We study a unitary version of the one-dimensional Anderson model, given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix. We fully characterize positivity and vanishing of the Lyapunov exponent for this model throughout the spectrum and for arbitrary distributions of the random phases. This includes Bernoulli distributions, where in certain cases a finite number of critical spectral values, with vanishing Lyapunov exponent, exists. We establish similar results for a unitary version of the random dimer model.
We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all ernergies in $(2,+\infty)$ except those in a discrete set, which leads to absence of absolutely continuous spectrum in $(2,+\infty)$. This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions.
We provide an explicit lower bound for the the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model. In particular, for the Anderson model on a strip of width $ W $ the lower bound is proportional to $ W^{-ε} $, for any $ ε>0 $. This bound is consistent with the fact that the lowest non-negative Lyapunov exponent is conjectured to have a lower bound proportional to $ W^{-1} $.
In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to Damanik--Sims--Stolz, and it covers a wider variety of random models. Along the way we note that a Large Deviation Theorem holds uniformly on compacts.
We study one-dimensional, continuum Bernoulli-Anderson models with general single-site potentials and prove positivity of the Lyapunov exponent away from a discrete set of critical energies. The proof is based on Fürstenberg's Theorem. The set of critical energies is described explicitly in terms of the transmission and reflection coefficients for scattering at the single-site potential. In examples we discuss the asymptotic behavior of generalized eigenfunctions at critical energies.
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in $\ell^2(\mathbb{Z})$ whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typical…
The proof of Anderson localization for the 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based in part on the multi-scale analysis. Later, in the 90s, it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. In this paper we present a short proof along these lines, for the Anderson mode…
It is proven that the inverse localization length of an Anderson model on a strip of width $L$ is bounded above by $L/λ^2$ for small values of the coupling constant $λ$ of the disordered potential. For this purpose, a formalism is developed in order to calculate the bottom Lyapunov exponent associated with random products of large symplectic matrices perturbatively in the coupling constant of the randomness.
We study two models of Anderson-type random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed four-by-four symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criterion by Gol'dsheid and Margulis (i.e. Zariski denseness of the group generated by the transfer matrices in the group of symplectic matrices) to prove positivity of both leading Lyapunov exponents for most energies. In each case this implies almost sure absence of absolutely continuous spectrum (at all energies…
We study the parabolic Anderson model in $(1+1)$ dimensions with nearest neighbor jumps and space-time white noise (discrete space/continuous time). We prove a contour integral formula for the second moment and compute the second moment Lyapunov exponent. For the model with only jumps to the right, we prove a contour integral formula for all moments and compute moment Lyapunov exponents of all orders.
The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we…
The eigenvalues of the Hatano--Nelson non-Hermitian Anderson matrices, in the spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity parameter, are shown to be real and exponentially close to the Hermitian eigenvalues. This complements previous results, according to which the eigenvalues in the spectral regions in which the non-Hermiticity parameter exceeds the Lyapunov exponent are aligned on curves in the complex plane.
We prove dynamical and spectral localization at all energies for the discrete generalized Anderson model via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schrödinger operators, to the case where a single random variable determines the potential on a block of an arbitrary, but fixed, size $α$. For this model, we also give a description of the almost sure spectrum as a set and prove uniform positivity of the Lyapunov exponents. In fact, regarding positivity of the Lyapunov exponents, we prove a stronger statement wh…
It is known that the eigenfunctions of a random Schrödinger operator on a strip decay exponentially, and that the rate of decay is not slower than prescribed by the slowest Lyapunov exponent. A variery of heuristic arguments suggest that no eigenfunction can decay faster than at this rate. We make a step towards this conjecture (in the case when the distribution of the potential is regular enough) by showing that, for each eigenfunction, the rate of exponential decay along any subsequence is strictly slower than the fastest Lyapunov exponent, and that there exists a subsequence along which it …
For the 2D matrix Langevin dynamics that corresponds to the continuous-time limit of the product of some $2 \times 2$ random matrices, the finite-time Lyapunov exponent can be written as an additive functional of the associated Riccati process submitted to some Langevin dynamics on the infinite periodic ring. Its large deviations properties can be thus analyzed from two points of view that are equivalent in the end by consistency but give different perspectives. In the first approach, one starts from the large deviations at level 2.5 for the joint probability of the empirical density and of th…
In this paper we consider the discrete one-dimensional Schroedinger operator with quasi-periodic potential v_n = λv (x + n ω). We assume that the frequency ωsatisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove - in the perturbative regime - that for large disorder λand for most frequencies ωthe operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density o…
This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements g…
In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $\operatorname{GL}_n(\mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.
