aging of asymmetric dynamics on the random energy model we show aging of glauber type dynamics on the random energy model in the sense that we obtain the annealed scaling limits of the clock process and of the age process the latter encodes the gibbs weight of the configuration occupied by the dynamics both limits are expressed in terms of stable subordinators

a zero thickness limit of multilayer structures a resonant tunnelling δ potential a zero thickness limit for two terminal and three terminal devices from the quantum electronics domain is analysed the study is focused on heterostructures composed of a single barrier with adjacent one or two prewells the point interactions obtained in this limit are shown to be described by a family of resonant diagonal matrices that connect the two sided boundary conditions at the device origin which are a subclass of the whole four parameter family of point interactions transmission through such a device is absent almost everywhere except at a few points whose number and position can be controlled by a gate voltage applied externally to the barrier subsystem it is remarkable that the existence of resonances in the zero thickness limit occurs only if a squeezing sequence is constructed in a like way in this case the limit describes adequately the resonant behaviour of a barrier well heterostructure with realistic parameters simple analytical expressions obtained for resonance sets are supported by direct numerical calculations of the transmission being in agreement with the results of experiments on semiconductor devices the zero thickness like limiting procedure can be used in the design of nanodevices or contacting quantum wires

approximation of quantum graph vertex couplings by scaled schrodinger operators on thin branched manifolds we discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds we show that if a neumann type laplacian on such manifolds is amended by suitable potentials the resulting schrodinger operators can approximate non trivial vertex couplings the latter include not only the couplings but also those with wavefunctions discontinuous at the vertex we work out the example of the symmetric couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal we conclude with a result that certain vertex couplings cannot be approximated by a pure laplacian

consistency of the group lasso and multiple kernel learning we consider the least square regression problem with regularization by a block l1 norm that is a sum of euclidean norms over spaces of dimensions larger than one this problem referred to as the group lasso extends the usual regularization by the l1 norm where all spaces have dimension one where it is commonly referred to as the lasso in this paper we study the asymptotic group selection consistency of the group lasso we derive necessary and sufficient conditions for the consistency of group lasso under practical assumptions such as model mis specification when the linear predictors and euclidean norms are replaced by functions and reproducing kernel hilbert norms the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection using tools from functional analysis and in particular covar iance operators we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate even when the necessary condition required for the non adaptive scheme is not satisfied

sparse second moment analysis for elliptic problems in stochastic domains we consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations assuming normal perturbations with small amplitude and known mean field and two point correlation function we derive using a second order shape calculus deterministic equations for the mean field and the two point correlation function of the random solution for a model dirichlet problem which are 3rd order accurate in the boundary perturbation size using a variational boundary integral equation formulation on the unperturbed nominal boundary and a wavelet discretization we present and analyze an algorithm to approximate the random solution s mean and its two point correlation function at essentially optimal order in essentially mathcal o n work and memory where n denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain

one dimensional schrodinger operators with singular potentials a schwartz distributional formulation using an extension of the hormander product of distributions we obtain an intrinsic formulation of one dimensional schrodinger operators with singular potentials this formulation is entirely defined in terms of standard schwartz distributions and does not require as some previous approaches the use of more general distributions or generalized functions we determine in the new formulation the action and domain of the schrodinger operators with arbitrary singular boundary potentials we also consider the inverse problem and obtain a general procedure for constructing the singular pseudo potential that imposes a specific set of local boundary conditions this procedure is used to determine the boundary operators for the complete four parameter family of one dimensional schrodinger operators with a point interaction finally the and potentials are studied in detail and the corresponding schrodinger operators are shown to coincide with the norm resolvent limit of specific sequences of schrodinger operators with regular potentials

spectra of graph neighborhoods and scattering let g e e 0 be a family of e thin riemannian manifolds modeled on a finite metric graph g for example the e neighborhood of an embedding of g in some euclidean space with straight edges we study the asymptotic behavior of the spectrum of the laplace beltrami operator on g e as e 0 for various boundary conditions we obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions uniformly for k ce 1 in terms of scattering data on a non compact limit space we then use this to determine the quantum graph which is to be regarded as the limit object in a spectral sense of the family g e our method is a direct construction of approximate eigenfunctions from the scattering and graph data and the use of a priori estimates to show that all eigenfunctions are obtained in this way

