probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

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

structure and perimeters of percolation clusters site percolation clusters are simulated at the percolation threshold on the square lattice an algorithm for walks around each cluster is used to obtain information on its fractal geometry the fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it if the vacant perimeter sites have only nearest neighbour connectivity then the perimeter has dimension de 1 37 or 0 03 instead of that of the hull dh 1 75 the authors also measured the mass of internal dangling sites surrounded by the backbone s blobs singly and doubly connected sites the number of blobs and the average linear distance between entry into the exit from a blob

the dynamics of some discrete population models in this paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation a ii 1 aa f a e converge as n 00 in e a is a constant real number with 0 0 sufficiently small thus formally equation e comes from x t h x t h yx t d x t r for small h if we set z f x ht then the preceding equation for t n and r h m becomes yh 1 1 h y n hd y n m which is of the form e where a 1 yh and f u hd u u 2 0 clearly h 0 is small if and only if a e 0 1 1069 h

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

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

moment of inertia of doubly connecting bonds in two dimensional bond percolation by applying a theorem from conformal field theory to bond percolation in two dimensions the author obtains an equation for the second moment i of bonds which doubly connect a percolation cluster i a 5 square root 3 27 pi 2 p pc 2 the equation is exact as the occupancy probability p approaches the critical probability pc from either side a is a calculable lattice dependent number which for a square lattice equals 1 2

a regular model for cluster numbers and structure just below percolation threshold a two dimensional deterministic fractal model is proposed to imitate the geometric texture just below percolation threshold the model is constructed for bond percolation via a rule of bond occupation on the square lattice it show the typical percolation behaviour as a function of a parameter p p is the bond concentration the model describes the approach towards the threshold below criticality the critical bond concentration pc and the correlation length exponent nu are found the scaling property of the cluster size distribution is studied below the threshold the critical concentration and critical exponents are shown to agree with those derived from the regular model proposed for the geometric texture above the threshold the scaling relations for the critical exponents are shown to be satisfied

exact enumeration of self avoiding walks on critical percolation clusters in 2 7 dimensions we study self avoiding walks on critical percolation clusters by means of a recently developed exact enumeration method which can handle walks of several thousand steps we had previously presented results for the two and three dimensional cases here we take a wider perspective and vary the system s dimensions up to d 7 beyond the supposed upper critical dimension of these results may serve to check analytical predictions and help understand how the medium s fractal structure impacts on the walks scaling behavior

a regular model for bond percolation the approach towards the threshold a two dimensional regular model is constructed for bond percolation via a rule of bond deletions on the square lattice the model describes the approach towards the threshold the critical bond concentration pc and the correlation length exponent nu are found the fractal geometry of the infinite percolating network and the scaling property of the cluster size distribution are studied above the threshold the infinite percolating network is shown to be reconstructed by the mandelbrot koch curve proposed as the non random model for the infinite cluster at criticality

on stabilization of equilibria using predictive control with and without pulses consider a chaotic difference equation x n 1 f x n we focus on the problem of control of chaos using a prediction based control pbc method if f has a unique positive equilibrium it is proved that global stabilization of this equilibrium can be achieved under mild assumptions on the map f if f has several positive equilibria we demonstrate that more than one equilibrium can be stabilized simultaneously we also show that it is still possible to stabilize an unstable equilibrium using a strategy of control with pulses that is the control is only applied after a fixed number of iterations we illustrate our main results with several examples mainly from population dynamics

percolation and spreading of damage in a simplified kauffman model abstract a simplified version of the kauffman cellular automaton is introduced as in the usual kauffman model there is a transition between a frozen phase and a chaotic phase where damage may spread we associate the onset of chaos in this model with a percolation transition of certain rules occurring in the model it seems to be in a different universality class from the usual kauffman cellular automaton

a note on coniglio s lemma it is proved that coniglio s 1981 1982 lemma holds for any pair of points in any graph in the low density series expansion of the cut edge weighted pair connectedness

