# Weak-disorder expansion for the Anderson model on a tree

@article{Miller1994WeakdisorderEF, title={Weak-disorder expansion for the Anderson model on a tree}, author={Jeffrey D. Miller and Bernard Derridda}, journal={Journal of Statistical Physics}, year={1994}, volume={75}, pages={357-388} }

We show how certain properties of the Anderson model of a tree are related to the solutions of a nonlinear integral equation. Whether the wave function is extended or localized, for example, corresponds to whether or not the equation has a complex solution. We show how the equation can be solved in a weakdisorder expansion. We find that, for small disorder strength λ, there is an energyEc(λ) above which the density of states and the conducting properties vanish to all orders in perturbation…

## 34 Citations

On the Anderson transition on the Bethe lattice

- Physics
- 2018

We study the Anderson model on the Bethe lattice by working directly with real energies $E$. We show that the criterion for the stability of the populations leads to the same results as other, more…

Anderson transition on the Bethe lattice: an approach with real energies

- Physics, Mathematics
- 2018

We study the Anderson model on the Bethe lattice by working directly with propagators at real energies E. We introduce a novel criterion for the localization–delocalization transition based on the…

Surprises in the phase diagram of the Anderson model on the Bethe lattice

- Physics, Mathematics
- 2012

rst for which an energy regime of extended states and a separate regime of localized states could be established. In this paper, we review recently discovered surprises in the phase diagram. Among…

The Anderson model on the Bethe lattice : Lifshitz Tails

- 2014

This paper is devoted to the study of the (discrete) Anderson Hamiltonian on the Bethe lattice, which is an infinite tree with constant vertex degree. The Hamiltonian we study corresponds to the sum…

Anderson localization on the Cayley tree: multifractal statistics of the transmission at criticality and off criticality

- Physics
- 2011

In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off…

Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions

- Physics
- 2009

For Anderson localization on the Cayley tree, we study the statistics of various observables as a function of the disorder strength W and the number N of generations. We first consider the Landauer…

Localization in quasiperiodic chains: A theory based on convergence of local propagators

- PhysicsPhysical Review B
- 2021

Quasiperiodic systems serve as fertile ground for studying localisation, due to their propensity already in one dimension to exhibit rich phase diagrams with mobility edges. The deterministic and…

Anderson localization in strongly coupled disordered electron–phonon systems

- Physics
- 2004

Based on the statistical dynamic mean-field theory, we investigate, in a generic model for a strongly coupled disordered electron–phonon system, the competition between polaron formation and Anderson…

Two critical localization lengths in the Anderson transition on random graphs

- Physics, Mathematics
- 2019

We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are…

Absolutely Continuous Spectrum in the Anderson Model on the Bethe Lattice

- Mathematics
- 1994

We prove that the spectrum of the Anderson Hamiltonian Hλ = −∆ + λV on the Bethe Lattice is absolutely continuous inside the spectrum of the Laplacian, if the disorder λ is sufficiently small. More…

## References

SHOWING 1-10 OF 46 REFERENCES

Weak disorder expansion of the invariant measure for the one-dimensional Anderson model

- Physics
- 1988

We show that the formal perturbation expansion of the invariant measure for the Anderson model in one dimension has singularities at all energiesE0=2 cosπ(p/q); we derive a modified expansion near…

Density of states of the random-hopping model on a Cayley tree.

- Physics, MedicinePhysical review. B, Condensed matter
- 1985

The density of states of the random-hopping model on a Cayley tree Y is calculated with a Gaussian distribution of hopping matrix elements in the energy range E &E„where E, is a critical energy described below.

Decay rate of the Green function in a random potential on the Bethe lattice and a criterion for localization

- Mathematics
- 1993

It is shown that in the tight-binding Anderson model on the Bethe lattice the exponential decay rate of the Green function can be obtained for arbitrary energies and arbitrary disorder. Analytical…

Localization transition in the Anderson model on the Bethe lattice: Spontaneous symmetry breaking and correlation functions

- Physics
- 1991

Abstract We present the complete analytical solution of the Anderson model on the Bethe lattice. Within the scope of the supersymmetric approach the delocalization transition manifests itself as a…

Self-consistent theory of localization. II. Localization near the band edges

- Physics
- 1974

For pt. I see abstr. A43970 of 1973. The solution of the integral equation which arises in the self-consistent theory of localization has been explored for a Cauchy distribution of site energies, for…

Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions

- Chemistry
- 1984

We describe a method which gives the weak disorder expansion (03BB ~ 0) of the Lyapounov exponent 03B3(E) of a discretized one-dimensional Schrodinger equation 03C8n+1 + 03C8n-1 + 03BBVn03C8n =…

Localization at large disorder and at extreme energies: An elementary derivations

- Mathematics
- 1993

The work presents a short proof of localization under the conditions of either strong disorder (λ > λ0) or extreme energies for a wide class of self adjoint operators with random matrix elements,…

Electrons in disordered systems and the theory of localization

- Physics
- 1974

Abstract This paper gives a review of the theory of noninteracting electrons in a static disordered lattice. The introductory section gives a brief survey of the main aspects of the problem and of…

A selfconsistent theory of localization

- Physics
- 1973

A new basis has been found for the theory of localization of electrons in disordered systems. The method is based on a selfconsistent solution of the equation for the self energy in second order…

Anderson localisation on a Cayley tree: a new model with a simple solution

- Physics
- 1990

A new model is introduced for wave propagation on a disordered Cayley tree. The model has sufficient simplifying features (related to the absence of time-reversal symmetry and to phase randomisation)…