stability of the absolutely continuous spectrum of random schrodinger operators on tree graphs the subject of this work is random schrodinger operators on regular rooted tree graphs with stochastically homogeneous disorder the operators are of the form h t u v acting in l2 with t the adjacency matrix u a radially periodic potential and v a random potential this includes the only class of homogeneously random operators for which it was proven that the spectrum of h exhibits an absolutely continuous ac component a results established by a klein for weak disorder in case u 0 and v given by iid random variables on our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder the method yields the continuity of the ac spectral density of h at 0 the latter is shown to converge in the l 1 sense over closed borel sets in which h 0 has no singular spectrum the analysis extends to random potentials whose values at different sites need not be independent assuming only that their joint distribution is weakly correlated across different tree branches

l p norms and support of eigenfunctions on graphs this article is concerned with properties of delocalization for eigenfunctions of schrodinger operators on large finite graphs more specifically we provide l p norm estimates for such eigenfunctions and show that they have a large support our estimates hold for any fixed possibly irregular graph in prescribed energy regions and also for certain sequences of graphs such as n lifts

on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

decay rate of the green function in a random potential on the bethe lattice and a criterion for localization 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 results in the case of the lorentzian distribution of site energies are presented as an application of these results a criterion for the localized region on the corresponding regular lattice on the level of the bethe peierls approximation is proposed our criterion yields exact results in the one dimensional limit and yields correct band edges for the hypercubic lattice in the vanishing limit of disorder the mobility edge trajectory obtained by our criterion is given by an elliptic curve in the case of the lorentzian distribution and its shape is found to be in qualitatively good agreement with that obtained by the finite size scaling method in the three dimensional system

regularity of the invariant measure and of the density of states in the one dimensional anderson model abstract let h 1 2 v on l2 z where v x x z are i i d r v s with common probability distribution let h t e itvd v let n e be the integrated density of states and let e be the invariant measure at energy e it is proven 1 1 let h be d times differentiable on 0 with bounded derivatives such that h j t 0 as t for j 0 1 d then i n e is a cd function ii if d 3 e is absolutely continuous for all e r and d e dx is of class c d 1 2 1 jointly in e and x 2 2 let h be differentiable on 0 with h j t o e b t for j 0 1 and some b 0 then d e dx is jointly analytic in e and x

local kesten mckay law for random regular graphs we study the adjacency matrices of random d regular graphs with large but fixed degree d in the bulk of the spectrum 2 sqrt d 1 varepsilon 2 sqrt d 1 varepsilon down to the optimal spectral scale we prove that the green s functions can be approximated by those of certain infinite tree like few cycles graphs that depend only on the local structure of the original graphs this result implies that the kesten mckay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors our method is based on estimating the green s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs

analyticity of the density of states in the anderson model on the bethe lattice let h 1 2 v on l2 b whereb is the bethe lattice andv x xeb are i i d r v s with common probability distribution it is shown that for distributions sufficiently close to the cauchy distribution the density of states e is analytic in a strip about the real axis

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

expansion of random graphs new proofs new results we present a new approach to showing that random graphs are nearly optimal expanders this approach is based on recent deep results in combinatorial group theory it applies to both regular and irregular random graphs let gamma be a random d regular graph on n vertices and let lambda be the largest absolute value of a non trivial eigenvalue of its adjacency matrix it was conjectured by alon combinatorica 6 2 83 96 1986 that a random d regular graph is almost ramanujan in the following sense for every varepsilon 0 lambda 0 a a s lambda rho varepsilon where rho is the spectral radius of the universal cover of omega when omega is regular we get a bound of rho 0 84 and for an arbitrary omega we prove a nearly optimal upper bound of sqrt 3 rho this is a substantial improvement upon all known results by friedman linial puder lubetzky sudakov vu and addario berry griffiths