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

position space renormalisation group approach to fractal dimensions of an infinite cluster its backbone and cutting bonds for percolation a position space renormalisation group method is presented to study the cluster structure of an infinite cluster at the percolation threshold fractal dimensions of an infinite cluster its backbone and cutting bonds are determined by the small cell method for two dimensional bond percolation it is shown that the fractal dimension of its cutting bonds agrees with the inverse of the connectedness length exponent at the fixed point

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

droplet structure in ising and potts models the structure of a recently introduced droplet in the q state potts model is analysed the authors derive exact relations from which it follows that the incipient infinite droplet at the critical temperature is a self similar fractal made of links and blobs just as recently found in random percolation the number of links nlinks between two points separated by a distance of the order b is given by nlinks varies as b1 nu b q where nu b q is the connectedness length exponent when t tc q and pb is used as an independent variable this result and the available calculations of nu b q indicate that the number of links decreases as q increases an analysis of the structure of the usual clusters made of nearest neighbour sites in a given potts site configuration is also made in particular it shows that for d 2 the incipient infinite cluster is made only of blobs and no links

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

techniques for the control of chaos abstract the concepts of chaos and its control are reviewed both are discussed from an experimental as well as a theoretical viewpoint examples are then given of the control of chaos in adiverse set of experimental systems current and future applications are discussed

anomalous acoustic behaviour and backbone structure of percolation clusters a real space renormalisation group method is used to investigate acoustic properties of percolation lattices in the vicinity of the percolation threshold scaling form an expressions for critical exponents x y describing the sound velocity are derived the existence of an anomalous dispersion relation is elucidated dimensionalities dt and dl which represent the geometrical structure of cluster backbones are introduced the exponents x y and the dimensionalities dt dl are expressed in terms of other exponents nu beta t s and beta b for percolation and their explicit values are evaluated from known estimates for nu beta t s and beta b a geometrical picture of the backbone structure is obtained in terms of the dimensionalities

stabilisation of difference equations with noisy prediction based control abstract we consider the influence of stochastic perturbations on stability of a unique positive equilibrium of a difference equation subject to prediction based control these perturbations may be multiplicative x n 1 f x n l n 1 f x n x n n 0 1 if they arise from stochastic variation of the control parameter or additive x n 1 f x n f x n x n l n 1 n 0 1 if they reflect the presence of systemic noise we begin by relaxing the control parameter in the deterministic equation and deriving a range of values for the parameter over which all solutions eventually enter an invariant interval then by allowing the variation to be stochastic we derive sufficient conditions less restrictive than known ones for the unperturbed equation under which the positive equilibrium will be globally a s asymptotically stable i e the presence of noise improves the known effectiveness of prediction based control finally we show that systemic noise has a blurring effect on the positive equilibrium which can be made arbitrarily small by controlling the noise intensity numerical examples illustrate our results

explicit invariant measures for products of random matrices we construct explicit invariant measures for a family of infinite products of random independent identically distributed elements of sl 2 c the matrices in the product are such that one entry is gamma distributed along a ray in the complex plane when the ray is the positive real axis the products are those associated with a continued fraction studied by letac seshadri z wahr verw geb 62 1983 485 489 who showed that the distribution of the continued fraction is a generalised inverse gaussian we extend this result by finding the distribution for an arbitrary ray in the complex right half plane and thus compute the corresponding lyapunov exponent explicitly when the ray lies on the imaginary axis the matrices in the infinite product coincide with the transfer matrices associated with a one dimensional discrete schrodinger operator with a random gamma distributed potential hence the explicit knowledge of the lyapunov exponent may be used to estimate the exponential rate of localisation of the eigenstates

cumulant renormalisation group and its application to the incipient infinite cluster in percolation the authors introduce the cumulant method into the context of position space renormalisation group the advantage of this method over previous procedures is that one need not systematically evaluate all 2n states of each kadanoff cell where n is the number of elements sites or bonds in the cell they illustrate this method by calculating for bond percolation with an eight bond cell the critical exponents characterizing the following quantities i the mean number of backbone bonds ii the mean number of bonds in the minimum and maximum paths along the backbone iii the mean number of singly connected red bonds and iv the mean number of bonds in all the self avoiding walks connecting the two extreme points they derive previously obtained results of the conductivity and resistivity

