supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

products of random matrices and generalised quantum point scatterers to every product of 2 2 matrices there corresponds a one dimensional schrodinger equation whose potential consists of generalised point scatterers products of random matrices are obtained by making these interactions and their positions random we exhibit a simple one dimensional quantum model corresponding to the most general product of matrices in sl 2 we use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

absorbing state phase transitions with quenched disorder quenched disorder in the sense of the harris criterion is generally a relevant perturbation at an absorbing state phase transition point here using a strong disorder renormalization group framework and effective numerical methods we study the properties of random fixed points for systems in the directed percolation universality class for strong enough disorder the critical behavior is found to be controlled by a strong disorder fixed point which is isomorph with the fixed point of random quantum ising systems in this fixed point dynamical correlations are logarithmically slow and the static critical exponents are conjecturedly exact for one dimensional systems the renormalization group scenario is confronted with numerical results on the random contact process in one and two dimensions and satisfactory agreement is found for weaker disorder the numerical results indicate static critical exponents which vary with the strength of disorder whereas the dynamical correlations are compatible with two possible scenarios either they follow a power law decay with a varying dynamical exponent like in random quantum systems or the dynamical correlations are logarithmically slow even for a weak disorder for models in the parity conserving universality class there is no strong disorder fixed point according to our renormalization group analysis

a sublogarithmic approximation for tollbooth pricing on trees an instance of the tollbooth problem consists of an undirected network and a collection of single minded customers each of which is interested in purchasing a fixed path subject to an individual budget constraint the objective is to assign a per unit price to each edge in a way that maximizes the collective revenue obtained from all customers the revenue generated by any customer is equal to the overall price of the edges in her desired path when this cost falls within her budget otherwise that customer will not purchase any edge our main result is a deterministic algorithm for the tollbooth problem on trees whose approximation ratio is o log m log log m where m denotes the number of edges in the underlying graph this finding improves on the currently best performance guarantees for trees and up until recently also on the best ratio for paths commonly known as the highway problem an additional interesting consequence is a computational separation between tollbooth pricing on trees and the o

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

anomalous diffusion localization aging and subaging effects in trap models at very low temperature we study in detail the dynamics of the one dimensional symmetric trap model via a real space renormalization procedure which becomes exact in the limit of zero temperature in this limit the diffusion front in each sample consists of two delta peaks which are completely out of equilibrium with each other the statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit t 0 we first compute disorder averages of one time observables such as the diffusion front the thermal width the localization parameters the two particle correlation function and the generating function of thermal cumulants of the position we then study aging and subaging effects our approach reproduces very simply the two different aging exponents and yields explicit forms for scaling functions of the various two time correlations we also extend the real space renormalization group method to include systematic corrections to the previous zero temperature procedure via a series expansion in t we then consider the generalized trap model with parameter alpha in 0 1 and obtain that the large scale effective model at low temperature does not depend on alpha in any dimension so that the only observables sensitive to alpha are those that measure the local persistence such as the probability to remain exactly in the same trap during a time interval finally we extend our approach at a scaling level for the trap model in d 2 and obtain the two relevant time scales for aging properties

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

beyond the flow decomposition barrier we introduce a new approach to the maximum flow problem this approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities our approach leads to an o min n 2 3 m 1 2 m log n 2 m log u time bound for a network with n vertices m arcs and integral arc capacities in the range 1 u this is a fundamental improvement over the previous time bounds we also improve bounds for the gomory hu tree problem the parametric flow problem and the approximate s t cut problem

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

non equilibrium dynamics of disordered systems understanding the broad continuum of relevant time scales via a strong disorder rg in configuration space we show that an appropriate description of the non equilibrium dynamics of disordered systems is obtained through a strong disorder renormalization procedure in configuration space that we define for any master equation with transitions rates between configurations the idea is to eliminate iteratively the configuration with the highest exit rate to obtain renormalized transition rates between the remaining configurations the multiplicative structure of the new generated transition rates suggests that for a very broad class of disordered systems the distribution of renormalized exit barriers defined as will become broader and broader upon iteration so that the strong disorder renormalization procedure should become asymptotically exact at large time scales we have checked numerically this scenario for the non equilibrium dynamics of a directed polymer in a two dimensional random medium