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

eigenvalues of graphs and a simple proof of a theorem of greenberg abstract in his ph d thesis greenberg proved that if x is the spectral radius of the universal cover x of a finite graph x then for each 0 a positive proportion depending only on x and of the eigenvalues of x have absolute value at least x in this paper we show that the same result holds true if we remove absolute from the previous result we also prove an analogue result for the smallest eigenvalues of x

ramanujan coverings of graphs abstract let g be a finite connected graph and let be the spectral radius of its universal cover for example if g is k regular then 2 k 1 we show that for every r there is an r covering a k a an r lift of g where all the new eigenvalues are bounded from above by it follows that a bipartite ramanujan graph has a ramanujan r covering for every r this generalizes the r 2 case due to marcus spielman and srivastava 26 every r covering of g corresponds to a labeling of the edges of g by elements of the symmetric group s r we generalize this notion to labeling the edges by elements of various groups and present a broader scenario where ramanujan coverings are guaranteed to exist in particular this shows the existence of richer families of bipartite ramanujan graphs than was known before inspired by 26 a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups the core argument of this component is taken from 27 another important ingredient of our proof is a new generalization of the matching polynomial of a graph we define the r th matching polynomial of g to be the average matching polynomial of all r coverings of g we show this polynomial shares many properties with the original matching polynomial for example it is real rooted with all its roots inside

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

the large connectivity limit of the anderson model on tree graphs we consider the anderson localization problem on the infinite regular tree within the localized phase we derive a rigorous lower bound on the free energy function recently introduced by aizenman and warzel using a finite volume regularization we also derive an upper bound on this free energy function this yields upper and lower bounds on the critical disorder such that all states at a given energy become localized these bounds are particularly useful in the large connectivity limit where they match confirming the early predictions of abou chacra anderson and thouless

finite trotter approximation to the averaged mean square distance in the anderson model we prove that a finite trotter approximation to the averaged mean square distance traveled by a particle in a disordered system on a lattice d exhibits at most a diffusive behavior in dimensions d 3 as long as the fourier transform of the single site probability hat mu is in l2

extended states in the anderson model on the bethe lattice abstract we prove that the anderson hamiltonianh von the bethe lattice has extended states for small disorder more precisely given any closed intervalicontained in the interior of the spectrum of the laplacian on the bethe lattice we prove that for small disorderh has purely absolutely continuous spectrum iniwith probability one i e ac h i iand pp h i sc h i with probability one and its integrated density of states is continuously differentiable on the intervali

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

temporal evolution of the domain structure in a poisson voronoi nucleation and growth transformation results for one and three dimensions the distribution of spatial domain structures originated during one and three dimensional poisson voronoi transformations are computed analytically extending the recently obtained results for the two dimensional case the presented method gives a full description of the developed microstructure and is valid for tessellations of any dimensionality the temporal and spatial dependences of the domain structure are completely discriminated and separated showing the existence of geometric configurations independent of time a single computation of the probability distribution of these geometric configurations allows us to calculate the total free boundary and size probability distributions at any desired time the obtained results show full agreement with stochastic simulations and reproduce completely the previously existing partial results a discussion about the potential applications of the method to the calculation of other geometrical properties and the characteristics of the final static structure leading to a gamma distribution of sizes is also presented

on the size distribution of poisson voronoi cells abstract poisson voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices although this particular space tessellation is intensively studied by mathematicians in two and three dimensional 3d spaces there is no exact result known for the size distribution of voronoi cells motivated by the simple form of the distribution function in the 1d case a simple and compact analytical formula is proposed for approximating the voronoi cell s size distribution function in the practically important 2d and 3d cases as well denoting the dimensionality of the space by d d 1 2 3 the f y const y 3 d 1 2 exp 3 d 1 y 2 compact form is suggested for the normalized cell size distribution function by using large scale computer simulations the viability of the proposed distribution function is studied and critically discussed

