive regimes depending on the energy and the disorder strength. In the strong disorder regime, the eigenfunctions are localized and the local spectral statistics are Poisson. In the weak disorder regime, the eigenfunctions are delocalized and the local statistics coincide with those of a Gaussian matrix ensemble.

For the Anderson model, a fundamental open question is to establish the metal-insulator transition, i.e. to show that in $d\geqslant 3$ dimensions the eigenfunctions of $-\Delta+\lambda V$ are delocalized for small disorder $\lambda$. The localization regime at large disorder or near the spectral edges has been well understood by Fröhlich and Spencer with the multiscale technique , and later by Aizenman and Molchanov by the fractional moment method ; many other works have since contributed to this field. In particular, it has been established that the local eigenvalue statistics are Poisson and that the eigenfunctions are exponentially localized with an upper bound on the localization length that diverges as the energy parameter approaches the presumed phase transition point .

The progress in the delocalization regime has been much slower. For the Bethe lattice, corresponding to the infinite-dimensional case, delocalization has been established in . In finite dimensions only partial results are available. The existence of an absolutely continuous spectrum (i.e. extended states) has been shown for a rapidly decaying potential, corresponding to a scattering regime . Diffusion has been established for a heavy quantum particle immersed in a phonon field in $d\geqslant 4$ dimensions . For the original Anderson Hamiltonian with a small coupling constant $\lambda$, the eigenfunctions have a localization length of at least $\lambda^{-2}$ (see ). The time and space scale $\lambda^{-2}$ corresponds to the kinetic regime where the quantum evolution can be modelled by a linear Boltzmann equation . Beyond this time scale the dynamics is diffusive. This has been established in the scaling limit $\lambda\to 0$ up to time scales $t\sim\lambda^{-2-\kappa}$ with an explicit $\kappa>0$ in . There are no rigorous results on the local spectral statistics of the Anderson model, but it is conjectured – and supported by numerous arguments in the physics literature, especially by supersymmetric methods (see ) – that the local correlation function of the eigenvalues of the finite volume Anderson model follows the GOE statistics in the thermodynamic limit.

Due to their mean-field character, Wigner matrices are simpler to study than the Anderson model and they are always in the delocalization regime. The complete delocalization of the eigenvectors was proved in . The local spectral statistics in the bulk are universal, i.e. they follow the statistics of the corresponding Gaussian ensemble (GOE, GUE, GSE), depending on the symmetry type of the matrix (see for explicit formulas). For an arbitrary single entry distribution, bulk universality has been proved recently in for all symmetry classes. A different proof was given in for the Hermitian case.

Supersymmetric methods offer a very attractive approach to study the delocalization transition in band matrices but the rigorous control of the functional integrals away from the saddle points is difficult and it has been performed only for the density of states . Effective models that emerge near the saddle points can be more accessible to rigorous mathematics. Recently Disertori, Spencer and Zirnbauer studied a related statistical mechanics model that is expected to reflect the Anderson localization and delocalization transition for real symmetric band matrices. They proved a quasi-diffusive estimate for the two-point correlation functions in a three dimensional supersymmetric hyperbolic nonlinear sigma model at low temperatures . Localization was also established in the same model at high temperatures .

We also mention that band matrices are not the only possible interpolating models to mimic the metal-insulator transition. Other examples include the Anderson model with a spatially decaying potential and a quasi one-dimensional model with a weak on-site potential for which a transition in the sense of local spectral statistics has been established in .

The main result of this paper is that the quantum dynamics of the $d$-dimensional band matrix is given by a superposition of heat kernels up to time scales $t\ll W^{d/3}$. Although diffusion is expected to hold up to time $t\sim W^{2}$ for $d=1$ and up to any time for $d\geqslant 3$ (assuming the thermodynamic limit has been taken), our method can follow the quantum dynamics only up to $t\ll W^{d/3}$. The threshold exponent $d/3$ originates in technical estimates on certain Feynman graphs; going beyond the exponent $d/3$ would require a further resummation of certain four-legged subdiagrams (see Section 11).

Finally, we remark that our method also yields a bound on the largest eigenvalue of a band matrix; see Theorem 3.4 in the forthcoming paper for details.

The problem of diffusion for random band matrices originated from several discussions with H.T. Yau and J. Yin. The authors are especially grateful to J. Yin for various insights and for pointing out an improvement in the counting of the skeleton diagrams.

The Setup

the number of points at distance at most $W$ from the origin. In the following we tacitly make use of the obvious relation $M\sim\;CW^{d}$. For notational convenience, we use both $W$ and $M$ in the following.

a cube with side length $N$ centred around the origin. Here $[\cdot]$ denotes integer part. We regard $\Lambda_{N}$ as periodic, i.e. we equip it with periodic addition and the periodic distance

Unless otherwise stated, all summations $\sum_{x}$ are understood to mean $\sum_{x\in\Lambda_{N}}$.

We consider random matrices $H^{\omega}\equiv H$ whose entries $H_{xy}$ are indexed by $x,y\in\Lambda_{N}$. Here $\omega$ denotes the running element in probability space. The large parameter of the model is the band width $W$. We shall always assume that $N\geqslant WM^{1/6}$. Under this condition all our results hold uniformly in $N$.

We assume that $H$ is either Hermitian or symmetric. The entries $H_{xy}$ satisfying $1\leqslant\lvert x-y\rvert\leqslant W$ are i.i.d. (with the obvious restriction that $H_{yx}=\overline{H_{xy}}\!\,$). In the Hermitian case they are uniformly distributed on a circle of appropriate radius in the complex plane,

If $\lvert x-y\rvert\notin[1,W]$ then $H_{xy}=0$. An important consequence of our assumptions (2.1a) and (2.1b) is

We remark that the assumption that the matrix entries have the special form (2.1a) or (2.1b) is not necessary for our results to hold. We make it here because it greatly simplifies our proof. The reason for this is that, as observed by Feldheim and Sodin , the condition (2.2) allows one to obtain a simple algebraic expression for the nonbacktracking powers of $H$; see Lemma 5.2.

