localization on quantum graphs with random vertex couplings we consider schrodinger operators on a class of periodic quantum graphs with randomly distributed kirchhoff coupling constants at all vertices we obtain necessary conditions for localization on quantum graphs in terms of finite volume criteria for some energy dependent discrete hamiltonians these conditions hold in the strong disorder limit and at the spectral edges

characterization of the spectrum of the landau hamiltonian with delta impurities we consider a random schrodinger operator in an external magnetic field the random potential consists of delta functions of random strengths situated on the sites of a regular two dimensional lattice we characterize the spectrum in the lowest n landau bands of this random hamiltonian when the magnetic field is sufficiently strong depending on n we show that the spectrum in these bands is entirely pure point that the energies coinciding with the landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length by relating the hamiltonian to a lattice operator we are able to use the aizenman molchanov method to prove localization

on bernoulli decompositions for random variables concentration bounds and spectral localization as was noted already by a n kolmogorov any random variable has a bernoulli component this observation provides a tool for the extension of results which are known for bernoulli random variables to arbitrary distributions two applications are provided here i an anti concentration bound for a class of functions of independent random variables where probabilistic bounds are extracted from combinatorial results and ii a proof based on the bernoulli case of spectral localization for random schrodinger operators with arbitrary probability distributions for the single site coupling constants for a general random variable the bernoulli component may be defined so that its conditional variance is uniformly positive the natural maximization problem is an optimal transport question which is also addressed here

non perturbative anderson localization in heavy tailed potentials via large deviations moment analysis we study a class of anderson hamiltonians with heavy tailed independent and identically distributed random potential on graphs with sub exponential growth of the balls and of the number of self avoiding paths connecting pairs of points we show that for a class of marginal distributions anderson localization occurs non perturbatively i e for any nonzero amplitude of the potential like in one dimensional systems the proof is based on the moment analysis of the green functions via large deviations estimates

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

an optimal wegner estimate and its application to the global continuity of the integrated density of states for random schrodinger operators we prove that the integrated density of states ids of random schrodinger operators with anderson type potentials on l 2 r d for d geq1 is locally holder continuous at all energies with the same holder exponent 0 alpha leq1 as the conditional probability measure for the single site random variable as a special case we prove that if the probability distribution is absolutely continuous with respect to lebesgue measure with a bounded density then the ids is lipschitz continuous at all energies the single site potential u in l 0 infty r d must be nonnegative and compactly supported the unperturbed hamiltonian must be periodic and satisfy a unique continuation principle we also prove analogous continuity results for the ids of random anderson type perturbations of the landau hamiltonian in two dimensions all of these results follow from a new wegner estimate for local random hamiltonians with rather general probability measures

a note on fractional moments for the one dimensional continuum anderson model we give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one dimensional continuum anderson model via the fractional moments method this follows via exponential decay of fractional moments of the green function which is shown to hold at arbitrary energy and for any single site distribution with bounded compactly supported density

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

localization bounds for multiparticle systems we consider the spectral and dynamical properties of quantum systems of n particles on the lattice mathbb z d of arbitrary dimension with a hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite range interaction two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction it is established here that for all n there are regimes of high disorder and or weak enough interactions for which the system exhibits spectral and dynamical localization the localization is expressed through bounds on the transition amplitudes which are uniform in time and decay exponentially in the hausdorff distance in the configuration space the results are derived through the analysis of fractional moments of the n particle green function and related bounds on the eigenfunction correlators

exponential dynamical localization for the almost mathieu operator we prove that the exponential moments of the position operator stay bounded for the supercritical almost mathieu operator with diophantine frequency

spectral transition for random quantum walks on trees we define and analyze random quantum walks on homogeneous trees of degree q geq 3 such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in mathbb c q hopping on the neighboring sites of the tree in the presence of static disorder the one time step random unitary evolution operator of the particle depends on a unitary matrix c in u q which monitors the strength of the disorder we prove for any q that there exist open sets of matrices in u q for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum thereby showing the existence of a spectral transition driven by c in u q for q in 3 4 we establish properties of the spectral diagram which provide a description of the spectral transition