energy dynamics in the sinai model abstract we study the time dependent potential energy w t u x 0 u x t of a particle diffusing in a one dimensional random force field the sinai model using the real space renormalization group method rsrg we obtain the exact large time limit of the probability distribution of the scaling variable w w t t ln t this distribution exhibits a non analytic behaviour at w 1 these results are extended to a small non zero applied field using the constrained path integral method we moreover compute the joint distribution of energy w t and position x t at time t in presence of a reflecting boundary at the starting point with possibly some drift in the direction the rsrg very simply yields the one time and aging two time behaviour of this joint probability it exhibits differences in behaviour compared to the unbounded motion such as analyticity relations with some magnetization distributions in the 1d spin glass are discussed

proliferation assisted transport in a random environment we investigate the competition between barrier slowing down and proliferation induced superdiffusion in a model of population dynamics in a random force field numerical results in d 1 suggest that a new intermediate diffusion behaviour appears we introduce the idea of proliferation assisted barrier crossing and give a flory like argument to understand qualitatively this non trivial diffusive behaviour a renormalization group analysis close to the critical dimension dc 2 confirms that the random force fixed point is unstable and flows towards an uncontrolled strong coupling regime

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

approximating the revenue maximization problem with sharp demands we consider the revenue maximization problem with sharp multi demand in which m indivisible items have to be sold to n potential buyers each buyer i is interested in getting exactly d i items and each item j gives a benefit v ij to buyer i we distinguish between unrelated and related valuations in the former case the benefit v ij is completely arbitrary while in the latter each item j has a quality q j each buyer i has a value v i and the benefit v ij is defined as the product v i q j the problem asks to determine a price for each item and an allocation of bundles of items to buyers with the aim of maximizing the total revenue that is the sum of the prices of all the sold items the allocation must be envy free that is each buyer must be happy with her assigned bundle and cannot improve her utility we first prove that for related valuations the problem cannot be approximated to a factor o m 1 e for any e 0 unless p np and that such result is asymptotically tight in fact we provide a simple m approximation algorithm even for unrelated valuations we then focus on an interesting subclass of proper instances that do not contain buyers a priori known not being able to receive any item for such instances we design an interesting 2 approximation algorithm and show that no 2 e approximation is possible for any 0 e 1 unless p np we observe that it is possible to efficiently check if an instance is proper and if discarding useless buyers is allowed an instance can be made proper in polynomial time without worsening the value of its optimal solution

single parameter scaling in one dimensional localization revisited the variance of the lyapunov exponent is calculated exactly in the one dimensional anderson model with random site energies distributed according to the cauchy distribution we find a new significant scaling parameter in the system and derive an exact analytical criterion for single parameter scaling which differs from the commonly used condition of phase randomization the results obtained are applied to the kronig penney model with the potential in the form of periodically positioned ensuremath delta functions with random strength

finite current stationary states of random walks on one dimensional lattices with aperiodic disorder abstract stationary states of random walks with finite induced drift velocity on one dimensional lattices with aperiodic disorder are investigated by scaling analysis three aperiodic sequences the thue morse tm the paperfolding pf and the rudin shapiro rs sequences are used to construct the aperiodic disorder these are binary sequences composed of two symbols a and b and the ratio of the number of as to that of bs converges to unity in the infinite sequence length limit but their effects on diffusional behavior are different for the tm model the stationary distribution is extended as in the case without current and the drift velocity is independent of the system size for the pf model and the rs model as the system size increases the hierarchical and fractal structure and the localized structure respectively are broken by a finite current and changed to an extended distribution if the system size becomes larger than a certain threshold value correspondingly the drift velocity is saturated in a large system while in a small system it decreases as the system size increases

statistics of finite time lyapunov exponents in a random time dependent potential the sensitivity of trajectories over finite time intervals t to perturbations of the initial conditions can be associated with a finite time lyapunov exponent lambda obtained from the elements m ij of the stability matrix m for globally chaotic dynamics lambda tends to a unique value the usual lyapunov exponent lambda infinity for almost all trajectories as t is sent to infinity but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity we compute for a particle moving in a randomly time dependent one dimensional potential how the distribution function p lambda t approaches the limiting distribution p lambda infinity delta lambda lambda infinity our method also applies to the tail of the distribution which determines the growth rates of moments of m ij the results are also applicable to the problem of wave function localization in a disordered one dimensional potential