the starry sky model of the percolation phase transition iii a new equilibrium equation and two kinds of the special elastic nodes abstract the equilibrium lame equation in two dimensional and three dimensional spaces can be presented as conductivity in magnetic field equation with lorenz force expression this allows us to apply the perturbation theory and generalize the lame equation and obtain a new equilibrium equation with all responses of compression and tension it means that compression in one direction leads to tension in the perpendicular direction and vice versa if the boundaries are fixed the solution in recurrent integral form for contributions to all orders of poisson ratio is then obtained zero order solution of lame equilibrium equation expresses the poisson ratio as ve p pc k k 0 3 and creates the special elastic nodes with maximum local deformation the first order solution gives mean values of effective lateral displacements or forces when lateral boundaries are fixed and expresses the bulk modulus as ke p pc 2t g where g 0 6 is a hall coefficient critical exponent it creates the special active elastic nodes where the whole sample microrotates around a new universality class of elasticity problem belongs to the hall coefficient universality class therefore the elasticity backbone coincides with the hall coefficient backbone with the fractal dimension d f b 2 25

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

stabilization of cycles for difference equations with a noisy pf control difference equations such as a ricker map for an increased value of the parameter experience instability of the positive equilibrium and transition to deterministic chaos to achieve stabilization various methods can be applied proportional feedback control suggests a proportional reduction of the state variable at every kth step first if k 1 a cycle is stabilized rather than an equilibrium second the equation can incorporate an additive noise term describing the variability of the environment as well as multiplicative noise corresponding to possible deviations in the control intensity the present paper deals with both issues it justifies a possibility of getting a stable blurred k cycle presented examples include the ricker model as well as equations with unbounded f such as the bobwhite quail population models though the theoretical results justify stabilization for either multiplicative or additive noise only numerical simulations illustrate that a blurred cycle can be stabilized when both multiplicative and additive noises are involved

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

pbc based pulse stabilization of periodic orbits abstract we investigate prediction based schemes to stabilize periodic solutions to potentially chaotic systems of periodic difference equations using pulses at times being a multiple of the period by introducing the concept of a stability domain we obtain precise information on the possibility to stabilize given solutions to avoid destabilizing bifurcations as well as on the magnitude of the required control

a new percolation statistic with unusual properties in bond percolation on lattice with p pc there are unoccupied bonds which if occupied would immediately join the backbone in general when such a bond is occupied several tag end bonds may join the backbone as well the author defines ni to be the number of bonds that would join the backbone if such a bond i were occupied and n to be the average of ni taken over all such bonds on the lattice the author derives a formula for n and finds that for all lattices near pc n gamma bpc p pc 1 where gamma b is the exponent for the divergence of the backbone fraction identical formulae apply to site percolation

logarithmic corrections for the percolative properties of the four state potts model clusters in the potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state droplets can be obtained by adding bonds with probability p 1 exp k between the sites in a cluster where k is the potts model s inverse temperature when q to 4 renormalization group rg fixed points describing clusters and droplets will coalesce leading to logarithmic corrections we calculate the precise form of these corrections used in a differential rg method our predictions are then tested using extensive monte carlo calculations theory and the simulations are found to be in excellent agreement

segment distribution in self similar patterns and backbone structure of percolation clusters a distribution function of segments in self similar patterns is investigated and determined from the fractal dimensionality d of the pattern and that d of the segments on the basis of scaling assumptions the critical behaviour of the segment distribution is studied critical exponents are introduced and expressed in terms of d and d the general theory is applied to link distribution in the backbone of an infinite percolation cluster critical exponents for link distribution are also introduced and evaluated explicitly