asymptotic spectrum for the dirichlet laplacian in thin deformed tubes with scaled geometry it is known that geometric properties do not play a role in the asymptotics of the eigenvalues of the laplacian restricted to thin dirichlet deformed tubular regions in this work it is shown that by properly scaling the geometric properties of the tubes a limit where the geometry is on equal footing with the deformation function is obtained such a result gives an indication of how subtle the confining limits can be

coupling in the singular limit of thin quantum waveguides we analyze the problem of approximating a smooth quantum waveguide with a quantum graph we consider a planar curve with compactly supported curvature and a strip of constant width around the curve we rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve consisting of one vertex and two infinite straight edges i e a broken line we discuss the convergence of the laplacian with dirichlet boundary conditions on the strip in a suitable sense and we obtain two possible limits the laplacian on the line with dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the laplacian on the line the first case generically occurs and corresponds to the decoupling of the two components of the limit curve while in the second case a coupling takes place we present also two families of curves which give rise to coupling

fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

nonexponential relaxation in the random energy model abstract we solve a general class of master equations for a system consisting of exponentially distributed random energy levels the equations differ in the choice of transition rates the original master equation by de dominicis et al is included as a special case another special case has random transition rates representing random energy barriers the relaxation is found to be of power law or stretched exponential type in all cases

a tomography of the grem beyond the rem conjecture in a companion paper we proved that in a large class of gaussian disordered spin systems the local statistics of energy values near levels n 1 2 with 1 2 are described by a simple poisson process in this paper we address the issue as to whether this is optimal and what will happen if 1 2 we do this by analysing completely the gaussian generalised random energy models grem we show that the rem behaviour persists up to the level c n where c denotes the critical temperature we show that beyond this value the simple poisson process must be replaced by more and more complex mixed poisson point processes

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

branched quantum wave guides with dirichlet boundary conditions the decoupling case we consider a family of open sets m which shrinks with respect to an appropriate parameter to a graph under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted dirichlet spectrum of m converges to the spectrum of the differential laplacian on the graph with dirichlet boundary conditions at the vertices i e a graph operator without coupling between different edges the smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood the lower bound is given by the first transversal mode of the edge neighbourhood we also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a laplacian acting on the edge with an additional potential coming from the curvature

convergence of spectra of uniformly fattened open book structures abstract we consider a stratified compact surface m in r 3 which we call an open book structure whose singular points form simple closed smooth curves and its neighborhood m which we call a fattened open book structure assuming a transversality condition at the singularities and absence of zero dimensional strata i e corners we show that the discrete spectrum of the neumann laplacian in m converges when 0 to the spectrum of a differential operator on m similar results have been obtained before for the case of fattened graphs i e m being one dimensional in the case of a smooth surfaces m the problem has been studied well however having singularities along strata of lower dimensions significantly complicates considerations as in the quantum graph case such considerations are triggered by various applications

a remark on krein s resolvent formula and boundary conditions we prove an analogue of krein s resolvent formula expressing the resolvents of self adjoint extensions in terms of boundary conditions applications to quantum graphs and systems with point interactions are discussed

quenched localisation in the bouchaud trap model with slowly varying traps we consider the quenched localisation of the bouchaud trap model on the positive integers in the case that the trap distribution has a slowly varying tail at infinity our main result is that for each n in 2 3 ldots there exists a slowly varying tail such that quenched localisation occurs on exactly n sites as far as we are aware this is the first example of a model in which the exact number of localisation sites are able to be tuned according to the model parameters key intuition for this result is provided by an observation about the sum max ratio for sequences of independent and identically distributed random variables with a slowly varying distributional tail which is of independent interest