a lower wegner estimate and bounds on the spectral shift function for continuum random schrodinger operators abstract we prove a strictly positive locally uniform lower bound on the density of states dos of continuum random schrodinger operators on the entire spectrum i e we show that the dos does not have a zero within the spectrum this follows from a lower wegner estimate for finite volume continuum random schrodinger operators we assume throughout iid random variables and the single site distribution having a lebesgue density bounded from below on its support the main mathematical novelty in this paper are pointwise in energy bounds on the expectation of the spectral shift function at all energies for these operators where we mainly focus on perturbations corresponding to a change from dirichlet to neumann boundary conditions along the boundary of a cube we show that the bound scales with the area of the hypersurface where the boundary conditions are changed we also prove bounds on the averaged spectral shift function for perturbations by bounded and compactly supported multiplication operators

self organized criticality attributed to a central limit like convergence effect abstract self organized criticality is a hypothesis used to explain the origin of 1 f noise and other scaling behaviors despite being proposed nearly 30 years ago no consensus exists as to its exact definition or mathematical mechanism s recently a model for 1 f noise was proposed based on a family of statistical distributions known as the tweedie exponential dispersion models these distributions are characterized by an inherent scale invariance that manifests as a variance to mean power law called fluctuation scaling they also serve as foci of convergence in a limit theorem on independent and identically distributed distributions fluctuation scaling can be modeled by self similar stochastic processes that relate the variance to mean power law to 1 f noise through their correlation structure a hypothesis is proposed whereby the effects of self organized criticality are mathematically modeled by the tweedie distributions and their convergence behavior as applied to self similar stochastic processes sandpile model fluctuations are shown to manifest 1 f noise fluctuation scaling and to conform to the tweedie compound poisson distribution the tweedie models and their convergence theorem allow for a mechanistic explanation of 1 f noise and fluctuation scaling in phenomena conventionally attributed to self organized criticality thus providing a paradigm shift in our understanding of these phenomena

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

dynamical localization for discrete anderson dirac operators we establish dynamical localization for random dirac operators on the d dimensional lattice with d in left 1 2 3 right in the three usual regimes large disorder band edge and 1d these operators are discrete versions of the continuous dirac operators and consist in the sum of a discrete free dirac operator with a random potential the potential is a diagonal matrix formed by different scalar potentials which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support we prove the exponential decay of fractional moments of the green function for such models in each of the above regimes i e j throughout the spectrum at larger disorder jj for energies near the band edges at arbitrary disorder and jjj in dimension one for all energies in the spectrum and arbitrary disorder dynamical localization in theses regimes follows from the fractional moments method the result in the one dimensional regime contrast with one that was previously obtained for 1d dirac model with bernoulli potential

internal lifshitz tails for schrodinger operators with random potentials in this short note we prove that for periodic schrodinger operators perturbed by an anderson type random potential with long range single site potential the density of states always show a lifshitz tail at a band edge thus we correct a mistake made in an earlier paper f klopp duke math j 98 335 396 1999

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

asymptotic ergodicity of the eigenvalues of random operators in the localized phase we prove that for a general class of random operators the family of the unfolded eigenvalues in the localization region is asymptotically ergodic in the sense of minami spectra of random operators and related topics 2011 minami conjectured this to be the case for discrete anderson model in the localized regime we also provide a local analogue of this result from the asymptotics ergodicity one can recover the statistics of the level spacings as well as a number of other spectral statistics our proofs rely on the analysis developed in germinet and klopp spectral statistics for random schrodinger operators in the localized regime 2010

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

exponential scaling limit of the single particle anderson model via adaptive feedback scaling we propose a twofold extension of the germinet klein bootstrap multi scale analysis bmsa for the anderson models on graphs first we show with the help of a single scaling algorithm that power law decay bounds at some initial scale imply an asymptotically exponential decay of eigenfunctions efs and of ef correlators efcs even on graphs of polynomial growth which do not fulfill the uniform scalability condition required for the existing bmsa techniques we also show that the exponential scaling limit of the efs and efcs holds true for a class of marginal distributions of the random potential with regularity lower than holder continuity of any positive order

many body localization in the droplet spectrum of the random xxz quantum spin chain abstract we study many body localization properties of the disordered xxz spin chain in the ising phase disorder is introduced via a random magnetic field in the z direction we prove a strong form of dynamical exponential clustering for eigenstates in the droplet spectrum for any pair of local observables separated by a distance l the sum of the associated correlators over these states decays exponentially in l in expectation this exponential clustering persists under the time evolution in the droplet spectrum our result applies to the large disorder regime as well as to the strong ising phase at fixed disorder with bounds independent of the support of the observables

the lp theory of the spectral shift function the wegner estimate and the integrated density of states for some random operators we develop the l p theory of the spectral shift function for p 1 applicable to pairs of self adjoint operators whose difference is in the trace ideal p for 0 p 1 this result is a key ingredient of a new short proof of the wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators the proof yields the correct volume dependence of the upper bound this implies the local holder continuity of the integrated density of states at energies in the unperturbed spectral gap under an additional condition of the single site potential local holder continuity is proved at all energies this new wegner estimate together with other standard results establishes exponential localization for a new family of models for additive and multiplicative perturbations