In some quasi-one-dimensional weakly disordered media, impurities are large and rare rather than small and dense. For an Anderson model with a low density of strong impurities, a perturbation theory in the impurity density is developed for the Lyapunov exponent and the density of states. The Lyapunov exponent grows linearly with the density. Anomalies of the Kappus-Wegner type appear for all rational quasi-momenta even in lowest order perturbation theory.
It is classical, following Furstenberg's theorem on positive Lyapunov exponent for products of random SL$(2, \mathbb R)$ matrices, that the one dimensional random Schrödinger operator has Anderson localization at arbitrary disorder. This paper proves a nonlinear analogue, thereby establishing a KAM-type persistence result for a non-integrable system.
Anomalies are known to appear in the perturbation theory for the one-dimensional Anderson model. A systematic approach to anomalies at critical points of products of random matrices is developed, classifying and analysing their possible types. The associated invariant measure is calculated formally. For an anomaly of so-called second degree, it is given by the groundstate of a certain Fokker-Planck equation on the unit circle. The Lyapunov exponent is calculated to lowest order in perturbation theory with rigorous control of the error terms.
We study the asymptotics of front speeds of the reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity and zero mean stationary ergodic Gaussian shear advection on the entire plane. By exploiting connections of KPP front speeds with the almost sure Lyapunov exponents of the parabolic Anderson problem, and with the homogenized Hamiltonians of Hamilton-Jacobi equations, we show that front speeds enhancement is quadratic in the small root mean square (rms) amplitudes of white in time zero mean Gaussian shears, and it grows at the order of the large rms amplitudes. Howe…
A random phase property establishing a link between quasi-one-dimensional random Schroedinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the full hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special m…
We consider unitary analogs of $d-$dimensional Anderson models on $l^2(\Z^d)$ defined by the product $U_ω=D_ωS$ where $S$ is a deterministic unitary and $D_ω$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman-Molchanov to get exponential estimates on fractional moments of the matrix elements of $U_ω(U_ω-z)^{-1}$, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements…
We show that Lyapunov exponents and stability exponents are equal in the case of product of $i.i.d$ isotropic(also known as bi-unitarily invariant) random matrices. We also derive aysmptotic distribution of singular values and eigenvalues of these product random matrices. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of above result, we show probability that product of $n$ $i.i.d$ real isotropic random matrices has all eigenvalues real goes to one as $n \to \infty$. Also, in the …
We consider the simple random walk on Z^d, d > 2, evolving in a potential of the form βV, where (V(x), x \in Z^d) are i.i.d. random variables taking values in [0,+\infty), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -Δ + βV.
We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can also be considered as a unitary version of the Anderson model. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density.
We study the one-dimensional discrete Schrödinger operator with the skew-shift potential $2λ\cos\left(2π\left(\binom{j}{2} ω+jy+x\right)\right)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $λ>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(λ)$ at small $λ$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(λ)$ is fully consistent with $L(λ)$ being positive and satisf…
We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix $T_0$. Focusing on the eigenvalues of $T_0$ of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required $T_0$ to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some result by Valko and Virag showi…
We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_ω=D_ωS$ where $S$ is a deterministic unitary and $D_ω$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of $U_ω$ is pure point almost surely for all values of the parameter of $S$. We provide similar results for unitary operators defined on $l^2(\N)$ together with an application to orthogonal polynomials on the unit circle. We get almo…
For products $P_N$ of $N$ random matrices of size $d \times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{μ_i\}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements, and extended to the general variance case for $μ_1$, methods known for the computation of $\lim_{N \to \infty} \langle μ_i \rangle$ are used to compute the large $N$ form of the variances of the exponents. Analogous calculations are performed in the case that the matrices making up $P_N$ are products of sub-blocks of random unitary matrices with Haar measure.…
Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitely exhibited for the Anderson model.
A perturbative formula for the lowest Lyapunov exponent of an Anderson model on a strip is presented. It is expressed in terms of an energy dependent doubly stochastic matrix, the size of which is proportional to the strip width. This matrix and the resulting perturbative expression for the Lyapunov exponent are evaluated numerically. Dependence on energy, strip width and disorder strength are thoroughly compared with the results obtained by the standard transfer matrix method. Good agreement is found for all energies in the band of the free operator and this even for quite large values of the…
We consider the distribution function $P(|ψ|^{2})$ of the eigenfunction amplitude at the center-of-band (E=0) anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with $|ψ|^{2}$ much larger than the inverse typical localization length $\ell_{0}$. Using the solution to the generating function $Φ_{an}(u,ϕ)$ found recently in our works we find the ALS probability distribution $P(|ψ|^{2})$ at $|ψ|^{2}\ell_{0} >> 1$. As an auxiliary prelimi…