on the dynamics of trap models in z d we consider trap models in z d these are stochastic processes in a random environ ment as follows the environment is given by a family x x 2 z d of positive iid random variables in the basin of attraction of an stable law 0 1 given our process is a continuous time markov pure jump process whose jump chain is a in principle generic random walk in z d d 1 independent of andrepresents the holding time averages of the continuous time process we may think of the sites of z d as traps and ofx as the depth of trap x we are interested in the trap process namely the process that associates to time t the depth of the currently visited trap our first result is the convergence of the law of that process under a suitable scaling the limit process is given by the jumps of a certain stable subordinator at the inverse of another stable subor dinator correlated with the first subordinator for that result the requirements on the underlying random walk are a the validity of a law of large numbers for its range and b the slow variation at infinity of the tail of the distribution of its time of return to the origin this includes all transient random walks as well as all random walks in d 2 and also many one dimensional random walks we then derive aging results for our process namely scaling limits for some two time correlation functions of the process the scaling limit result mentioned above is an integrated result with respect to the environment un der an additional condition on the size of the intersection of the ranges of two independent copies of the underlying random walk roughly saying that it is relatively small compared with the range we derive a stronger scaling limit result roughly stating that it holds in probability with respect to the environment with that additional condition we also strengthen the aging results from the integrated version mentioned above to convergence in probability with respect to the environment

a hardy inequality in twisted waveguides we show that twisting of an infinite straight three dimensional tube with non circular cross section gives rise to a hardy type inequality for the associated dirichlet laplacian as an application we prove certain stability of the spectrum of the dirichlet laplacian in locally and mildly bent tubes namely it is known that any local bending no matter how small generates eigenvalues below the essential spectrum of the laplacian in the tubes with arbitrary cross sections rotated along a reference curve in an appropriate way in the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non empty discrete spectrum

the arcsine law as a universal aging scheme for trap models we give a general proof of aging for trap models using the arcsine law for stable subordinators this proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape we apply this proof to aging for trap models on large two dimensional tori and for trap dynamics of the random energy model on a broad range of time scales c 2006 wiley periodicals inc

convergence of clock processes in random environments and ageing in the p spin sk model we derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible extending recent results by gayrard 2010 2011 forthcoming based on general criterion for convergence of sums of dependent random variables due to durrett and resnick ann probab 6 1978 829 846 we demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the p spin sk model we prove that on a wide range of time scales the clock process converges to a stable subordinator almost surely with respect to the environment we also show that a time time correlation function converges to the arcsine law for this subordinator almost surely this improves recent results of ben arous bovier and cern comm math phys 282 2008 663 695 that obtained similar convergence results in law with respect to the random environment

biased random walk on critical galton watson trees conditioned to survive we consider the biased random walk on a critical galton watson tree conditioned to survive and confirm that this model with trapping belongs to the same universality class as certain one dimensional trapping models with slowly varying tails indeed in each of these two settings we establish closely related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs

limit theorems for sums of random exponentials we study limiting distributions of exponential sums open image in new window as t n where x i are i i d random variables two cases are considered a ess sup x i 0 and b ess sup x i we assume that the function h x log p x i x case b or h x log p x i 1 x case a is regularly varying at with index 1 case b or 0 case a the appropriate growth scale of n relative to t is of the form open image in new window where the rate function h0 t is a certain asymptotic version of the function open image in new window case b or open image in new window case a we have found two critical points 1 2 below which the law of large numbers and the central limit theorem respectively break down for 0 2 under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable with characteristic exponent 0 2 and skewness parameter 1

on symmetric random walks with random conductances on z d we study models of continuous time symmetric z d valued random walks in random environments one of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent we consider the case of independent conductances with a polynomial tail near 0 and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box

on δ like potential scattering on star graphs we discuss the potential scattering on the noncompact star graph the schrodinger operator with the short range potential localized in a neighborhood of the graph vertex is considered we study the asymptotic behavior of the corresponding scattering matrix in the zero range limit it has been known for a long time that in dimension 1 there is no non trivial hamiltonian with the distributional potential i e the potential acts as a totally reflecting wall several authors have in recent years studied the scattering properties of the regularizing potentials e 2q x e approximating the first derivative of the dirac delta function a non zero transmission through the regularized potential has been shown to exist as e 0 we extend these results to star graphs with the point interaction which is an analog of the potential on the line we prove that generically such a potential on the graph is opaque we also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs this set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary sturm liouville problem associated with q on the graph