exact solutions for the statistics of extrema of some random 1d landscapes application to the equilibrium and the dynamics of the toy model abstract the real space renormalization group rsrg method introduced previously for the brownian landscape is generalized to obtain the joint probability distribution of the subset of the important extrema at large scales of other one dimensional landscapes for a large class of models we give exact solutions obtained either by the use of constrained path integrals in the continuum limit or by solving the rsrg equations via an ansatz which leads to the liouville equation we apply in particular our results to the toy model energy landscape which consists in a quadratic potential plus a brownian potential which describes among others the energy of a single domain wall in a 1d random field ising model rfim in the presence of a field gradient the measure of the renormalized landscape is obtained explicitly in terms of airy functions and allows to study in detail the boltzmann equilibrium of a particle at low temperature as well as its non equilibrium dynamics for the equilibrium we give results for the statistics of the absolute minimum which dominates at zero temperature and for the configurations with nearly degenerate minima which govern the thermal fluctuations at very low temperature for the dynamics we compute the distribution over samples of the equilibration time or equivalently the distribution of the largest barrier in the system we also study the properties of the rare configurations presenting an anomalously large equilibration time which govern the long time dynamics we compute the disorder averaged diffusion front which interpolates between the kesten distribution of the sinai model at short rescaled time and the reaching of equilibrium at long rescaled time finally the method allows to describe the full coarsening i e many domain walls of the 1d rfim in a field gradient as well as its equilibrium

random walks in random environments random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems especially in the non reversible situation since they began to be studied in the 1970s we review the model available results and techniques and point out several gaps in the understanding of these processes

a numerical approach to copolymers at selective interfaces we consider a model of a random copolymer at a selective interface which undergoes a localization delocalization transition in spite of the several rigorous results available for this model the theoretical characterization of the phase transition has re mained elusive and there is still no agreement about several important issues for example the behavior of the polymer near the phase transition line from a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known in this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram our main results include various numerical observations that suggest that the critical line lies strictly in between the two bounds a rigorous statistical test based on concentration inequalities and super additivity for determining whether a given point of the phase diagram is in the localized phase this is applied in particular to show that with a very low level of error the lower bound does not coincide with the critical line an analysis of the precise asymptotic behavior of the partition function in the de localized phase with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a brownian scaling a new proof of the lower bound on the critical line this proof relies on a characteri zation of the localized regime which is more appealing for interpreting the numerical data

near integrable systems a two dimensional circular quantum billiard with unusual boundary conditions introduced by berry and dennis 2008 j phys a math theor 41 135203 is considered in detail it is demonstrated that most of its eigenfunctions are strongly localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with neumann boundary conditions deviations from strong localization are also discussed these results agree well with numerical calculations

heat kernel expansion for fermionic billiards in an external magnetic field using seeley s heat kernel expansion the authors compute the asymptotic density of states of the dirac operator coupled to a magnetic field on a two dimensional manifold with boundary fermionic billiard local boundary conditions compatible with vector current conservation depend on a free parameter alpha it is shown that the perimeter correction identically vanishes for alpha 0 in that case the next order constant term is found to be proportional to the euler characteristic of the manifold these results are independent of the external magnetic field and of the shape of the billiard provided the boundary is sufficiently smooth for the flat circular billiard the constant term is found to be 1 2 in agreement with a numerical result of berry and mondragon

random walks in a random environment on a strip a renormalization group approach we present a real space renormalization group scheme for the problem of random walks in a random environment on a strip which includes the one dimensional random walk in a random environment with bounded non nearest neighbour jumps we show that the model renormalizes to an effective one dimensional random walk with nearest neighbour jumps and conclude that sinai scaling is valid in the recurrent case while in the sub linear transient phase the displacement grows as a power of the time