1 ƒ noise in the distribution of prime numbers the fourier transform of the signal given by the number of primes contained in the successive intervals of equal length l 216 65 536 up to n 238 2 749 1011 was performed it turns out that the power spectrum displays the 1 behaviour with the exponent 1 64 this slope does not depend on the length of the sampled intervals which suggests some kind of self similarity in the distribution of primes

multifractality attributed to dual central limit like convergence effects abstract multifractals can be defined as fractal systems that express a range of fractal dimensions the origins of multifractality in time series data have conventionally been attributed to fat tailed probability distributions and to long range correlations multifractal sequences can be generated from the eigenvalue deviations of the gaussian unitary and orthogonal ensembles of random matrix theory these deviations can be resolved into component monofractal sequences governed by the tweedie compound poisson distribution a statistical model that expresses a variance to mean power law related to long range correlations fully multifractal descriptions of these deviations can be constructed provided that the parameter of the compound poisson model related to fractal dimension varies in accordance with an asymmetric laplace distribution both the tweedie compound poisson distribution and the asymmetric laplace distribution serve as foci of convergence in limit theorems on independent and identically distributed random variables the hypothesis that multifractal sequences can be attributed to mathematical convergence effects that have as their focus these two statistical models is proposed

regularity of the density of states of random schrodinger operators in this paper we solve a long standing open problem for random schrodinger operators on l 2 mathbb r d with i i d single site random potentials we allow a large class of free operators including magnetic potential however our method of proof works only for the case when the random potentials satisfy a complete covering condition we require that the supports of the random potentials cover mathbb r d and the bump functions that appear in the random potentials form a partition of unity for such models we show that the density of states dos is m times differentiable in the part of the spectrum where exponential localization is valid if the single site distribution has compact support and has holder continuous m 1 st derivative the required holder continuity depends on the fractional moment bounds satisfied by appropriate operator kernels our proof of the random schrodinger operator case is an extensions of our proof for anderson type models on ell 2 mathbb g mathbb g a countable set with the property that the cardinality of the set of points at distance n from any fixed point grows at some rate in n alpha alpha 0 this condition rules out the bethe lattice where our method of proof works but the degree of smoothness also depends on the localization length a result we do not present here even for these models the random potentials need to satisfy a complete covering condition the anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition

finite volume fractional moment criteria for anderson localization a technically convenient signature of localization exhibited by discrete operators with random potentials is exponential decay of the fractional moments of the green function within the appropriate energy ranges known implications include spectral localization absence of level repulsion strong form of dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases we present a family of finite volume criteria which under some mild restrictions on the distribution of the potential cover the regime where the fractional moment decay condition holds the constructive criteria permit to establish this condition at spectral band edges provided there are sufficient lifshitz tail estimates on the density of states they are also used here to conclude that the fractional moment condition and thus the other manifestations of localization are valid throughout the regime covered by the multiscale analysis in the converse direction the analysis rules out fast power law decay of the green functions at mobility edges

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

an eigensystem approach to anderson localization abstract we introduce a new approach for proving localization pure point spectrum with exponentially decaying eigenfunctions dynamical localization for the anderson model at high disorder in contrast to the usual strategy we do not study finite volume green s functions instead we perform a multiscale analysis based on finite volume eigensystems eigenvalues and eigenfunctions information about eigensystems at a given scale is used to derive information about eigensystems at larger scales this eigensystem multiscale analysis treats all energies of the finite volume operator at the same time establishing level spacing and localization of eigenfunctions in a fixed box with high probability a new feature is the labeling of the eigenvalues and eigenfunctions by the sites of the box