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

quantum graphs as holonomic constraints we consider the dynamics on a quantum graph as the limit of the dynamics generated by a one particle hamiltonian in r2 with a potential having a deep strict minimum on the graph when the width of the well shrinks to zero for a generic graph we prove convergence outside the vertices to the free dynamics on the edges for a simple model of a graph with two edges and one vertex we prove convergence of the dynamics to the one generated by the laplacian with dirichlet boundary conditions in the vertex

the adiabatic limit of schrodinger operators on fibre bundles we consider schrodinger operators h delta g varepsilon v on a fibre bundle m mathop rightarrow limits pi b with compact fibres and a metric g varepsilon that blows up directions perpendicular to the fibres by a factor varepsilon 1 gg 1 we show that for an eigenvalue lambda of the fibre wise part of h satisfying a local gap condition and every n in mathbb n there exists a subspace of l 2 m that is invariant under h up to errors of order varepsilon n 1 the dynamical and spectral features of h on this subspace can be described by an effective operator on the fibre wise lambda eigenspace bundle mathcal e rightarrow b giving detailed asymptotics for h

local energy statistics in disordered systems a proof of the local rem conjecture recently bauke and mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model here we give necessary conditions for this hypothesis to be true which we show to be satisfied in wide classes of examples short range spin glasses and mean field spin glasses of the sk type we also show that under certain conditions the conjecture holds even if energy levels that grow moderately with the volume of the system are considered

point interactions of the dipole type defined through a three parametric power regularization a family of point interactions of the dipole type is studied in one dimension using a regularization by rectangles in the form of a barrier and a well separated by a finite distance the rectangles and the distance are parametrized by a squeezing parameter e 0 with three powers and describing the squeezing rates for the barrier the well and the distance respectively this parametrization allows us to construct a whole family of point potentials of the dipole type including some other point interactions such as e g potentials varying the power it is possible to obtain in the zero range limit the following two cases i the limiting potential is opaque the conventional result obtained earlier by some authors or ii this potential admits a resonant tunneling the opposite result obtained recently by other authors the structure of resonances if any also depends on a regularizing sequence the sets of the space where a non zero resonant or non resonant transmission occurs are found for all these cases in the zero range limit the transfer matrix is shown to be of the form with real parameters and g depending on a regularizing sequence those cases when 1 and g 0 mean that the corresponding potential is accompanied by an effective potential

universality of the rem for dynamics of mean field spin glasses we consider a version of glauber dynamics for a p spin sherrington kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the n dimensional hypercube we show that for all p 3 and all inverse temperatures 0 there exists a constant p 0 such that for all exponential time scales exp n with p the properly rescaled clock process time change process converges to an stable subordinator where 2 1 moreover the dynamics exhibits aging at these time scales with a time time correlation function converging to the arcsine law of this stable subordinator in other words up to rescaling on these time scales that are shorter than the equilibration time of the system the dynamics of p spin models ages in the same way as the rem and by extension bouchaud s rem like trap model confirming the latter as a universal aging mechanism for a wide range of systems the sk model the case p 2 seems to belong to a different universality class

the spectrum of the hilbert space valued second derivative with general self adjoint boundary conditions abstract we consider a large class of self adjoint elliptic problems associated with the second derivative acting on a space of vector valued functions we present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems the first more general one allows to replace a secular equation which is well known in some special cases by an abstract rank condition the second one though available in general seems to apply particularly well to a specific boundary condition the sometimes dubbed anti kirchhoff condition in the literature that arises in the theory of differential operators on graphs it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities in particular some results on the symmetry of the spectrum in either case

an efron stein inequality for nonsymmetric statistics on etablit l analogue d un resultat d efron et stein 1981 que l on demontre a l aide d une technique d espace de hilbert introduite par vitale 1984