resonant enhancement of anderson localization analytical approach we study localization properties of the eigenstates and wave transport in a one dimensional system consisting of a set of barriers and or wells of fixed thickness and random heights the inherent peculiarity of the system resulting in the enhanced anderson localization is the presence of the resonances emerging due to the coherent interaction of the waves reflected from the interfaces between the wells and or barriers our theoretical approach allows to derive the localization length in infinite samples both out of the resonances and close to them we examine how the transport properties of finite samples can be described in terms of this length it is shown that the analytical expressions obtained by standard methods for continuous random potentials can be used in our discrete model in spite of the presence of resonances that cannot be described by conventional theories all our results are illustrated with numerical data manifesting an excellent agreement with the theory

maximum bipartite flow in networks with adaptive channel width traditionally network optimization problems assume that each link in the network has a fixed capacity recent research in wireless networking has shown that it is possible to design networks where the capacity of the links can be changed adaptively to suit the needs of specific applications in particular one gets a choice of having a few high capacity outgoing links or many low capacity ones at any node of the network this motivates us to have a re look at classical network optimization problems and design algorithms to solve them in this new framework in particular we consider the problem of maximum bipartite flow which has been studied extensively in the fixed capacity network model one of the motivations for studying this problem arises from the need to maximize the throughput of an infrastructure wireless network comprising base stations one set of vertices in the bipartition and clients the other set of vertices in the bipartition we show that this problem has a significantly different combinatorial structure in this new network model from the fixed capacity one while there are several polynomial time algorithms for the maximum bipartite flow problem in traditional networks we show that the problem is np hard in the new model in fact our proof extends to showing that the problem is apx hard we complement our lower bound by giving two algorithms for solving the problem approximately the first algorithm is deterministic and achieves an approximation factor of o logn where n is the number of nodes in the network while the second algorithm is randomized and achieves an approximation factor of ee 1

fluctuations of random matrix products and 1d dirac equation with random mass we study the fluctuations of certain random matrix products pi n m n cdots m 2m 1 of mathrm sl 2 mathbb r describing localisation properties of the one dimensional dirac equation with random mass in the continuum limit i e when matrices m n s are close to the identity matrix we obtain convenient integral representations for the variance gamma 2 lim n rightarrow infty mathrm var ln pi n n the case studied exhibits a saturation of the variance at low energy varepsilon along with a vanishing lyapunov exponent gamma 1 lim n rightarrow infty ln pi n n leading to the behaviour gamma 2 gamma 1 sim ln 1 varepsilon rightarrow infty as varepsilon rightarrow 0 our continuum description sheds new light on the kappus wegner band center anomaly

on a non linear fluctuation theorem for the ageing dynamics of disordered trap models we consider the dynamics of the disordered trap model which is known to be completely out of equilibrium and to present strong localization effects in its ageing phase we are interested in the influence of an external force when it is applied from the very beginning at t 0 or only after a waiting time tw we obtain a non linear fluctuation theorem for the corresponding one time and two time diffusion fronts in any given sample which implies the following consequences i for fixed times there exists a linear response regime where the fluctuation dissipation relation or einstein relation is valid even in the ageing time sector in contrast with other ageing disordered systems ii for a fixed waiting time and fixed external field the validity of the linear response regime is limited in time by a characteristic time depending on the external force iii in the non linear response regime there exists a very simple relation for the asymmetry in the position for the one time and two times disorder averaged diffusion fronts in the presence of an external force in contrast to other models of random walks in random media the present non linear fluctuation theorem is a consequence of a special dynamical symmetry of the trap model

localisation and universal fluctuations in ultraslow diffusion processes we study ultraslow diffusion processes with logarithmic mean squared displacement msd comparison of annealed renewal continuous time random walks ctrws with logarithmic waiting time distribution and sinai diffusion in quenched random landscapes reveals striking similarities despite the great differences in their physical nature in particular they exhibit a weakly non ergodic disparity of the time averaged and ensemble averaged msds remarkably for the ctrw we observe that the fluctuations of time averages become universal with an exponential suppression of mobile trajectories we discuss the fundamental connection between the golosov localization effect and non ergodicity in the sense of the disparity between ensemble averaged msd and time averaged msd