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

the non backtracking spectrum of the universal cover of a graph a non backtracking walk on a graph h is a directed path of directed edges of h such that no edge is the inverse of its preceding edge non backtracking walks of a given length can be counted using the non backtracking adjacency matrix b indexed by h s directed edges and related to ihara s zeta function we show how to determine b s spectrum in the case where h is a tree covering a finite graph we show that when h is not regular this spectrum can have positive measure in the complex plane unlike the regular case we show that outside of b s spectrum the corresponding green function has periodic decay ratios the existence of such a ratio system can be effectively checked and is equivalent to being outside the spectrum we also prove that the spectral radius of the non backtracking walk operator on the tree covering a finite graph is exactly gr where gr is the growth rate of the tree this further motivates the definition of the graph theoretical riemann hypothesis proposed by stark and terras st finally we give experimental evidence that for a fixed finite graph h a ran dom lift of large degree has non backtracking new spectrum near that of h s universal cover this suggests a new generalization of alon s second eigenvalue conjecture

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

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

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

weak disorder expansion for the anderson model on a tree 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 there is an energyec above which the density of states and the conducting properties vanish to all orders in perturbation theory we compute pertubatively the position of the lineec which begins in the limit of zero disorder at the band edge of the pure system inside the band of the pure system the density of states and conducting properties can be computed perturbatively this expansion breaks down nearec because of small denominators we show how it can be resummed by choosing the appropriate scaling of the energy for energies greater thanec we show that nonperturbative effects contribute to the density of states but we have been unable to tell whether they also contribute to the conducting properties

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

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

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

quantum ergodicity on regular graphs we give three different proofs of the main result of anantharaman and le masson duke math j 164 4 723 765 2015 establishing quantum ergodicity a form of delocalization for eigenfunctions of the laplacian on large regular graphs of fixed degree these three proofs are much shorter than the original one quite different from one another and we feel that each of the four proofs sheds a different light on the problem the goal of this exploration is to find a proof that could be adapted for other models of interest in mathematical physics such as the anderson model on large regular graphs regular graphs with weighted edges or possibly certain models of non regular graphs a source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs

levy khintchine random matrices and the poisson weighted infinite skeleton tree we study a class of hermitian random matrices which includes wigner matrices heavy tailed random matrices and sparse random matrices such as adjacency matrices of erd os renyi random graphs with pn 1 our n n random matrices have real entries which are i i d up to symmetry the distribution of entries depends on n and we require sums of rows to converge in distribution it is then well known that the limit distribution must be infinitely divisible we show that a limiting empirical spectral distribution lsd exists and via local weak convergence of associated graphs the lsd corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many poisson weighted infinite trees using a backbone structure of special edges called cords to infinity one example covered by the results are matrices with i i d entries having infinite second moments but normalized to be in the gaussian domain of attraction in this case the limiting graph is n rooted at 1 so the lsd is the semi circle law the results also extend to self adjoint complex matrices and wishart matrices

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

absolutely continuous spectrum for the anderson model on a product of a tree with a finite graph abstract we prove the almost sure existence of absolutely continuous spectrum at low disorder for the anderson model on the simplest example of a product of a regular tree with a finite graph this graph contains loops of unbounded size

spectra of lifted ramanujan graphs abstract a random n lift of a base graph g is its cover graph h on the vertices n v g where for each edge uv in g there is an independent uniform bijection and h has all edges of the form i u i v a main motivation for studying lifts is understanding ramanujan graphs and namely whether typical covers of such a graph are also ramanujan let g be a graph with largest eigenvalue 1 and let be the spectral radius of its universal cover friedman 2003 12 proved that every new eigenvalue of a random lift of g is o 1 2 1 1 2 with high probability and conjectured a bound of o 1 which would be tight by results of lubotzky and greenberg 1995 15 linial and puder 2010 17 improved friedman s bound to o 2 3 1 1 3 for d regular graphs where 1 d and 2 d 1 this translates to a bound of o d 2 3 compared to the conjectured 2 d 1 here we analyze the spectrum of a random n lift of a d regular graph whose nontrivial eigenvalues are all at most in absolute value we show that with high probability the absolute value of every nontrivial eigenvalue of the lift is o log this result is tight up to a logarithmic factor and for d 2 3 e it substantially improves the above upper bounds of friedman and of linial and puder in particular it implies that a typical n lift of a ramanujan graph is nearly ramanujan