Scaling and results

up to higher order terms in $t$. Thus $\sum_{x\neq 0}\varrho(t,x)$ is an $O(1)$ quantity, separated away from zero, indicating that the distance from the origin is of $O(W)$ for times $t\sim O(1)$.

In time $t$ the particle performs $O(t)$ jumps of size $O(W)$. We expect that the jumps are approximately independent and the trajectory is a random walk consisting of $O(t)$ steps with size $O(W)$ each. Thus, the typical distance from the origin is of order $t^{1/2}W$. We rescale time and space $(t,x)\mapsto(T,X)$ so as to make the macroscopic quantities $T$ and $X$ of order one, i.e. we set

where $W$ and $\eta$ are two large parameters. Ideally, one would like to study the long time limit $\eta\to\infty$ for a fixed $W$. In this case, however, we know that the dynamics cannot be diffusive for $d=1$. Indeed, as explained in the introduction, it is expected that the motion cannot be diffusive for distances larger than $W^{2}$; this has in fact been proved for distances larger than $W^{8}$. Thus we have to consider a scaling limit where $\eta$ and $W$ are related and they tend simultaneously to infinity. To that end we choose an exponent $\kappa>0$ and set $\eta\equiv\eta(W)\mathrel{\mathop{:}}=W^{d\kappa}$.

Our first main result establishes that $\varrho(t,x)$ behaves diffusively up to time scales $t=O(W^{d\kappa})$ if $\kappa<1/3$.

uniformly in $N\geqslant W^{1+d/6}$ and $0\leqslant T\leqslant T_{0}$. Here

Let $\varepsilon>0$ and $0<\kappa<1/3$. Then

Theorem 3.3 implies that the fraction of eigenvectors subexponentially localized on scales $W^{1+\kappa d/2}$ converges to zero in probability.

For fixed $\gamma>0$ and $K>0$ define the random subset of eigenvectors

Main ideas of the proof

This can be seen as follows. The expectation

After the Chebyshev transform, we need to compute expectations

The main work consists of proving that the non-ladder diagrams are negligible. Similarly to the basic idea of , the non-ladder diagrams are classified according to their combinatorial complexity. The large number of complex diagrams is offset by their small value, expressed in terms of powers of $W$. Conversely, diagrams containing large pieces of ladder subdiagrams have a relatively large contribution but their number is small.

More precisely, focusing only on the pairing diagrams in the Hermitian case, it is easy to see that ladder subdiagrams are marginal for power counting. We define the skeleton of a graph by collapsing parallel ladder rungs (called bridges) into a single rung. We show that the value of a skeleton diagram is given by a negative power of $M\sim CW^{d}$ that is proportional to the size of the skeleton diagram. This is how the dimension $d$ enters our estimate. We then sum up all possible ladder subdiagrams corresponding to a given skeleton. Although the ladder subdiagrams do not yield additional $W$-powers, they represent classical random walks for which dispersive bounds are available, rendering them summable. The restriction $t\ll W^{d/3}$ comes from summing up the skeleton diagrams. In Section 11 we present a critical skeleton that shows that this restriction is necessary without further resummation or a more refined classification of complex graphs.

The path expansion

Here $U_{k}$ denotes the Chebyshev polynomial of the second kind, defined through

for $k=0,1,2,\dots$. The Chebyshev polynomials satisfy the orthogonality relation

Therefore the coefficients $\alpha_{k}(t)$ are given by

The coefficient $\alpha_{k}(t)$ can be evaluated explicitly using the standard identities (see )

Here $T_{k}$ denotes the Chebyshev polynomial of the first kind and $J_{k}$ the Bessel function of the first kind; they are defined through

If $k=2l$ is even we may therefore compute

If $k=2l+1$ is odd a similar calculation yields

as follows from the orthonormality of the Chebyshev polynomials.

2 Expansion in terms of nonbacktracking paths

For $n=0,1,2,\dots$ let $H^{(n)}$ denote the $n$-th nonbacktracking power of $H$. It is defined by

where $\sum^{\prime}$ means sum under the restriction $x_{i}\neq x_{i+2}$ for $i=0,\dots,n-2$. We call this restriction the nonbacktracking condition.

The following key observation is due to Bai and Yin .

The nonbacktracking powers of $H$ satisfy

For the convenience of the reader we give the simple proof. The cases $n=0,1,2$ are easily checked. Moreover,

Notice that in the last step we used (2.2). ∎

Feldheim and Sodin have observed that (5.5) is reminiscent of the recursion relation for the Chebyshev polynomials of the second kind. Let us abbreviate $\widetilde{U}_{n}(\xi)\mathrel{\mathop{:}}=U_{n}(\xi/2)$. Then we have (see e.g. )

Comparing this to Lemma 5.2, we get, following ,

Solving for $\widetilde{U}_{n}(H)$ yields

with the convention that $H^{(n)}=0$ for $n<0$. Therefore Lemma 5.1 yields

Graphical representation

For ease of presentation, we assume throughout the proof of Theorem 3.1 (Sections 6 – 8) that we are in the Hermitian case (2.1a). How to extend our arguments to cover the symmetric case (2.1b) is described in Section 9.

Expanding in nonbacktracking paths yields a graphical expansion. Let us write $H^{(n)}_{0x}H^{(n^{\prime})}_{x0}$ as a sum over paths $x_{0},x_{1},\dots,x_{n+n^{\prime}-1},x_{0}$, where $x_{0}=0$ and $x_{n}=x$. Such a path is graphically represented as a loop of $n+n^{\prime}$ vertices belonging to the set $\mathcal{V}_{n,n^{\prime}}\mathrel{\mathop{:}}=\{0,\dots,n+n^{\prime}-1\}$; see Figure 6.1. Vertices $i\in\mathcal{V}_{n,n^{\prime}}$ satisfying the nonbacktracking condition (i.e. $x_{i-1}\neq x_{i+1}$) are drawn using black dots; other vertices are drawn using white dots.