multiscaling of noise induced parametric instability we describe the statistical properties of growth rates of a linear oscillator driven by a parametric noise we show that in general the fluctuations of local lyapunov exponents are non gaussian and demonstrate multiscaling analytical calculations of the generalized lyapunov exponents are complemented with approximative and numerical results this allows us to identify the parameter range where the deviations from the gaussian statistics become important

localization of random walks in one dimensional random environments we consider a random walk on the one dimensional semi lattice 0 1 2 we prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment the size of this neighbourhood is estimated the limit parameters of the walks are also determined

exact asymptotic results for persistence in the sinai model with arbitrary drift we obtain exact asymptotic results for the disorder averaged persistence of a brownian particle moving in a biased sinai landscape we employ a method that maps the problem of computing the persistence to the problem of finding the energy spectrum of a single particle quantum hamiltonian which can be subsequently found our method allows us analytical access to arbitrary values of the drift bias thus going beyond the previous methods that provide results only in the limit of vanishing drift we show that on varying the drift the persistence displays a variety of rich asymptotic behaviors including in particular interesting qualitative changes at some special values of the drift

mean first passage and residence times of random walks on asymmetric disordered chains an algebraic derivation is presented which yields the exact solution of the mean first passage and mean residence times of a one dimensional asymmetric random walk for quenched disorder two models of disorder are analytically treated both absorbing absorbing and reflecting absorbing boundaries are considered particularly the interplay between asymmetry and disorder is studied

moderate deviations for diffusions with brownian potentials we present precise moderate deviation probabilities in both quenched and annealed settings for a recurrent diffusion process with a brownian potential our method relies on fine tools in stochastic calculus including kotani s lemma and lamperti s representation for exponential functionals in particular our result for quenched moderate deviations is in agreement with a recent theorem of comets and popov probab theory related fields 126 2003 571 609 who studied the corresponding problem for sinai s random walk in random environment

persistence of randomly coupled fluctuating interfaces we study the persistence properties in a simple model of two coupled interfaces characterized by heights h 1 and h 2 respectively each growing over a d dimensional substrate the first interface evolves independently of the second and can correspond to any generic growing interface e g of the edwards wilkinson or of the kardar parisi zhang variety the evolution of h 2 however is coupled to h 1 via a quenched random velocity field in the limit d ensuremath rightarrow 0 our model reduces to the matheron de marsily model in two dimensions for d 1 our model describes a rouse polymer chain in two dimensions advected by a transverse velocity field we show analytically that after a long waiting time t 0 ensuremath rightarrow ensuremath infty the stochastic process h 2 at a fixed point in space but as a function of time becomes a fractional brownian motion with a hurst exponent h 2 1 ensuremath ensuremath beta 1 2 where ensuremath beta 1 is the growth exponent characterizing the first interface the associated persistence exponent is shown to be ensuremath theta s 2 1 ensuremath h 2 ensuremath beta 1 2 these analytical results are verified by numerical simulations

on the localization transition of random copolymers near selective interfaces in this note we consider the de localization transition for random directed 1 1 dimensional copolymers in the proximity of an interface separating selective solvents we derive a rigorous lower bound on the free energy this yields a substantial improvement on the bounds from below on the critical line that were known so far our result implies that the critical curve does not lie below the critical curve set forth by monthus eur phys j b 13 111 130 2000 on the base of a renormalization group analysis we discuss this result in the light of the rigorous and non rigorous approaches present in the literature and by making an analogy with a particular asymptotics of a disordered wetting model we propose a simplified framework in which the question of identifying the critical curve as well as understanding the nature of the transition may be approached

random walk in a two dimensional self affine random potential properties of the anomalous diffusion phase at small external force we study the dynamical response to an external force f for a particle performing a random walk in a two dimensional quenched random potential of hurst exponent h 1 2 we present numerical results on the statistics of first passage times that satisfy closed backward master equations we find that there exists a zero velocity phase in a finite region of the external force 0lfl f c where the dynamics follows the anomalous diffusion law x t ensuremath sim ensuremath xi f t ensuremath mu f the anomalous exponent 0l ensuremath mu f l1 and the correlation length ensuremath xi f vary continuously with f in the limit of vanishing force f ensuremath rightarrow 0 we measure the following power laws the anomalous exponent vanishes as ensuremath mu f ensuremath propto f a with a ensuremath simeq 0 6 instead of a 1 in dimension d 1 and the correlation length diverges as ensuremath xi f ensuremath propto f ensuremath ensuremath nu with ensuremath nu ensuremath simeq 1 29 instead of ensuremath nu 2 in dimension d 1 our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model where the traps are characterized by a typical length ensuremath xi f along the direction of the force and by a typical barrier 1 ensuremath mu f the fact that these traps are smaller in linear size and in depth than in dimension d 1 means that the particle uses the transverse direction to find lower barriers