topological proof for the alexander orbach conjecture this paper advocates an unconventional analytical approach to studying the fractal geometry of percolation at the threshold which is based on the most general methods of the differential topology our particular interest concentrates on the alexander orbach ao conjecture j phys france lett 43 l625 1982 which assigns the universal mean field value 4 3 to the spectral fractal dimension stackrel ifmmode tilde else fi d at the percolation threshold for all embedding euclidean dimensions n greater than one i e ng 2 using the topological arguments we show that the ao conjecture might be improved for the relatively low embedding dimensions 2l nl 5 for which the analytical result stackrel ifmmode tilde else fi d 1 327 ifmmode pm else textpm fi 0 001 is proposed instead of the original ao estimate 4 3 meanwhile we assume that the exact value stackrel ifmmode tilde else fi d 4 3 holds for all ng 6 as it follows directly from the well known mean field theory the improved value of stackrel ifmmode tilde else fi d ensuremath approx 1 327 for 2l nl 5 is obtained from an analysis of the basic topological properties of the percolating fractal sets at the threshold of percolation we show that these properties could be investigated fruitfully with the introduction of the concept of the fractal manifold which might serve as an effective instrument when considering the topology of the fractal objects our results indicate that the proposed value of stackrel ifmmode tilde else fi d ensuremath approx 1 327 for the spectral dimension at the percolation threshold has the fundamental topological origin related to the most general features of the fractal geometry of percolation at criticality we argue that the constraint 2l nl 5 on the topological dimension n of the embedding euclidean space is the direct consequence of the famous whitney theorem which establishes the embedding properties of manifolds from the viewpoint of their dimensionality a simple topological condition that identifies the threshold of percolation is obtained for 2l nl 5 the particular topological restrictions implied throughout the present study are discussed and the important issue of contractibility of the fractal manifolds is pointed out

a new class of point interactions in one dimension we present a class of self adjoint extensions of the symmetric operator c 0 r1 0 which correspond formally to perturbations of the laplacian by pseudopotentials involving 2 these operators which provide new examples of generalized point interactions in the sense of seba are defined by the boundary conditions 0 e z 0 r 0 0 ez r 0 0 for z c r r we calculate their spectra resolvents and scattering matrices and show that they can be realized as limits of schrodinger operators with local short range potentials

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

transition between strong and weak disorder regimes for the optimal path abstract we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path l opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight i exp ar i where r i is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finite a there is a crossover network size n a such that for n n a the scaling behavior of l opt is in the strong disorder regime while for n n a the scaling behavior is in the weak disorder regime we derive the scaling relation between n a and a with the help of simulations and also present an analytic derivation of the relation

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

self avoiding walks on diluted lattices near the percolation threshold using the node link picture of the infinite cluster near the percolation threshold a crossover of the critical behaviour of the self avoiding walks on the random lattice near the percolation threshold or just above threshold is suggested and a scaling function of this crossover is proposed the author compares the result of this qualitative argument with a monte carlo simulation result

self organized critical scaling at surfaces at dissipative boundaries models of self organized criticality show peculiar scalings different from the bulk ones in the distributions characterizing avalanches for abelian models with dirichlet boundary conditions evidence of this is obtained by a mean field approach to semi infinite sandpiles and by numerical simulations in two and three dimensions on the other hand within the mean field description closed neumann conditions restore bulk scaling exponents also at the border numerical results are consistent with this property also at finite d

branching interfaces with infinitely strong couplings a hierarchical froth model of the interface of a random q state potts ferromagnet in 2d is studied by recursive methods a fraction p of the nearest neighbour bonds is made inaccessible to domain walls by infinitely strong ferromagnetic couplings energetic and geometric scaling properties of the interface are controlled by zero temperature fixed distributions for the directed percolation threshold the interface behaves as for p 0 and scaling supports random ising q 2 critical behaviour for all q s at three regimes are obtained for different rates of ferro versus antiferromagnetic couplings with rates above a threshold value the interface is linear fractal dimension and its energy fluctuations scale with length as with when the threshold is reached the interface branches at all scales and is fractal with thus at dilution modifies both low temperature interfacial properties and critical scaling below threshold the interface becomes a probe of the backbone geometry backbone fractal dimension which even controls energy fluctuations numerical determinations of directed percolation exponents on diamond hierarchical lattice are also presented