There are $n+n^{\prime}$ oriented edges $e_{0},\dots,e_{n+n^{\prime}-1}$ defined by $e_{i}\mathrel{\mathop{:}}=(i,i+1)$ (here, and in the following, $\mathcal{V}_{n,n^{\prime}}$ is taken to be periodic). We denote by $\mathcal{E}_{n,n^{\prime}}\mathrel{\mathop{:}}=\{e_{0},\dots,e_{n+n^{\prime}-1}\}$ the set of edges. In Figure 6.1 the edges are oriented clockwise. Each vertex has an outgoing and an incoming edge, and each edge $e$ has an initial vertex $a(e)$ and final vertex $b(e)$. Moreover, we order the edges using their initial vertices.

The two last products implement the nonbacktracking condition. We define the unordered pair of labels corresponding to the edge $e$ through

where the summation is restricted to label configurations yielding the lumping $\Gamma$.

Next, observe that the expectation of a monomial $\prod_{y,z}(H_{yz})^{\nu_{yz}}$ is nonzero if and only if $\nu_{yz}=\nu_{zy}$ for all $y,z$ (here we only use that the law of the matrix entries is invariant under rotations of the complex plane). In particular, $V_{x}(\Gamma)$ vanishes if one lump $\gamma\in\Gamma$ is of odd size. Defining the subset $\mathscr{G}_{n,n^{\prime}}\subset\widetilde{\mathscr{G}}_{n,n^{\prime}}$ of lumpings whose lumps are of even size, we find that

We summarize the key properties of $\mathscr{G}_{n,n^{\prime}}$.

Let $\Gamma\in\mathscr{G}_{n,n^{\prime}}$. Then each lump $\gamma\in\Gamma$ is of even size. Moreover, any two edges $e,e^{\prime}\in\gamma$ in the same lump $\gamma$ are separated by either at least two edges or a vertex in $\{0,n\}$ (nonbacktracking property).

Note that the expectation in (6.1) is equal to

where $\bar{n}\mathrel{\mathop{:}}=\frac{n+n^{\prime}}{2}$. In particular, $V_{x}(\Gamma)\geqslant 0$.

An important subset of lumpings of $\mathcal{E}_{n,n^{\prime}}$ is the set of pairings, $\mathscr{P}_{n,n^{\prime}}\subset\mathscr{G}_{n,n^{\prime}}$, which contains all lumpings $\Gamma$ satisfying $\lvert\gamma\rvert=2$ for all $\gamma\in\Gamma$. We call two-element lumps $\sigma\in\mathcal{P}_{n,n^{\prime}}$ bridges. Given a pairing $\Gamma\in\mathscr{P}_{n,n^{\prime}}$, we say that $e$ and $e^{\prime}$ are bridged (in $\Gamma$) if there is a $\sigma\in\Gamma$ such that $\sigma=\{e,e^{\prime}\}$. Bridges are represented graphically by drawing a line, for each $\{e,e^{\prime}\}\in\Gamma$, from the edge $e$ to $e^{\prime}$; see Figure 6.2. Thus a pairing $\Gamma\in\mathscr{P}_{n,n^{\prime}}$ is the edge set of a graph whose vertex set is $\mathcal{E}_{n,n^{\prime}}$. If $\Gamma$ is a pairing, each bridge $\sigma\in\Gamma$ has a unique partition $\pi_{\sigma}$ of its edges, so that the expression (6.1) for $V_{x}(\Gamma)$ may be rewritten in the simpler form

The main contribution to the expansion is given by the ladder pairing $L_{n}\in\mathscr{P}_{n,n}$. It is defined as

The ladder is represented graphically in Figure 6.3.

The non-ladder lumpings

In this section we estimate the contribution of the non-ladder lumpings and show that it vanishes in the limit $W\to\infty$. Let $\mathscr{G}_{n,n^{\prime}}^{*}\subset\mathscr{G}_{n,n^{\prime}}$ denote the set of non-ladder lumpings, i.e. $\mathscr{G}^{*}_{n,n^{\prime}}\mathrel{\mathop{:}}=\mathscr{G}_{n,n^{\prime}}$ if $n\neq n^{\prime}$ and $\mathscr{G}^{*}_{n,n}\mathrel{\mathop{:}}=\mathscr{G}_{n,n}\setminus\{L_{n}\}$. Similarly, let $\mathscr{P}_{n,n^{\prime}}^{*}\mathrel{\mathop{:}}=\mathscr{P}_{n,n^{\prime}}\cap\mathscr{G}^{*}_{n,n^{\prime}}$ denote the set of non-ladder pairings.

Let $0<\kappa<1/3$ and pick a $\beta$ satisfying $0<\beta<2/3-2\kappa$. Then there is a constant $C$ such that

for $W$ larger than some $W_{0}(T,\kappa)$ and $N\geqslant W^{1+d/6}$.

The rest of this section is devoted to the proof of Proposition 7.1.

Replacing the expectation in (6.1) with (6.2) we get

We start by estimating the sum over all lumpings $\Gamma\in\mathscr{G}_{n,n^{\prime}}^{*}$ in terms of a sum over all pairings $\Gamma\in\mathscr{P}_{n,n^{\prime}}^{*}$. Let us define

Let $\varrho_{\gamma}$ and $\pi_{\gamma}$ be given for each $\gamma\in\Gamma$. For each $\gamma$, pick any pairing $\Sigma_{\gamma}$ of $\gamma$ that is compatible with $\pi_{\gamma}$ in the sense that, for each bridge $\sigma\in\Sigma_{\gamma}$, the two edges of $\sigma$ belong to different subsets of $\pi_{\gamma}$. If $n=n^{\prime}$, we additionally require that not all $\Sigma_{\gamma}$’s are subsets of the Ladder $L_{n}$ (such a choice is always possible). Next, set $\varrho_{\sigma}=\varrho_{\gamma}$ for all $\sigma\in\Sigma_{\gamma}$. Note that each bridge $\sigma$ carries a unique partition $\pi_{\sigma}$. It is then easy to see that for any pairing $\Sigma_{\gamma}$ as above, we have