microscopic approach to random walks in the present paper the microscopic approach to random walk models is introduced for any particular model it provides a rigorous way to derive the transport equations for the macroscopic density of walking particles although it is not more complicated than the standard random walk framework it has virtually no limitations with respect to the initial distribution of particles as a consequence the transport equations derived with this method almost automatically give answers to such important problems as aging and two point probability distribution

the effect of boundaries on the spectrum of a one dimensional random mass dirac hamiltonian the average density of states dos of the one dimensional dirac hamiltonian with a random mass on a finite interval 0 l is derived our method relies on the eigenvalue distributions extreme value statistics problem which are obtained explicitly the well known dyson singularity is recovered above the crossover energy below c we find a log normal suppression of the average dos

asymptotic localization of stationary states in the nonlinear schrodinger equation the mapping of the nonlinear schrodinger equation with a random potential on the fokker planck equation is used to calculate the localization length of its stationary states the asymptotic growth rates of the moments of the wave function and its derivative for the linear schrodinger equation in a random potential are computed analytically and resummation is used to obtain the corresponding growth rate for the nonlinear schrodinger equation and the localization length of the stationary states

superaging correlation function and ergodicity breaking for brownian motion in logarithmic potentials we consider an overdamped brownian particle moving in a confining asymptotically logarithmic potential which supports a normalized boltzmann equilibrium density we derive analytical expressions for the two time correlation function and the fluctuations of the time averaged position of the particle for large but finite times we characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential and we support our predictions with extensive langevin simulations while the boltzmann measure is used to obtain stationary correlation functions we show how the non normalizable infinite covariant density is related to the superaging behavior

distribution of dynamical quantities in the contact process random walks and quantum spin chains in random environments we study the distribution of dynamical quantities in various one dimensional disordered models the critical behavior of which is described by an infinite randomness fixed point in the disordered contact process the survival probability mathcal p t is found to show multiscaling in the critical point meaning that mathcal p t t ensuremath ensuremath delta where the environment and time dependent exponent ensuremath delta has a universal limit distribution when t ensuremath rightarrow ensuremath infty the limit distribution is determined by the strong disorder renormalization group method analytically in the end point of a semi infinite lattice where it is found to be exponential while in the infinite system conjectures on its limiting behaviors for small and large ensuremath delta which are based on numerical results are formulated by the same method the survival probability in the problem of random walks in random environments is also shown to exhibit multiscaling with an exponential limit distribution in addition to this the imaginary time spin spin autocorrelation function of the random transverse field ising chain is found to have a form similar to that of survival probability of the contact process at the level of the renormalization approach consequently a relationship between the corresponding limit distributions in the two problems can be established finally the distribution of the spontaneous magnetization in this model is also discussed

stochastic aggregation scaling properties we study the scaling properties of stochastic aggregation processes in one dimension numerical simulations for both diffusive and ballistic transport show that the mass distribution is characterized by two independent non trivial exponents corresponding to the survival probability of particles and monomers the overall behaviour agrees qualitatively with the mean field theory this theory also provides a useful approximation for the decay exponents as well as the limiting mass distribution

exact results on sinai s diffusion we study the continuum version of sinai s problem of a random walker in a random force field in one dimension a method of stochastic representation is used to represent various probability distributions in this problem mean probability density function and first passage time distributions this method reproduces already known rigorous results and also confirms directly some recent results derived using approximation schemes we demonstrate clearly in the sinai scaling regime that the disorder dominates the problem and that the thermal distributions tend to zero one laws