field theoretic approaches to biconnectedness in percolating systems two field theoretic formulations for the percolation problem are presented from which the critical exponents describing the backbone of the infinite cluster at the percolation threshold are obtained at high spatial dimension d the order parameter exponent for the backbone beta 2 is given by beta 2 2 beta psi 2 nu where beta is the critical exponent for the density of the infinite cluster and psi 2 is a new crossover exponent in mean field theory psi 2 0 and for d 6 epsilon psi 2 2 epsilon 2 49 o epsilon 3 presumably psi 2 d is a smooth function of d for d d where numerical and theoretical work indicates that d is about 3 the result indicates that the fractal dimensionality of the backbone is given in terms of the percolation exponents as gamma nu psi 2

high field vs low field magneto transport in a percolating medium recent theoretical and numerical work on high field magneto transport in a percolating medium is described and compared to earlier work on weak field magneto transport in such systems while the weak field behavior is well described by the simple nodes links picture which ignores blobs and loops on a scale smaller than the percolation correlation length p the strong field behavior is extremely sensitive to those features the critical behavior at strong magnetic fields h near the percolation threshold is governed by competition between the usual h 0 fixed point and a new h fixed point which of those fixed points dominates the behavior is determined by the relative sizes of two characteristic lenghts the percolation correlation length p and a new magnetic field dependent length h

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

control of unstable steady states by time delayed feedback methods we show that time delayed feedback methods which have successfully been used to control unstable periodic orbits provide a tool to stabilize unstable steady states we present an analytical investigation of the feedback scheme using the lambert function and discuss effects of both a low pass filter included in the control loop and nonzero latency times associated with the generation and injection of the feedback signal

series expansion analysis of the backbone properties of two dimensional percolation clusters low density series expansions for the backbone properties of two dimensional bond percolation clusters are derived and analysed expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice earlier series work was confined to three properties of the square lattice and was to order the fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest monte carlo results the union intersection and length of the longest self avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for on the other hand the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection the triangular and square lattices are found to have sigificantly different dimensions the fractal dimension of the shortest path also appears to be non universal and we find for the square lattice and for the triangular lattice critical amplitude ratios are considered and found to be in agreement with theoretical inequalities

possibly exact fractal dimensions from conformal invariance based on an equivalence between the q state potts model and conformally invariant field theories the author speculates that all fractal dimensionalities of two dimensional percolation are given by the formula d 100 x2 48 where x is an integer the same formula should also apply to self avoiding walks and lattice animals this approach is extended to cluster weighted percolation his conjecture seems to agree with previous exact and numerical results except for the backbone exponent and possibly for lattice animals

exponent inequalities for the bulk conductivity of a hierarchical model the bulk conductivity p of the bond lattice in d is considered where the bonds have conductivity 1 with probabilityp or e 0 with probability 1 p various representations of the derivatives of p are developed these representations are used to analyze the behavior of p for e 0 near the percolation thresholdp c when the conducting backbone is assumed to have a hierarchical node link blob nlb structure this model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds exact asymptotics of frac d 2 sigma dp 2 for the nlb model are proven under some technical assumptions the proof employs a novel technique whereby frac d 2 sigma dp 2 for the nlb model with e 0 andp nearp c is computed using perturbation theory for p for two and three component resistor lattices aroundp 1 with a sequence of e s converging to 1 as one goes deeper in the hierarchy these asymptotics establish convexity of p for the nlb model nearp c and that its critical exponentt obeys the inequalities 1 t 2 ford 2 3 while 2 t 3 ford 4 the upper boundt 2 ind 3 which is realizable in the nlb class virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model

transport on percolation clusters with power law distributed bond strengths the simplest transport problem namely finding the maximum flow of current or maxflow is investigated on critical percolation clusters in two and three dimensions using a combination of extremal statistics arguments and exact numerical computations for power law distributed bond strengths of the type p ensuremath sigma ensuremath sim ensuremath sigma ensuremath ensuremath alpha assuming that only cutting bonds determine the flow the maxflow critical exponent v is found to be v ensuremath alpha d ensuremath 1 ensuremath nu 1 1 ensuremath ensuremath alpha this prediction is confirmed with excellent accuracy using large scale numerical simulation in two and three dimensions however in the region of anomalous bond capacity distributions 0l ensuremath alpha l 1 we demonstrate that due to cluster structure fluctuations it is not the cutting bonds but the blobs that set the transport properties of the backbone this blob dominance avoids a crossover to a regime where structural details the distribution of the number of red or cutting bonds would set the scaling the restored scaling exponents however still follow the simplistic red bond estimate this is argued to be due to the existence of a hierarchy of so called minimum cut configurations for which cutting bonds form the lowest level and whose transport properties scale all in the same way we point out the relevance of our findings to other scalar transport problems i e conductivity