Thus, by partitioning each $\gamma\in\Gamma$ into bridges, we see that each term in $\sum_{\Gamma\in\mathscr{G}_{n,n^{\prime}}^{*}}V_{x}(\Gamma)$ is bounded by a corresponding term in $\sum_{\Gamma\in\mathscr{P}_{n,n^{\prime}}^{*}}R_{x}(\Gamma)$. In fact, there is an overcounting arising from the different ways of partitioning $\gamma$ into bridges. ∎

Because of Lemma 7.2 we may restrict ourselves to pairings. We estimate $\sum_{\Gamma\in\mathscr{P}_{n,n^{\prime}}^{*}}R_{x}(\Gamma)$. If $\Gamma$ is a pairing we may write, just like (6.3), the expression (7.1) in the simpler form

2 Collapsing of parallel bridges

Let us introduce the set $\overline{\mathscr{P}}\!\,^{*}_{n,n^{\prime}}$, defined as the set of all non-ladder pairings of $\mathcal{E}_{n,n^{\prime}}$. Clearly, $\mathscr{P}^{*}_{n,n^{\prime}}$ is a proper subset of $\overline{\mathscr{P}}\!\,^{*}_{n,n^{\prime}}$ (due to the nonbacktracking condition of Lemma 6.1 which is imposed on pairings in $\mathscr{P}^{*}_{n,n^{\prime}}$).

Let $n,n^{\prime}\geqslant 0$ and $\Gamma\in\overline{\mathscr{P}}\!\,^{*}_{n,n^{\prime}}$. For any $i,j$, we say that the two bridges $\{e_{i},e_{j}\}$ and $\{e_{i+1},e_{j-1}\}$ of $\Gamma$ are parallel if $i+1,j\notin\{0,n\}$; see Figure 7.1. Two parallel bridges may be collapsed to obtain a new pairing $\Gamma^{\prime}$ of a smaller set of edges, in which the parallel bridges are replaced by a single bridge. More precisely: We obtain $\Gamma^{\prime}\in\overline{\mathscr{P}}\!\,^{*}_{m,m^{\prime}}$ from $\Gamma\in\overline{\mathscr{P}}\!\,^{*}_{n,n^{\prime}}$ by removing the vertices $i+1$ and $j$, by creating the edges $(i,i+2)$ and $(j-1,j+1)$, and by bridging them. Finally, we rename the vertices using the increasing integers $0,1,2,\dots,n+n^{\prime}-3$; by definition, the new name of the vertex $n$ is $m$, and $m^{\prime}$ is defined through $m+m^{\prime}+2=n+n^{\prime}$.

The converse operation of collapsing bridges, expanding bridges, is self-explanatory.

In the next lemma we iterate the above procedure $\Gamma\mapsto\Gamma^{\prime}$ until all parallel bridges have been collapsed.

Let $\Gamma\in\mathscr{P}^{*}_{n,n^{\prime}}$. Then there exist $m\leqslant n$, $m^{\prime}\leqslant n^{\prime}$, and a pairing $S(\Gamma)\in\overline{\mathscr{G}}\!\,^{*}_{m,m^{\prime}}$ containing no parallel bridges, such that $\Gamma$ may be obtained from $S(\Gamma)$ by successively expanding bridges. This defines $S(\Gamma)$ uniquely.

Successively collapse all parallel bridges in $\Gamma$; see Figure 7.2. The result is clearly independent of the order in which this is done.

We call the pairing $\Sigma=S(\Gamma)$ the skeleton of $\Gamma$. The set of skeleton pairings of the edges $\mathcal{E}_{m,m^{\prime}}$ is denoted by

Note that $\mathscr{S}^{*}_{m,m^{\prime}}$ is in general not a subset of $\mathscr{P}^{*}_{m,m^{\prime}}$. The following lemma summarizes the key properties of $\mathscr{S}^{*}_{m,m^{\prime}}$.

Each $\Sigma\in\mathscr{S}^{*}_{m,m^{\prime}}$ contains no parallel bridges.

Let $\Sigma\in\mathscr{S}^{*}_{m,m^{\prime}}$ and $\sigma=\{e,e^{\prime}\}\in\Sigma$. Then $e,e^{\prime}$ are adjacent only if $e\cap e^{\prime}\in\{0,m\}$.

If $\bar{m}\mathrel{\mathop{:}}=\frac{m+m^{\prime}}{2}=1$ then $\mathscr{S}^{*}_{m,m^{\prime}}=\emptyset$.

Statement (i) follows immediately from the definition of $S(\Gamma)$. Statement (ii) is a consequence of the nonbacktracking property of pairings in $\mathscr{P}^{*}_{n,n^{\prime}}$, i.e. Lemma 6.1. To see this, let $\Sigma\in\mathscr{S}^{*}_{m,m^{\prime}}$ be of the form $\Sigma=S(\Gamma)$ for some $\Gamma\in\mathscr{P}^{*}_{n,n^{\prime}}$. If $\Sigma=S(\Gamma)$ contains a bridge $\{e,e^{\prime}\}$ consisting of two consecutive edges $e,e^{\prime}$, then $\Gamma$ must also contain a bridge $\{f,f^{\prime}\}$ consisting of two consecutive edges $f,f^{\prime}$. If $e\cap e^{\prime}\notin\{0,m\}$, then $f\cap f^{\prime}\notin\{0,n\}$, in contradiction to Lemma 6.1. Statement (iii) is an immediate consequence of (ii) and the requirement that $L_{1}\notin\mathscr{S}^{*}_{1,1}$. ∎

3 Contribution of parallel bridges

For given $n$ and $n^{\prime}$, we estimate $\sum_{\Gamma\in\mathscr{P}^{*}_{n,n^{\prime}}}R_{x}(\Gamma)$ by summing over skeleton pairings $\Sigma$, followed by summing over all possible ways of expanding the bridges of $\Sigma$.