tweedie convergence a mathematical basis for taylor s power law 1 f noise and multifractality plants and animals of a given species tend to cluster within their habitats in accordance with a power function between their mean density and the variance this relationship taylor s power law has been variously explained by ecologists in terms of animal behavior interspecies interactions demographic effects etc all without consensus taylor s law also manifests within a wide range of other biological and physical processes sometimes being referred to as fluctuation scaling and attributed to effects of the second law of thermodynamics 1 f noise refers to power spectra that have an approximately inverse dependence on frequency like taylor s law these spectra manifest from a wide range of biological and physical processes without general agreement as to cause one contemporary paradigm for 1 f noise has been based on the physics of self organized criticality we show here that taylor s law when derived from sequential data using the method of expanding bins implies 1 f noise and that both phenomena can be explained by a central limit like effect that establishes the class of tweedie exponential dispersion models as foci for this convergence these tweedie models are probabilistic models characterized by closure under additive and reproductive convolution as well as under scale transformation and consequently manifest a variance to mean power function we provide examples of taylor s law 1 f noise and multifractality within the eigenvalue deviations of the gaussian unitary and orthogonal ensembles and show that these deviations conform to the tweedie compound poisson distribution the tweedie convergence theorem provides a unified mathematical explanation for the origin of taylor s law and 1 f noise applicable to a wide range of biological physical and mathematical processes as well as to multifractality

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

fixed points of automorphisms of free groups abstract the set of fixed points of an automorphism of a finitely generated free group is a finitely generated group settling a conjecture of g p scott s

from fixed energy localization analysis to dynamical localization an elementary path we review several techniques initiated by a remarkable work written by spencer j stat phys 51 1009 1019 1988 a quarter of century ago used and further developed in numerous subsequent researches we also describe a fairly general elementary derivation of spectral and strong dynamical anderson localization from the fixed energy analysis of the green s functions on locally finite graphs of polynomial growth obtained either by the multi scale analysis or by the fractional moment method this derivation goes in the same direction as the simon wolf method commun pure appl math 39 75 90 1986 but provides more quantitative estimates can be adapted to multi particle models and combined with a simplified variant of the germinet klein argument commun math phys 222 415 448 2001 results in an elementary proof of strong dynamical localization on arbitrary graphs of polynomial growth

localization for some continuous random hamiltonians in d dimensions abstract we prove the existence with probability one of an interval of pure point spectrum for some families of continuous random schrodinger operators in d dimensions for anderson like models with positive short range single site potentials we also prove that the corresponding eigenfunctions decay exponentially and that the integrated density of states is lipschitz continuous for the other families of random potentials that we study we show that the corresponding eigenfunctions decay faster than an inverse power of x which depends upon the decay rate of the single site potential to obtain these results we develop an extension of the classical aronszajn donoghue theory for a class of relatively compact perturbations and a spectral averaging method which extends kotani s trick to these more general families of operators

edge and impurity effects on quantization of hall currents we consider the edge hall conductance and show it is invariant under perturbations located in a strip along the edge decaying perturbations far from the edge are also allowed this enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the right half plane as a particular interesting case we put forward a general quantization formula for the difference of edge hall conductances in semi infinite samples with and without a confining wall it implies in particular that the edge hall conductance takes its ideal quantized value under a gap condition for the bulk hamiltonian or under some localization properties for a random bulk hamiltonian provided one first regularizes the conductance we shall discuss this regularization issue our quantization formula also shows that deviations from the ideal value occurs if a semi infinite distribution of impurity potentials is repulsive enough to produce current carrying surface states on its boundary

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

an adiabatic theorem for resonances we prove a robust extension of the quantum adiabatic theorem the theorem applies to systems that have resonances instead of bound states and to systems for which just an approximation to a bound state is known to demonstrate the theorem s usefulness in a concrete situation we apply it to shape resonances 2011 wiley periodicals inc

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

bounds on the spectral shift function and the density of states we study spectra of schrodinger operators on d first we consider a pair of operators which differ by a compactly supported potential as well as the corresponding semigroups we prove almost exponential decay of the singular values n of the difference of the semigroups as n and deduce bounds on the spectral shift function of the pair of operators

linear response theory for magnetic schrodinger operators in disordered media abstract we justify the linear response theory for an ergodic schrodinger operator with magnetic field within the noninteracting particle approximation and derive a kubo formula for the electric conductivity tensor to achieve that we construct suitable normed spaces of measurable covariant operators where the liouville equation can be solved uniquely if the fermi level falls into a region of localization we recover the well known kubo str eda formula for the quantum hall conductivity at zero temperature