the anderson transition new numerical results for the critical exponents abstract the results of our recent evaluation of the critical exponents of the localization length v and the conductivity s at the anderson transition for box and gaussian distributions of the random potential are reported data are evaluated not only in the centre of the band but also near the band edge near the band centre we obtain s v 1 4 0 2 and s v 0 9 0 3 for the box and the gaussian distribution respectively near the band edge we found it impossible to determine accurately the exponents

measure preserving words are primitive a word w 2 fk the free group on k generators is called primitive if it belongs to some basis of fk associated with w and a finite group g is the word map w g g g defined on the direct product of k copies of g we call w measure preserving if given uniform measure on g g the image of this word map induces uniform measure on g for every finite group g it is easy to see that every primitive word is measure preserving and several authors have conjectured that the two properties are in fact equivalent here we prove this conjecture the main ingredients of the proof include random coverings of stallings graphs algebraic extensions of free groups and mobius inversions our methods yield the stronger result that a subgroup of fk is measure preserving iff it is a free factor as an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of fk form a closed set in this topology

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

delocalization at small energy for heavy tailed random matrices we prove that the eigenvectors associated to small enough eigenvalues of an heavy tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity the delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of l 4 and l 2 norms of vectors in contrast as a consequence of a previous result for random matrices with sufficiently heavy tails the eigenvectors associated to large enough eigenvalues are localized according to the same criterion the proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix

some rigorous results for random and quasi periodic potentials abstract we review some mathematical results and conjectures for random and quasi periodic schrodinger operators the role of localization in certain nonlinear systems is also discussed in addition we present some general arguments which show that the correlation length exponent v 2 d for a wide class of d dimensional random systems including the dilute ising ferro magnet

properties of a three dimensional poisson voronoi tesselation a monte carlo study a complete statistical description of the properties of a cellular microstructure generated by a three dimensional poisson voronoi tesselation has been obtained by a rigorous computer simulation involving several hundred thousand cells a two parameter gamma distribution is found to be a good fit to the cell s face volume and surface area distributions for a sample size of several thousand cells or less a lognormal distribution can also be used to approximate these distributions the individual face area and edge length distributions are also obtained

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

a strong maximum principle for differential equations with nonstandard p x growth conditions abstract in this paper with some special technics we give a strong maximum principle for the equations with nonstandard p x growth conditions div x u u d x f x u 0 in where x s d x f x u satisfy some conditions

a supersymmetric transfer matrix and differentiability of the density of states in the one dimensional anderson model let h v on 2 z where v x 9 xez are d r v s with common probability distribution v let h t e itvdv v and let k e be the integrated density of states it is proven i if h is n times differentiable with h j t 0 l t for some 0 7 0 l n then k e is a cn function in particular if v has compact support and h t 0 l h 1 with 0 then k e is c this allows v to be singular continuous ii lfh t 0 e a tl for some 0 then k e is analytic in a strip about the real axis the proof uses the supersymmetric replica trick to rewrite the averaged green s function as a two point function of a one dimensional supersymmetric field theory which is studied by the transfer matrix method

absolutely continuous spectrum for the anderson model on a tree a geometric proof of klein s theorem we give a new proof of a version of klein s theorem on the existence of absolutely continuous spectrum for the anderson model on the bethe lattice at weak disorder

on the absolutely continuous spectrum of stark hamiltonians we study the spectral properties of the schrodinger operator with a constant electric field perturbed by a bounded potential it is shown that if the derivative of the potential in the direction of the electric field is smaller at infinity than the electric field then the spectrum of the corresponding stark operator is purely absolutely continuous in one dimension the absolute continuity of the spectrum is implied by just the boundedness of the derivative of the potential the sharpness of our criterion for higher dimensions is illustrated by constructing smooth potentials with bounded partial derivatives for which the corresponding stark operators have a dense point spectrum