The first two statements are obvious. The last follows from a standard local central limit theorem; see for instance the proof in . ∎

4 Orbits of vertices

Fix $\Gamma\in\mathscr{P}^{*}_{n,n^{\prime}}$. We observe that the product in (7.2) may be interpreted as an indicator function that fixes labels along paths of vertices. To this end, we define a map $\tau\equiv\tau_{\,\Gamma}$ on the vertex set $\mathcal{V}_{n,n^{\prime}}$. Start with a vertex $i\in\mathcal{V}_{n,n^{\prime}}$. Let $e$ be the outgoing edge of $i$ (i.e. $e=(i,i+1)$), and $e^{\prime}$ the edge bridged by $\Gamma$ to $e$. Then we define $\tau i$ as the final vertex of $e^{\prime}$ (i.e. $e^{\prime}=(\tau i-1,\tau i)$). Thus the product in (7.2) may be rewritten as

Starting from any vertex $i\in\mathcal{V}_{n,n^{\prime}}$ we construct a path $(i,\tau i,\tau^{2}i,\dots)$. In this fashion the set of vertices is partitioned into orbits of $\tau$; see Figure 7.4. Let $[i]\subset\mathcal{V}_{n,n^{\prime}}$ denote the orbit of the vertex $i\in\mathcal{V}_{n,n^{\prime}}$.

Next, let $Z^{*}(\Sigma)\mathrel{\mathop{:}}=Z(\Sigma)\setminus\{\}$ and define $L(\Sigma)\mathrel{\mathop{:}}=\lvert Z^{*}(\Sigma)\rvert$. The set $Z^{*}(\Sigma)$ is the set of orbits whose label is summed over in $\sum_{x}R_{x}(\Gamma)$. The following lemma gives an upper bound on $L(\Sigma)$. It states, roughly, that the number of orbits (or free labels) is bounded by $2\bar{m}/3$; we refer to it as the $2/3$ rule. Compare this bound with the trivial bound $L(\Sigma)\leqslant\bar{m}$, which would be sharp if $\Sigma$ were allowed to have parallel bridges.

Let $\Sigma\in\mathscr{S}_{m,m^{\prime}}^{*}$. Then $L(\Sigma)\leqslant\frac{2\bar{m}}{3}+\frac{1}{3}$.

Let $Z^{\prime}(\Sigma)\mathrel{\mathop{:}}=Z(\Sigma)\setminus\{,[m]\}$. We show that every orbit $\zeta\in Z^{\prime}(\Sigma)$ consists of at least 3 vertices. Let $i\in\mathcal{V}_{m,m^{\prime}}$ belong to $\zeta\in Z^{\prime}(\Sigma)$. Then, by Lemma 7.4 (ii), we have that $\tau i\neq i$. By assumption, $\tau i\notin\{0,m\}$. Hence $\tau^{2}i\neq i$, for otherwise $\Sigma$ would have two parallel bridges, in contradiction to Lemma 7.4 (i). Therefore the orbit of $\tau$ contains at least 3 vertices. Note that there are orbits containing exactly 3 vertices, as depicted in Figure 7.4.

The total number of vertices of $\Sigma$ not including the vertices and $m$ is $2\bar{m}-2$, so that we get

The claim follows from the bound $\lvert Z^{*}(\Sigma)\rvert\leqslant\lvert Z^{\prime}(\Sigma)\rvert+1$. ∎

There is a subset of bridges $\Sigma_{T}\subset\Sigma$ of size $\lvert\Sigma_{T}\rvert=L(\Sigma)$, such that, in the subgraph of $\Pi(\Sigma)$ with the edge set $\phi(\Sigma_{T})$, each orbit $\zeta\in Z^{*}(\Sigma)$ is connected to $$.

Starting from $\zeta_{0}=$, we construct a sequence of orbits $\zeta_{0},\zeta_{1},\dots,\zeta_{L(\Sigma)}$, and a sequence of bridges $\sigma_{1},\dots,\sigma_{L(\Sigma)}$, with the property that for all $k=1,\dots,L(\Sigma)$ there is a $k^{\prime}<k$ such that $\zeta_{k}$ and $\zeta_{k^{\prime}}$ are connected by $\phi(\sigma_{k})$.

Assume that $\zeta_{0},\dots,\zeta_{k-1}$ have already been constructed. Let $i$ be the smallest vertex of $\mathcal{V}_{m,m^{\prime}}\setminus(\zeta_{0}\cup\cdots\cup\zeta_{k-1})$. Then we set $\zeta_{k}=[i]$. By construction, the vertex $i-1$ belongs to an orbit $\zeta_{k^{\prime}}$ for some $k^{\prime}<k$. Set $\sigma_{k}$ to be the bridge containing $\{i-1,i\}$. Hence, by definition of $\Pi(\Sigma)$, we see that $\zeta_{k}$ and $\zeta_{k^{\prime}}$ are connected by $\phi(\sigma_{k})$.

The set $\Sigma_{T}$ is given by $\{\sigma_{1},\dots,\sigma_{L(\Sigma)}\}$. ∎

Because $\lvert\Sigma_{T}\rvert=L(\Sigma)$, the subgraph of $\Pi(\Sigma)$ with the edge set $\phi(\Sigma_{T})$ is a tree that connects all orbits in $Z^{*}(\Sigma)$ to $$. Let us call this tree$\mathcal{T}(\Sigma)$. Its root is$$.

Indeed, using Lemma 7.7 and $\bar{m}\geqslant 2$ we find

Since $N\geqslant WM^{1/6}$ and $M\sim CW^{d}$ we find

where we replaced $d$ with $1$ to obtain an upper bound. Thus we get

by Lemma 7.5. Continuing in this manner until we reach the root, we find

6 Sum over pairings

We may now estimate $\sum_{n+n^{\prime}=2p}\sum_{\Gamma\in\mathscr{P}_{n,n^{\prime}}^{*}}\sum_{x}R_{x}(\Gamma)$ for fixed $p$. Let first $p,m,m^{\prime}\geqslant 0$ and $\Sigma\in\mathscr{S}^{*}_{m,m^{\prime}}$. Then (7.9) yields

The sum on the right-hand side is equal to

This expresses the fact that the first edge of $\Sigma$ can be bridged with at most $(2\bar{m}-1)$ edges, the next remaining edge with at most $(2\bar{m}-3)$ edges, and so on. Therefore (7.3) and Lemma 7.4 (iii) yield

7 Conclusion of the proof

In this subsection we complete the proof of Proposition 7.1 by showing that the error

satisfies $E_{W}=o(1)$ as $W\to\infty$, uniformly in $N\geqslant W^{1+d/6}$.