phase diagram of the random frequency oscillator the case of ornstein uhlenbeck noise abstract we study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an ornstein uhlenbeck noise this system undergoes a noise induced bifurcation when the amplitude of the noise grows the critical curve that separates the absorbing phase from an extended non equilibrium steady state corresponds to the vanishing of the lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy we derive various expressions for this lyapunov exponent by using different approximation schemes this allows us to study quantitatively the phase diagram of the random parametric oscillator

generalized lyapunov exponents of the random harmonic oscillator cumulant expansion approach the cumulant expansion is used to estimate generalized lyapunov exponents of the random frequency harmonic oscillator three stochastic processes are considered gaussian white noise ornstein uhlenbeck and poisson shot noise in some cases nontrivial numerical difficulties arise these are mostly solved by implementing an appropriate importance sampling monte carlo scheme we analyze the relation between random frequency oscillators and many particle systems with pairwise interactions like the lennard jones gas

the limit distribution of sinai s random walk in random environment abstract recently sinai proved that if x n is a one dimensional random walk in random environment which is recurrent then log n 2 x n converges in distribution here we calculate the limit distribution explicitly

zero lyapunov exponent in the vicinity of the saddle node bifurcation point in the presence of noise we consider a behavior of the zero lyapunov exponent in the vicinity of the bifurcation point that occurs as the result of the interplay between dynamical mechanisms and random dynamics we analytically deduce the laws for the dependence of this lyapunov exponent on the control parameter both above and below the bifurcation point the developed theory is applicable both to the systems with the random force and to the deterministic chaotic oscillators we find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations we also discuss how the revealed regularities are expected to take place in other relevant physical circumstances

continuous approach for the random field ising chain we study the random field ising chain in the limit of strong exchange coupling in order to calculate the free energy we apply a continuous langevin type approach this continuous model can be solved exactly whereupon we are able to locate the crossover between an exponential and a power law decay of the free energy with increasing coupling strength in terms of magnetization this crossover restricts the validity of the linear scaling the known analytical results for the free energy are recovered in the corresponding limits the outcomes of numerical computations for the free energy are presented which confirm the results of the continuous approach we also discuss the validity of the replica method which we then utilize to investigate the sample to sample fluctuations of the finite size free energy

optimal envy free pricing with metric substitutability we study the unit demand envy free pricing problem faced by a profit maximizing seller with unlimited supply when there is metric substitutability among the items consumer i s value for item j is v i c i j and the substitution costs c i j form a metric our model is motivated by the observation that sellers often sell the same product at different prices in different locations and rational consumers optimize the tradeoff between prices and substitution costs while the general envy free pricing problem is hard to approximate we show that the problem of maximizing revenue with metric substitutability among items can be solved exactly in polynomial time we do this by first showing that in any optimal price vector the set of nodes that pay exactly their value uniquely determines which nodes buy an item and what price they pay and therefore the revenue we transform the problem of finding an optimal set of such nodes to an instance of weighted independent set on a perfect graph which can be solved in polynomial time by the strong perfect graph theorem proving the result we then analyze the computational tractability of various extensions to our model we begin with relaxing the metric substitutability requirement and show that when the substitution costs do not form a metric even if a 1 epsilon approximate triangle inequality holds the problem becomes np hard thus the triangle inequality characterizes the threshold at which the problem goes from tractable to hard we then relax assumptions on the supply and demand we consider restricting supplies to a subset of locations or the amount of supplies or allowing buyers to demand more than one unit in all cases the problem becomes np hard in addition the multiunit demand case illustrates an interesting paradoxical nonmonotonicity the optimal revenue the seller can extract can actually decrease when consumers demands increase we show the revenue maximization problem with multiunit demand is apx hard even for the simplest valuations with equal marginal values for all items up to the demand constraint and demands of at most 3