universality and extremal aging for dynamics of spin glasses on subexponential time scales we consider random hopping time rht dynamics of the sherrington kirkpatrick sk model and p spin models of spin glasses for any of these models and for any inverse temperature 0 we prove that on time scales that are subexponential in the dimension the properly scaled clock process time change process of the dynamics converges to an extremal process moreover on these time scales the system exhibits aging like behavior which we call extremal aging in other words the dynamics of these models ages as the random energy model rem does hence by extension this confirms bouchaud s rem like trap model as a universal aging mechanism for a wide range of systems that for the first time includes the sk model 2011 wiley periodicals inc

glauber dynamics of the random energy model i metastable motion on the extreme states we investigate the long time behavior of the glauber dynamics for the random energy model below the critical temperature we give very precise estimates on the motion of the process to and between the states of extremal energies we show that when disregarding time the consecutive steps of the process on these states are governed by a markov chain that jumps uniformly on all possible states the mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point a first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order t is itself of order t the estimates proven in this paper will furnish crucial input for a follow up paper where aging is analysed in full detail

an explicit realization of resonant tunnelling δ potentials a family of non trivial one point interactions is constructed by means of a two scale approximation with spatially symmetric piecewise constant functions the approximation is realized through two powers of a squeezing parameter the convergence is considered in terms of finite range transmission matrices which are given in an explicit form the limiting matrix that connects the two sided boundary conditions at the point of singularity appears to be the unity matrix defined at a countable set in the space of intensities of the potential everywhere beyond this resonance set the limiting point interaction separates the system into two subsystems with boundary conditions of the dirichlet type in other words any interaction of this family behaves as a fully transparent barrier on a resonance set and beyond this set as a fully reflecting wall among these interactions a special regularizing sequence that converges to the second derivative of dirac s delta function in the sense of distributions is found because of this example it is reasonable to term all the point interactions with the properties described above as resonant tunnelling potentials

on the existence of thermodynamics for the random energy model derrida s random energy model is considered almost sure andlp convergence of the free energy at any inverse temperature are proven rigorous upper and lower bounds to the finite size corrections to the free energy are given

variational bayes group sparse time adaptive parameter estimation with either known or unknown sparsity pattern in this paper we study the problem of time adaptive group sparse signal estimation from a bayesian viewpoint we propose two online variational bayes schemes that are specifically designed to estimate and track group sparse signals in time the proposed schemes address both the cases where the grouping information of the signal is either known or not for the case of known group sparsity pattern the proposed scheme builds on a novel hierarchical model for the bayesian adaptive group lasso utilizing the variational bayes framework update equations for all model parameters are given for both the batch and time adaptive estimation scenarios to address the case where the group sparsity pattern is unknown the hierarchical bayesian model of the former scheme is extended by organizing the penalty parameters of the bayesian lasso in a conditional autoregressive model intrinsic conditional autoregression is exploited to penalize the signal coefficients in a structured manner and thus obtain group sparse solutions automatically again a robust and computationally efficient online variational bayes estimator is developed capitalizing on the conjugacy of the proposed hierarchical bayesian formulation experimental results are reported that corroborate the superior estimation performance of the proposed online schemes when compared with state of the art methods

phase transition for the speed of the biased random walk on the supercritical percolation cluster we prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on d that is for each d 2 and for any supercritical parameter p pc we prove the existence of a critical strength for the bias such that below this value the speed is positive and above the value it is zero we identify the value of the critical bias explicitly and in the subballistic regime we find the polynomial order of the distance moved by the particle each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk a key element in proving our results is to understand that on large scales the particle trajectory is essentially one dimensional we prove such a dynamic renormalization statement in a much stronger form than was previously known 2013 wiley periodicals inc

universality of trap models in the ergodic time scale consider a sequence of possibly random graphs gn vn en n 1 whose vertices s have i i d weights wnx x vn with a distribution belonging to the basin of attraction of an stable law 0 1 let xnt t 0 be a continuous time simple random walk on gn which waits a emph mean wnx exponential time at each vertex x under considerably general hypotheses we prove that in the ergodic time scale this trap model converges in an appropriate topology to a k process we apply this result to a class of graphs which includes the hypercube the d dimensional torus d 2 random d regular graphs and the largest component of super critical erd os r enyi random graphs