We begin by deriving bounds on the coefficients $a_{n}(t)$.

The term $k=k^{\prime}=0$ yields $1$ by (5.4). The rest is equal, by (5.4), to

In order to prove (ii), we use the integral representation (see )

Let us first consider the case $t\leqslant n$. Then it is easy to see that $\frac{t^{n+2k}}{(n+2k)!}\leqslant\frac{t^{n}}{n!}$. Together with (7.14) this yields

If $t>n$ we have $\frac{t^{n}}{n!}\geqslant C$. Thus the bound (7.15) yields

Using the new variables $p\mathrel{\mathop{:}}=\bar{n}=\frac{n+n^{\prime}}{2}$ and $q\mathrel{\mathop{:}}=\frac{n-n^{\prime}}{2}$ we find from the definition (7.11)

Next, we observe that Lemma 7.9 (ii) implies that terms corresponding to $n,n^{\prime}\gg t=\eta T\sim CM^{\kappa}T$ are strongly suppressed. Thus we introduce a cutoff at $p=M^{\mu}$, where $\kappa<\mu<\frac{1}{3}$. Let us first consider the terms $p\leqslant M^{\mu}$. We need to estimate

by (7.10). For $p\leqslant M^{\mu}$ and $W$ large enough, the term in the square brackets is bounded by

Thus we find \bigl{(}{E^{\leqslant}_{W}}\bigr{)}^{2}\leqslant CM^{\mu-1/3}.

Let us now consider the case $p>M^{\mu}$, i.e. estimate

By (7.13) and the elementary inequality $\frac{p!}{(p-q)!}\leqslant\frac{(p+q)!}{p!}$ we have

by (7.10). Setting $\eta\sim CM^{\kappa}$ yields

Choosing $\mu=1/3-\beta$ (where, we recall, $0<\beta<2/3-2\kappa$) completes the proof of Proposition 7.1.

The ladder pairings

In this section we analyse the contribution of the ladder pairings, $\sum_{n\geqslant 0}\lvert a_{n}(\eta T)\rvert^{2}\,V_{x}(L_{n})$, and complete the proof of Theorem 3.1. (Recall that $\eta\mathrel{\mathop{:}}=W^{d\kappa}$ is the time scale.) Recalling the expression (6.3), and noting that in the case of the ladder the variables $x_{0},\dots,x_{n}$ determine the value of all variables $x_{0},\dots,x_{2n-1}$, we readily find

Throughout this section we assume that $\eta=W^{d\kappa}$ for some $\kappa<1/3$.

We perform a series of steps to simplify the expression (8.1). In a first step, we get rid of the last product.

Under the assumptions of Proposition 7.1 we have

Next, we estimate $\sum_{x}\lvert E_{x}^{1}\rvert$. We begin by observing that each partition $P$ of $\{0,\dots,n-1\}$ uniquely defines a partition $\Gamma(P)\in\mathscr{G}_{n,n}^{*}$. Indeed, each lump $p\in P$ gives rise to the lump $\gamma\in\Gamma(P)$ defined by $\gamma=\bigcup_{i\in p}\{e_{i},e_{2n-1-i}\}$. In particular, $\Gamma(P)\neq\Gamma(P^{\prime})$ if $P\neq P^{\prime}$. We now claim that

Invoking Proposition 7.1 completes the proof. ∎

In a second step, we get rid of the second to last product in (8.1), i.e. the nonbacktracking condition.

The expression in the square brackets is equal to

We introduce a cutoff at $n=M^{1/3}$. The part $n\leqslant M^{1/3}$ is bounded by

by Lemma 7.9 (i). The part $n>M^{1/3}$ is estimated using Lemma 7.9 (ii), exactly as in the estimate of $E_{W}^{>}$ in Section 7.7. ∎

Under the assumptions of Proposition 7.1 we have

The claim follows from Lemmas 8.1 and 8.2, combined with an argument identical to the proof of Lemma 7.9 (i) that allows us to replace $\lvert a_{n}(t)\rvert^{2}$ with $\lvert\alpha_{n}(t)\rvert^{2}$. We replaced the factor $\frac{1}{(M-1)^{n}}$ with $\frac{1}{M^{n}}$ by introducing a cutoff at $n=M^{1/3}$, exactly as in the proof of Lemma 8.2. ∎

In a third step, we use the central limit theorem to replace $P_{x}(n)$ with a Gaussian. Recall the definition of the heat kernel

where $[\cdot]$ denotes the integer part.

in the expectation in (8.3). The second resulting term is bounded by

This vanishes in the limit $W\to\infty$ by the central limit theorem, since $\frac{N}{W\sqrt{[\eta T]}}\to\infty$ by assumption.

The first term resulting from the partition is

by the same argument as above. Therefore we get

While the distribution $\lvert\alpha_{n}(t)\rvert^{2}$ has no limit as $t\to\infty$, it turns out that the rescaled distribution,

In order to prove this, we consider the integrated distribution

We now show that $F_{t}(\lambda)$ converges pointwise to $F(\lambda)=\int_{0}^{\lambda}f$. See Figure 8.1 for a graph of the functions $f_{t},f,F_{t},F$.

exists for all $\lambda\geqslant 0$ and satisfies

In order to conclude the proof of Theorem 3.1, we need the following result.

Indeed, Theorem 3.1 is an immediate consequence of Propositions 7.1 and 8.6. The rest of this section is devoted to the proof of Proposition 8.6.

We begin by observing that the family of probability measures defined by the densities $\{f_{t}\}_{t\geqslant 0}$ is tight, so that we may cut out values of $\lambda$ in the range $[0,\delta)\cup(1-\delta,\infty)$.

