Extending the effective-potential theory of M-matrices to random ensembles: resonances, sharp decay constants, weaker decoupling, and Poisson spectral statistics

For a real symmetric non-singular M-matrix $A$ (positive definite with non-positive off-diagonal entries), Filoche, Mayboroda, and Tao (2021) proved that the landscape vector $u=A^{-1}\mathbf{1}$ acts as an effective localization potential: the reciprocal $1/u$ defines an Agmon-type metric on the index set that governs exponential decay of eigenvectors away from the wells of $1/u$ and, under a separation hypothesis on the wells, the matrix decouples into blocks with inter-block coupling at most of order $e^{-2S/\sqrt{W_c}}$, where $S$ is the Agmon separation and $W_c$ the connectivity. The paper itself leaves open how far this deterministic mechanism reaches toward the random ensembles it is meant to describe. The open problem is to turn the landscape mechanism into a rigorous theory of typical localization and local spectral statistics in natural random ensembles. Five interlocking questions remain open: (1) no general criterion excludes strong inter-well resonances; (2) the deterministic decay constant of order $1/\sqrt{W_c}$ is believed non-sharp; (3) the theory requires the Z-/M-matrix sign structure, absent from natural ensembles such as Gaussian band matrices; (4) no rigorous derivation of Poisson local eigenvalue statistics from landscape-induced decoupling exists; (5) the separation hypothesis may be stronger than necessary.

Background

For a Schrödinger operator $-\Delta+V$ with a confining potential $V$, eigenfunctions with eigenvalue $E$ decay exponentially outside the classically allowed region $\{V\le E\}$; the classical Agmon method makes this quantitative through the Agmon distance, a geodesic distance associated with the metric $(V-E)_{+}$, where $x_{+}=\max(x,0)$. In disordered media this mechanism fails in its naive form: for an Anderson-type potential (random $V$ taking finitely many values on a lattice; Anderson localization is the confinement of eigenstates by disorder) the set $\{V\le E\}$ can percolate, leaving no room for Agmon decay, even though eigenfunctions at the bottom of the spectrum are exponentially localized. The localization landscape was introduced for this situation by Filoche and Mayboroda (2012): if $H\ge 0$ is the operator, the solution $u$ of $Hu=1$ (with $1$ the constant function) satisfies $|\varphi|\le E\,u$ for every eigenfunction $\varphi$ with eigenvalue $E$, and the reciprocal $1/u$ acts as an effective potential whose wells (the peaks of $u$) predict where eigenfunctions localize and at what exponential rate. Arnold, David, Jerison, Mayboroda, and Filoche (2016, 2019) proved a rigorous Agmon-type estimate for $-\Delta+V$ with the metric built from $(1/u-E)_{+}$, and identified the central remaining obstacle: resonances, that is, the possibility that tunneling between two comparable wells produces eigenfunctions spread over several wells.

The source paper (Filoche-Mayboroda-Tao, J. Math. Phys. 62, 041902, 2021) transports the landscape theory to finite matrices. A real symmetric $N\times N$ matrix $A=(a_{ij})$ is a Z-matrix if $a_{ij}\le 0$ for all $i\ne j$, and a non-singular M-matrix if it is additionally positive definite; such matrices arise as weighted graph Laplacians, as symmetric transition-rate matrices of reversible Markov chains, and in random matrix models; one application is many-body localization (the disorder-induced suppression of quantum transport in interacting particle systems, formulated on Fock space, the state space of arbitrarily many quantum particles), for which Balasubramanian-Liao-Galitski 2020 introduced a landscape. For a symmetric non-singular M-matrix the landscape vector $u=A^{-1}\mathbf{1}$ has non-negative entries, and the paper defines an effective potential $\bar v_i=(Au)_i/u_i=1/u_i$ (entrywise), a shifted potential $v_i=(\bar v_i-E)_{+}$, a well set $K_E=\{i:\bar v_i\le E\}$, and an Agmon-type pseudo-metric on the index set $[N]=\{1,\dots,N\}$:

$\rho(i,j)\;=\;\inf_{i_0=i,\dots,i_L=j}\ \sum_{\ell}\ \ln\Big(1+\sqrt{\tfrac{\sqrt{v_{i_\ell}\,v_{i_{\ell+1}}}}{|a_{i_\ell i_{\ell+1}}|}}\Big),$

where the infimum runs over paths along non-zero entries of $A$. Writing $W_c$ for the connectivity of $A$ (the maximal number of non-zero off-diagonal entries per row), the paper proves (Theorem 2.7) that every eigenvector $\varphi$ with eigenvalue $E$ obeys $\sum_k \varphi_k^2\, e^{2\rho(k,K_E)/\sqrt{W_c}}\,v_k\le W_c\max_{i,j}|a_{ij}|\,\|\varphi\|_2^2$, that is, exponential decay away from the wells at a rate governed by $\rho$ with the prefactor $1/\sqrt{W_c}$ in the exponent. Under a separation hypothesis (Hypothesis 5.1) - the well set at an energy threshold splits into disjoint wells $K_\ell$ with disjoint neighborhoods $\Omega_\ell$ whose inner boundaries stay at $\rho$-distance at least $S$ from their wells - the paper proves a decoupling theorem (Theorem 5.2): local eigenvectors of the restrictions $A|_{\Omega_\ell}$ and global eigenvectors of $A$ agree up to an error of order $e^{-2S/\sqrt{W_c}}$ (up to polynomial prefactors in $W_c$, $\max|a_{ij}|$, and a spectral-window parameter $\delta$), and correspondingly the eigenvalue counting function of $A$ matches that of the disjoint blocks up to the same error (Corollary 5.5).