thin waveguides with robin boundary conditions we consider the laplace operator in a thin three dimensional tube with a robin type condition on its boundary and study asymptotically the spectrum of such operator as the diameter of the tube s cross section becomes infinitesimal in contrast with the dirichlet condition case 2 we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section if that symmetry condition fails then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube s axis depending on the curvature and the rotation angle in the symmetric case the behavior of lower energy modes is shown to be ruled by a one dimensional sturm liouville problem involving an effective potential given in explicit form

asymptotics of dirichlet eigenvalues and eigenfunctions of the laplacian on thin domains in rd abstract we consider the laplace operator with dirichlet boundary conditions on a domain in r d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero this generalizes our previous results in two dimensions and as in that case allows us to obtain an approximation for dirichlet eigenvalues for a large class of domains under very mild assumptions as an application we derive a three term asymptotic expansion for the first eigenvalue of d dimensional ellipsoids

semiclassical analysis of constrained quantum systems we study the dynamics of a quantum particle in constrained by a strong potential force to stay within a distance of order in suitable units from a smooth n dimensional submanifold m we prove that in the semiclassical limit the evolution of the wavefunction is approximated in norm up to terms of order 1 2 by the evolution of a semiclassical wave packet centred on the trajectory of the corresponding classical constrained system

aging in two dimensional bouchaud s model let e x be a collection of i i d exponential random variables symmetric bouchaud s model on 2 is a markov chain x t whose transition rates are given by w xy exp e x if x y are neighbours in 2 we study the behaviour of two correlation functions x t w t x t w and x t x t w t t w t w t we prove the sub aging behaviour of these functions when 1

convergence to extremal processes in random environments and extremal ageing in sk models this paper extends recent results on ageing in mean field spin glasses on short time scales obtained by ben arous and gun commun pure appl math 65 77 127 2012 in law with respect to the environment to results that hold almost surely respectively in probability with respect to the environment it is based on the methods put forward in gayrard in aging in reversible dynamics of disordered systems ii emergence of the arcsine law in the random hopping time dynamics of the rem 2010 electron j probab 17 58 1 33 2012 and naturally complements bovier and gayrard in ann probab 2012

k processes scaling limit and aging for the trap model in the complete graph we study k processes which are markov processes in a denumerable state space all of whose elements are stable with the exception of a single state starting from which the process enters finite sets of stable states with uniform distribution we show how these processes arise in a particular instance as scaling limits of the trap model in the complete graph and subsequently derive aging results for those models in this context

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

convergence to fractional kinetics for random walks associated with unbounded conductances we consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail we prove that the scaling limit of this process is a fractional kinetics process that is the time change of a d dimensional brownian motion by the inverse of an independent stable subordinator we further show that the same process appears in the scaling limit of the non symmetric bouchaud s trap model

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

bound states in curved quantum waveguides a free quantum particle living on a curved planar strip of a fixed width d with dirichlet boundary conditions is studied it can serve as a model for electrons in thin films on a cylinder type substrate or in a curved quantum wire assuming that the boundary of is infinitely smooth and its curvature decays fast enough at infinity it is proved that a bound state with energy below the first transversal mode exists for all sufficiently small d a lower bound on the critical width is obtained using the birman schwinger technique

scaling limit for trap models on ℤd we give the quenched scaling limit of bouchaud s trap model in d 2 this scaling limit is the fractional kinetics process that is the time change of a d dimensional brownian motion by the inverse of an independent stable subordinator

scattering solutions in networks of thin fibers small diameter asymptotics small diameter asymptotics is obtained for scattering solutions in a network of thin fibers the asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph we calculate the lagrangian gluing conditions at vertices v in gamma for the problems on the limiting graph if the frequency of the incident wave is above the bottom of the absolutely continuous spectrum the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

recursive ell_ 1 infty group lasso we introduce a recursive adaptive group lasso algorithm for real time penalized least squares prediction that produces a time sequence of optimal sparse predictor coefficient vectors at each time index the proposed algorithm computes an exact update of the optimal l1 penalized recursive least squares rls predictor each update minimizes a convex but nondifferentiable function optimization problem we develop an on line homotopy method to reduce the computational complexity numerical simulations demonstrate that the proposed algorithm outperforms the l1 regularized rls algorithm for a group sparse system identification problem and has lower implementation complexity than direct group lasso solvers