Let $\varepsilon>0$. Then there is a $\delta>0$ and a $t_{0}\geqslant 0$ such that

as $\delta\to 0$. Choose $\delta>0$ small enough that the left-hand side of (8.6) is bounded by $\frac{\varepsilon}{2\lVert\varphi\rVert_{\infty}}$. Moreover, Proposition 8.5 also implies that there is a $t_{0}$ such that

Now by (8.4), Proposition 8.6 will follow if we can show

Lemma 8.7 implies that in order to prove (8.7) it suffices to prove

Let $\delta>0$. Then there is a $C>0$ such that $f_{t}(\lambda)\leqslant C$ for all $\lambda\in[\delta,1-\delta]$ and $t$ large enough.

We estimate this using the following result due to Krasikov (see , Theorem 2). Setting $\mu\mathrel{\mathop{:}}=(2\nu+1)(2\nu+3)$ and assuming that $\nu>-1/2$ and $t>\sqrt{\mu+\mu^{2/3}}/2$, we have the bound

Setting $\nu=[t\lambda]+1$ yields $\lvert J_{[t\lambda]+1}(t)\rvert^{2}\leqslant\frac{C}{t}$ for $\lambda\in(\delta,1-\delta)$ and $t$ large enough. This completes the proof. ∎

By Lemmas 8.8 and 8.4, it is enough to prove that

The proof of Proposition 8.6 is therefore completed by the following result.

The proof is a simple integration by parts. It is easy to check that on $[\delta,1-\delta]$ the function $g$ is smooth and its derivative is bounded. We find

Proposition 8.5 and dominated convergence yield the claim. ∎

Symmetric matrices

In this section we describe how to extend the argument of Sections 6 – 8 to the symmetric case (2.1b). While in the Hermitian case (2.1a) we had

Since the distribution of $H_{xy}$ is symmetric, Lemma 6.1 also holds in the symmetric case. However, (9.1) implies that there is no restriction on the order of the labels associated with an edge. Thus we replace (6.1) with

Next, we define the set $\mathscr{G}^{*}_{n,n^{\prime}}$ as the set of lumpings $\mathscr{G}_{n,n^{\prime}}$ without the complete ladder and the complete antiladder (see its definition below). It is easy to see that the analogue of Lemma 7.2 holds with

It therefore suffices to estimate the contribution of pairings $\Gamma\in\mathscr{P}^{*}_{n,n^{\prime}}$. We have that

Thus, the graphical representation of pairings has to be modified as follows. Each bridge $\sigma\in\Gamma$ carries a tag, straight or twisted, which arises from multiplying out the product in (9.3). Twisted bridges are graphically represented with dashed lines.

In order to find a good notion of combinatorial complexity of pairings, we define antiparallel bridges as follows. Two bridges $\{e_{i},e_{j}\}$ and $\{e_{i+1},e_{j+1}\}$ are antiparallel if $i+1,j+1\notin\{0,n\}$; see Figure 9.1. An antiladder is a sequence of bridges such that two consecutive bridges are antiparallel.

It is easy to see that, in addition to ladders whose rungs are straight bridges, antiladders whose rungs are twisted bridges have a leading order contribution.

The skeleton $\Sigma=S(\Gamma)$ of the pairing $\Gamma$ is obtained from $\Gamma$ by the following procedure. A pair of parallel straight bridges is collapsed to form a single straight bridge. A pair of antiparallel twisted bridges is collapsed to form a single twisted bridge. This is repeated until no parallel straight bridges or antiparallel twisted bridges remain. The resulting pairing is the skeleton $\Sigma=S(\Gamma)$; see Figure 9.2.

Thus we see that Lemma 7.3 holds. Moreover, Lemma 7.4 holds, provided that (i) is replaced with

Each $\Sigma\in\mathscr{S}_{m,m^{\prime}}^{*}$ contains no parallel straight bridges and no antiparallel twisted bridges.

Crucially, Lemma 7.7 remains valid for such tagged skeletons. This can be easily seen using the orbit construction of the proof of Lemma 7.7, combined with (i’).

Finally, the complete ladder pairing yields (3.1). The complete antiladder is subleading, as its contribution vanishes unless $x=0$.

Delocalization: proofs of Theorem 3.3 and Corollary 3.4

In this section we show how to derive Theorem 3.3 from Theorem 3.1, and derive Corollary 3.4 as a consequence.

We use an argument due to Chen showing that diffusive motion implies delocalization of the vast majority of eigenvectors.

for any $\zeta>0$. Next, we observe that the norm in the first term may be bounded by 1:

by translation invariance. Note that this estimate holds uniformly in $t$.

for $X\neq 0$ and $L(1,X)$ is continuous at $X=0$, a simple limiting argument shows that Theorem 3.1 implies

Setting $\zeta=\sqrt{\varepsilon}$ completes the proof. ∎

Pick an intermediate exponent $\widetilde{\kappa}$ satisfying $\kappa<\widetilde{\kappa}<1/3$ and abbreviate

where $\delta>0$ is some small constant to be chosen later. Using $(a+b)^{\gamma}\leqslant(2a)^{\gamma}+(2b)^{\gamma}$ we find

Choosing $\delta<2^{-\gamma}$ therefore yields

Critical pairings

In this section we give an example family of pairings which are critical in the sense that they saturate the 2/3 rule (Lemma 7.7). This implies that extending our results beyond time scales of order $W^{d/3}$ requires either a further resummation of pairings or a more refined classification of graphs in terms of their deviation from the 2/3 rule.

Let $k\geqslant 1$ and consider the skeleton pairing $\Sigma_{k}$ defined in Figure 11.1. It is a critical pairing in the sense that all orbits not containing the vertices $0,m$ consist of $3$ vertices.

It is easy to see that for $\Sigma_{k}$ we have

As shown in the Section 7 (see (7.13)), the coefficients $a_{n}(t)$ essentially vanish if $n>(1+o(1))t$. Setting $t=M^{\kappa}$ thus means restricting the summation to $n,n^{\prime}\leqslant M^{\kappa}$.

which diverges as $W\to\infty$ if $\kappa>1/3$. Hence a control of the error term at time scales $\kappa$ larger than $1/3$ would require further resummation of such critical pairings.