The motivation is probabilistic. For random band matrices - symmetric $N\times N$ matrices whose entries are independent up to symmetry and vanish when $|i-j|>W$ - physics predicts and mathematics conjectures a transition at $W^2\approx N$: for $W^2\ll N$ the eigenvectors are localized on the scale $W^2$ and the local eigenvalue statistics converge to a Poisson point process (the model of independent random points, the spectral signature of localization), while for $W^2\gg N$ the eigenvectors delocalize and the statistics are of Wigner-Dyson (universal random-matrix) type. Standard Gaussian band matrices have sign-varying off-diagonal entries and are almost surely not positive definite, so they fall outside the M-matrix theory; the paper proposes sign-modified ensembles (off-diagonal entries supported on the negative half-line, plus a positive multiple of the identity to restore positive definiteness) and conjectures, in the spirit of universality, that landscape-governed localization should persist for $W\ll N^{1/2}$. The paper's own numerics on such ensembles show exponential decay following the Agmon prediction down to amplitudes of order $e^{-40}$, with the observed decay slope $c$ apparently not equal to the proven $1/\sqrt{W_c}$ scale (suggesting the analysis is not optimal), and with strong inter-well resonances observed to be rare but not excluded by any general criterion. The paper poses Poisson local eigenvalue statistics in a localized regime with growing band width as a target phenomenon, and it identifies landscape-induced decoupling together with the eigenvalue-counting comparison of Corollary 5.5 as the approach it envisions toward reaching it.

Problem Statement

Extend the effective-potential (landscape) theory of Filoche-Mayboroda-Tao from its deterministic Z-/M-matrix core to a rigorous theory of typical localization and local spectral statistics in natural random ensembles. Five interlocking questions, all explicitly left open by the source paper, define the problem; components (1), (2) and (4) constitute the passage from deterministic upper bounds to ensemble-typical statements, while (3) and (5) sharpen and generalize the deterministic core on which that passage rests. A rigorous answer to any one of these five components, taken on its own, is a complete resolution of the task: the components are independent research problems, and answering one does not presuppose progress on the others.

(1) Resonances. Provide a general criterion - deterministic, or holding with high probability in a specified random ensemble - guaranteeing the near-absence of strong inter-well resonances: no eigenvector near the bottom of the spectrum distributes non-trivial $\ell^2$ mass across two or more Agmon-separated wells of the effective potential $1/u$. Here "strong inter-well resonance" is read quantitatively: a criterion must exclude, for some explicitly stated spectral-window width $\delta>0$ above the bottom of the spectrum, mass fraction $0<\theta\le\tfrac12$, and finite Agmon separation $S>0$, every eigenvector with eigenvalue within $\delta$ of the bottom that carries at least a fraction $\theta$ of its $\ell^2$ mass in each of two wells of $1/u$ at Agmon distance at least $S$; the solver may fix $\delta,\theta,S$ as part of the criterion (they may depend on the ensemble parameters, but not on the particular realization), provided they are stated explicitly and the criterion is non-vacuous - the required exclusion must not hold automatically for every matrix, that is, the excluded resonance configurations must actually be possible. In particular $\theta>\tfrac12$, a vanishing spectral window $\delta$, and an unbounded Agmon separation $S$ all trivialize the exclusion and are not admissible. The source theorems only imply that such a resonance requires coupling of order $e^{-2S/\sqrt{W_c}}$; they do not exclude it, and numerics suggest it is rare.

(2) Sharp constants. Determine the optimal decay rate of eigenvectors in the landscape Agmon metric: either improve the deterministic factor $1/\sqrt{W_c}$ in the exponent of Theorem 2.7 (equivalently the coupling $e^{-2S/\sqrt{W_c}}$ in Theorem 5.2), or prove a matching lower bound showing that a constant of this order cannot be uniformly improved - at least for natural random M-matrix ensembles such as the sign-modified band matrices, for which numerics indicate the true decay slope differs from $1/\sqrt{W_c}$.

(3) Beyond the sign structure. Either extend the landscape/Agmon eigenvector decay and decoupling theorems to matrix ensembles without the Z-/M-matrix sign structure (for example standard Gaussian band matrices, which are sign-varying and not positive definite), or rigorously justify the sign-modified ensembles proposed in the paper: prove that for such ensembles (off-diagonal entries supported on the negative half-line plus a diagonal shift ensuring positive definiteness) the landscape $u=A^{-1}\mathbf{1}$ typically produces isolated single wells, so that the deterministic theorems yield typical single-well localization for band widths $W\ll N^{1/2}$, as conjectured.

(4) Poisson statistics. Derive rigorously the Poisson local eigenvalue statistics for a natural random ensemble of the considered type (sign-modified or standard Gaussian band matrices) in a localized regime with band width growing with $N$. Any rigorous derivation for such an ensemble in a growing-width localized regime answers this component. Already-known results do not suffice: full Poisson convergence is known only for fixed width, and limit-point identifications only in strongly localized sub-regimes.

(5) Weaker separation hypotheses. Find weaker geometric or combinatorial conditions than Hypothesis 5.1 - in particular, weaken or remove axiom (iii), which requires the inner boundaries of the well neighborhoods (the points of $\Omega_\ell$ joined by a non-zero entry of $A$ to a point outside $\Omega_\ell$) to stay at Agmon distance at least $S$ from their wells - under which effective decoupling (exponentially small coupling in the separation) still holds.

References

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