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

on the statistics of area size in two dimensional thick voronoi diagrams cells of voronoi diagrams in two dimensions are usually considered as having edges of zero width however this is not the case in several experimental situations in which the thickness of the edges of the cells is relatively large in this paper the concept of a thick voronoi tessellation that is with edges of non zero width is introduced and the statistics of cell areas as thickness changes are analyzed

spectral theory of one channel operators and application to absolutely continuous spectrum for anderson type models abstract a one channel operator is a self adjoint operator on l 2 g for some countable set g with a rank 1 transition structure along the sets of a quasi spherical partition of g jacobi operators are a very special case in essence there is only one channel through which waves can travel across the shells to infinity this channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells not all of the transfer matrices are defined for some countable set of energies still many theorems from the world of jacobi operators are translated to this setup the results are then used to show absolutely continuous spectrum for the anderson model on certain finite dimensional graphs with a one channel structure this result generalizes some previously obtained results on antitrees

structural stability for variable exponent elliptic problems i the p x laplacian kind problems abstract we study the structural stability i e the continuous dependence on coefficients of solutions of the elliptic problems under the form b u n div a n x u n f n the equation is set in a bounded domain of r n and supplied with the homogeneous dirichlet boundary condition on here b is a non decreasing function on r and a n x n is a family of applications which verifies the classical leray lions hypotheses but with a variable summability exponent p n x 1 p p n p the need for varying p x arises for instance in the numerical analysis of the p x laplacian problem uniqueness and existence for these problems are well understood by now we apply the stability properties to further generalize the existence results the continuous dependence result we prove is valid for weak and for renormalized solutions notice that besides being interesting on its own the renormalized solutions framework also permits us to deduce optimal convergence results for the weak solutions our technique avoids the use of a fixed duality framework like the w 0 1 p x w 1 p x duality and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself in a local or in a non local way the sequel of this paper will be concerned with well posedness of some p u laplacian kind problems and with existence of solutions to elliptic systems with variable solution dependent growth exponent

on the preservation of absolutely continuous spectrum for schrodinger operators abstract we present general principles for the preservation of a c spectrum under weak perturbations the schrodinger operators on the strip and on the caley tree bethe lattice are considered

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

p x laplacian equations satisfying cerami condition the goal of this paper is to prove the existence and multiplicity of solutions for the p x laplacian equations without the well known ambrosetti rabinowitz type growth conditions the critical point theorems with cerami condition are used

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

domain size distribution in a poisson voronoi nucleation and growth transformation we present an analytical result for the evolution of the domain size distribution during the growth of simultaneously nucleated domains the final stage of this transformation is the well known poisson voronoi tessellation the method can be easily extended to the calculation of the probability distribution of any other geometric characteristic as far as we know it is the first time that an exact result is given for this classic system

a new algorithm for three dimensional voronoi tessellation abstract a new efficient algorithm for computing the voronoi tessellation in three dimensions is presented the voronoi tessellation is obtained by constructing delaunay tetrahedra the complete set of these tetrahedra corresponds to that of a single voronoi polyhedron

the spectrum of the random environment and localization of noise we consider random walk on a mildly random environment on finite transitive d regular graphs of increasing girth after scaling and centering the analytic spectrum of the transition matrix converges in distribution to a gaussian noise an interesting phenomenon occurs at d 2 as the limit graph changes from a regular tree to the integers the noise becomes localized open image in new window the graphs of the noise covariance structure for d 4 3 2 1 from above

a self consistent approach to localization transition abstract the anderson transition is investigated using a self consistent approximation in analogy with mean field approximations in classical spin systems mobility edge trajectories in a three dimensional disordered system with various distributions of the site energies are obtained the present results are qualitatively in good agreement with the results obtained by using the finite size scaling method