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

breaking supersymmetry in a one dimensional random hamiltonian the one dimensional supersymmetric random hamiltonian where x is a gaussian white noise of zero mean and variance g presents particular spectral and localization properties at low energy a dyson singularity in the integrated density of states idos n e 1 ln2e and a delocalization transition related to the behavior of the lyapunov exponent inverse localization length vanishing like e 1 ln e as e 0 we study how this picture is affected by breaking supersymmetry with a scalar random potential h hsusy v x where v x is a gaussian white noise of variance in the limit g3 a fraction of states n 0 g ln2 g3 migrate to the negative spectrum and the lyapunov exponent reaches a finite value 0 g ln g3 at e 0 the exponential lifshits tail of the idos for e is studied in detail and is shown to involve a competition between the two noises x and v x whatever the larger is this analysis relies on analytic results for n e and e obtained by two different methods a stochastic method and the replica method the problem of extreme value statistics of eigenvalues is also considered distribution of the nth excited state energy the results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation creation local rates

individual energy level distributions for one dimensional diagonal and off diagonal disorder we study the distribution of the n th energy level for two different one dimensional random potentials this distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function we first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum it is demonstrated that in the limit of a large system l the distribution of the n th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables in the second part we consider the case of a supersymmetric random hamiltonian potential v x x 2 x we study first the case of x being a white noise with zero mean it is in particular shown that the ground state energy which behaves on average like exp l 1 3 in agreement with previous work is not a self averaging quantity in the limit l as is seen in the case of diagonal disorder then we consider the case when x has a non zero mean value

random walk in a one dimensional random medium abstract the random walk of a particle in a one dimensional random medium is examined by means of the equivalent transfer rates technique in the discrete as well as in the continuous version of the model the probability distributions of the energy dependent equivalent transfer rates are found analytically either by a matching procedure in the discrete case or exactly in the continuous model both discrete and continuous models are shown to belong to the same universality class the average probability of presence of the particle at its initial point is then computed for a non zero global bias it decreases at large times according to a negative power law with an exponent depending on disorder and bias when there is no global bias sinai s model the decay at large times follows a logarithmic law

random walkers in one dimensional random environments exact renormalization group analysis sinai s model of diffusion in one dimension with random local bias is studied by a real space renormalization group which yields exact results at long times the effects of an additional small uniform bias force are also studied we obtain analytically the scaling form of the distribution of the position x t of a particle the probability of it not returning to the origin and the distributions of first passage times in an infinite sample as well as in the presence of a boundary and in a finite but large sample we compute the distribution of the meeting time of two particles in the same environment we also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward these quantities obey multifractal scaling characterized by generalized persistence exponents ensuremath theta g which we compute in the presence of a small bias the number of returns to the origin becomes finite characterized by a universal scaling function which we obtain the full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained as well as detailed analytical information about correlations between directions and times of successive jumps the two time distribution of the positions of a particle x t and x t ensuremath with tg t ensuremath is also computed exactly it is found to exhibit aging with several time regimes characterized by different behaviors in the unbiased case for t ensuremath t ensuremath ensuremath sim t ensuremath ensuremath alpha with ensuremath alpha g1 it exhibits a mathrm ln t mathrm ln t ensuremath scaling with a singularity at coinciding rescaled positions x t x t ensuremath this singularity is a novel feature and corresponds to particles that remain in a renormalized valley for closer times ensuremath alpha l1 the two time diffusion front exhibits a quasiequilibrium regime with a mathrm ln t ensuremath t ensuremath mathrm ln t ensuremath behavior which we compute the crossover to a t t ensuremath aging form in the presence of a small bias is also obtained analytically rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail connections with the green function of a one dimensional schr odinger problem and quantum spin chains are discussed

spectral analysis of sinai s walk for small eigenvalues sinai s walk can be thought of as a random walk on z with random potential v with v weakly converging under diffusive rescaling to a two sided brownian motion we consider here the generator l n of sinai s walk on n n o z with dirichlet conditions on n n by means of potential theory for each h 0 we show the relation between the spectral properties of l n for eigenvalues of order o exp h n and the distribution of the h extrema of the rescaled potential v n x v nx n defined on 1 1 information about the h extrema of v n is derived from a result of neveu and pitman concerning the statistics of h extrema of brownian motion as first application of our results we give a proof of a refined version of sinai s localization theorem

random walks and branching processes in correlated gaussian environment we study persistence probabilities for random walks in correlated gaussian random environment first studied by oshanin rosso and schehr from the persistence results we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters more precisely we obtain estimates on the tail distribution of its total population size of its maximum population and of its extinction time