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

two body scattering on a graph and application to simple nanoelectronic devices the two body system on a graph with one junction is considered the effective three body scattering problem turns out to be exactly solvable for pointwise interactions additional degrees of freedom corresponding to a dynamics of some structure e g an atomic cluster located in the junction point of common contact of three thin electrodes are considered these degrees of freedom bring effective energy dependent interaction into the effective schrodinger equation in the scattering channel the wave function of the system is constructed in the explicit form using the extension theory methods the obtained results are applied to the qualitative description of a simple three electrode nanoelectronic device the perturbation theory approach based on the analysis of the liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function

aging of the metropolis dynamics on the random energy model we study the metropolis dynamics of the simplest mean field spin glass model the random energy model we show that this dynamics exhibits aging by showing that the properly rescaled time change process between the metropolis dynamics and a suitably chosen fast markov chain converges in distribution to a stable subordinator the rescaling might depend on the realization of the environment but we show that its exponential growth rate is deterministic

metastability in stochastic dynamics of disordered mean field models we study a class of markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding gibbs measures we derive the analog of the wentzell freidlin theory in this case showing that any transition can be decomposed with probability exponentially close to one into a deterministic sequence of admissible transitions for these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor the distributions of the rescaled transition times are shown to converge to the exponential distribution we exemplify our results in the context of the random field curie weiss model

sparls the sparse rls algorithm we develop a recursive l1 regularized least squares sparls algorithm for the estimation of a sparse tap weight vector in the adaptive filtering setting the sparls algorithm exploits noisy observations of the tap weight vector output stream and produces its estimate using an expectation maximization type algorithm we prove the convergence of the sparls algorithm to a near optimal estimate in a stationary environment and present analytical results for the steady state error simulation studies in the context of channel estimation employing multipath wireless channels show that the sparls algorithm has significant improvement over the conventional widely used recursive least squares rls algorithm in terms of mean squared error mse moreover these simulation studies suggest that the sparls algorithm with slight modifications can operate with lower computational requirements than the rls algorithm when applied to tap weight vectors with fixed support

glauber dynamics of the random energy model ii aging below the critical temperature we investigate the long time behavior of the glauber dynamics for the ran dom energy model below the critical temperature we establish that for a suitably chosen timescale that diverges with the size of the system one can prove that a natural auto correlation function exhibits aging moreover we show that the long time asymptotics of this function coincide with those of the so called rem like trap model proposed by bouchaud and dean our results rely on very precise estimates on the distribution of transition times of the process between different states of extremely low energy

on the spectrum and weakly effective operator for dirichlet laplacian in thin deformed tubes abstract we study the laplacian in deformed thin bounded or unbounded tubes in r 3 i e tubular regions along a curve r s whose cross sections are multiplied by an appropriate deformation function h s 0 one of the main requirements on h s is that it has a single point of global maximum we find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero it is shown that such behaviors are not influenced by some geometric features of the tube such as curvature torsion and twisting and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator

diffusion processes in thin tubes and their limits on graphs the present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary respectively reflecting at the boundary and corresponding processes in the limit where the thin tubular domains are shrinking to graphs the methods we use are probabilistic ones for shrinking we use big potentials respectively reflection on the boundary of tubes we show that there exists a unique limit process and we characterize the limit process by a second order differential generator acting on functions defined on the limit graph with kirchhoff boundary conditions at the vertices

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator

aging in metropolis dynamics of the rem a proof we consider metropolis dynamics of the random energy model we prove that the classical two time correlation function that allows one to establish aging converges almost surely to the arcsine law distribution function as predicted in the physics literature in the optimal domain of the time scale and temperature parameters where this result can be expected to hold to do this we link the two time correlation function to a certain continuous time clock process which after proper rescaling is proven to converge to a stable subordinator almost surely in the random environment and in the fine j 1 topology of skorohod this fine topology then enables us to deduce from the arcsine law for stable subordinators the asymptotic behavior of the two time correlation function that characterizes aging