In the estimates of the preceding paragraph we did not make full use of the heat kernel decay associated with each skeleton bridge. For simplicity, the following discussion is restricted to $d=1$ (it may be easily extended to higher dimensions; in fact some estimates are better in higher dimensions). Using Lemma 7.5, we may improve (11.1) to

this is a simple consequence of the heat kernel bound of Lemma 7.5 and the fact that each six-block of $\Sigma_{k}$ contains two bridges in $\Sigma_{k}\setminus(\Sigma_{k})_{T}$ for which we may apply the bound (7.7b) (in which we drop the unimportant second term for simplicity). Now (11.2) is bounded by

which is summable for $\kappa<2/5$. Note, however, that the factor $k^{-6k}$ from (11.1) has been replaced with the larger factor $k^{-4k}$. Recall that the factor $k^{-6k}$ is used to cancel the combinatorics $\bar{m}!\sim k^{6k}$ arising from the summation over all skeletons. In the present example this small factor is not needed, as the family $\{\Sigma_{k}\}$ is small. It is clear, however, that a systematic application of this approach requires a more refined classification of skeletons in terms of how much they deviate from the 2/3 rule. One expects that the number of skeletons saturating the 2/3 rule is small, and that they are therefore amenable to estimates of type (11.3). Conversely, most of the $\bar{m}!$ skeletons are expected to deviate strongly from the 2/3 rule, so that their greater number is compensated by their small individual contributions.

for $\lvert x-y\rvert\ll N$. Thus a correct lower bound on the contribution of each skeleton graph should have taken into account this additional decay as well. A somewhat lenghtier calculation shows that with the asymptotics (11.4) the estimate (11.2) may be improved to

In conclusion: Our estimates rely on an indiscriminate application of the 2/3 rule to all skeleton pairings; going beyond time scales of order $W^{d/3}$ would require either (i) a refined classification of the skeleton pairings in terms of how much they deviate from the 2/3 rule, combined with a systematic use of the bound (7.7b) on all bridges in $\Sigma\setminus\Sigma_{T}$; or (ii) a further resummation of graphs in order to exploit cancellations. The approach (i) can be expected to reach at most times of order $W^{2/5}$ for $d=1$.

Appendix A Proof of Proposition 8.5

Note first that $F$ is monotone nondecreasing and satisfies $0\leqslant F(\lambda)\leqslant 1$, as follows from (5.4). Hence it is enough to prove (8.5a) for $\lambda\in(0,1)$.

For the following it is convenient to replace $F_{t}$ with $\widetilde{F}_{t}$, defined by

By Lemma 8.8 we have $F_{t}(\lambda)-\widetilde{F}_{t}(\lambda)=o(1)$ as $t\to\infty$.

Thus let $\lambda\in(0,1)$ be fixed. From (5.2) we find

We now claim that the limit $t\to\infty$ of the first two terms of (A.1) vanish by a stationary phase argument. Let us write the first term of (A.1) as $R_{t}^{1}+R_{t}^{2}$, where

where $\{\xi\}\mathrel{\mathop{:}}=\xi-[\xi]\in[0,1)$. One readily finds the bounds

A standard stationary phase argument therefore yields $\lim_{t\to\infty}R_{t}^{1}=0$.

where $\mathcal{P}$ denotes principal value. We now show that $\widetilde{F}^{0}_{t}(\lambda)=o(1)$. Indeed, the expression in square brackets in the definition of $\widetilde{F}_{t}^{0}(\lambda)$ is equal to $-1$. Exactly as above we therefore conclude that $\widetilde{F}_{t}^{0}(\lambda)=O(t^{-1/2})$.

which vanishes in the limit $t\to\infty$ by the above saddle point argument (the expression in the square brackets is an entire analytic function, and the phase $\cos\theta-\cos\theta^{\prime}+\lambda\theta-\lambda\theta^{\prime}$ has the four nondegenerate saddle points defined by $\sin\theta=\sin\theta^{\prime}=\lambda$).

In a second step, we choose a scale $2/5<\varepsilon<1/2$ and introduce a cutoff in $\lvert\theta-\theta^{\prime}\rvert$ at $t^{-\varepsilon}$. Thus we have

Let us abbreviate $D_{t}\;\mathrel{\mathop{:}}=\;\{(\theta,\theta^{\prime})\in[0,\pi]^{2}\,:\,\lvert\theta-\theta^{\prime}\rvert>t^{-\varepsilon}\}$. The second term on the right-hand side of (A.2) is equal to

In the domain $D_{t}$ the phase $\phi$ has two stationary points defined by $\sin\theta=\sin\theta^{\prime}=\lambda$ and $\theta\neq\theta^{\prime}$. For all $(\theta,\theta^{\prime})$ not in some fixed neighbourhood of these stationary points and satisfying $\lvert\theta-\theta^{\prime}\rvert>t^{-\varepsilon}$, we have the bound

for some constant $C>0$ depending on $\lambda$, and large enough $t$. Thus a standard saddle point analysis shows that (A.3) is of the order $t^{-1/2}+t^{2\varepsilon-1}=o(1)$.

In a third step, we analyse the first term on the right-hand side of (A.2). We introduce the new coordinates

It is easy to check that, for $v\in[-a_{t,u},a_{t,u}]$, we have

In a fourth step, we analyse $I_{t}(u)$ using contour integration. Abbreviate $b\mathrel{\mathop{:}}=\lambda-\sin u$. Let us assume that $u$ satisfies $b\neq 0$. Then, setting $z=\lvert b\rvert tv$, we find

Let us consider the case $b>0$. Using the identity

where $\gamma$ is the arc $\{bta_{t,u}(\cos\varphi,\sin\varphi)\,:\,\varphi\in[0,\pi]\}$. The absolute value of the integral is bounded by

which is bounded uniformly in $t$ and $b\neq 0$, and vanishes in the limit $t\to\infty$ for all $b\neq 0$. The case $b<0$ is treated in the same way. In summary, we have, for each $u$ satisfying $\sin u\neq\lambda$, that

A similar (in fact easier) analysis yields

This completes the proof of Proposition 8.5.

References