regular random fractals and the n parameter model two models of fractal objects are presented the first model represents a new class of fractals which is intermediate to statistical fractals and exact fractals in that each configuration of the random ensemble has an exact and adjustable fractal dimension the second model has a continuously adjustable backbone dimensions and can be used to directly generate conducting backbones

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

a monte carlo renormalisation approach to fractal dimensions of infinite cluster backbone and cutting bonds for percolation a monte carlo renormalisation group method is presented to study the cluster structure of an infinite cluster at the percolation threshold applying this technique to bond percolation on the square lattice fractal dimensions of an infinite cluster its backbone and cutting bonds are calculated this method converges very rapidly with an increase in the size of the cell the author estimates that the fractal dimension of an infinite cluster d 1 88 or 0 02 the fractal dimension of its backbone db 1 62 or 0 02 and the fractal dimension of its cutting bonds dc 0 75 or 0 01 up to the scale factor b 10 in excellent agreement with the large cell monte carlo simulation result

a new approach to the problem of disordered harmonic chains abstract in the problem of harmonic chains with random masses the exponential growth rate and the spectral distribution function are shown to be the real and imaginary part of dyson s characteristic function along its cut in the complex plane a new function satisfying a certain integral equation is introduced from which the characteristic function follows immediately the functions introduced by dyson schmidt and matsuda and ishii are recovered as special cases and it is shown that schmidt s distributions w n converge as n relations between dyson s and schmidt s functions are obtained finally the same method is applied to the problem of harmonic chains with random variables k n m n and k n 1 m n and to the tight binding electron model with random site energies

exact exponent relations for random resistor diode networks investigates the percolation properties of a random network of non directed bonds resistors and arbitrarily oriented directed bonds diodes for the square lattice there is a multicritical line which connects the isotropic percolation threshold with a network fully occupied by randomly oriented diodes random manhattan along this line symmetry and invariance properties are used to demonstrate that the correlation length exponents are constant furthermore a lattice independent relation is proved which shows that at the isotropic percolation threshold the correlation length diverges with the same exponent when the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes

scaling of optimal path lengths distribution in complex networks we study the distribution of optimal path lengths in random graphs with random weights associated with each link disorder with each link i we associate a weight i expari where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we suggest in an analogy with the average length of the optimal path that the distribution of optimal path lengths has a universal form that is controlled by the expression 1 pc a where is the optimal path length in strong disorder a and pc is the percolation threshold this relation is supported by numerical simulations for erd s renyi and scale free graphs we explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network

a fractal model for band structures just above percolation threshold a two dimensional regular fractal model is proposed to imitate the geometric texture of percolating networks just above percolation threshold the system is self similar fractal on length scales smaller than the connectedness length but becomes homogeneous non fractal square lattice on larger length scales the scaling of vibrational density of states and band structures are studied by using exact renormalisation group methods it is shown that an infinite number of narrow bands appear to have self similar structures of typical cantor sets at low frequency

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

spreading and backbone dimensions of 2d percolation the author presents results of high statistics simulations of the spreading of 2d percolation and of backbones of 2d percolation clusters while the algorithm employed for the spreading is more or less standard for identifying backbones the author uses a new and very efficient recursive algorithm the author uses new critical exponents to test previous conjectures

fractals in two dimensions and conformal invariance abstract the exact fractal properties of critical geometrical systems in two dimensions are briefly reviewed they are generically described by an infinite discrete spectrum of scaling dimensions associated with underlying conformal operators applications involve polymers percolation brownian motion and ising clusters a seemingly essential distinction between critical phenomena spectra and multifractal ones is given

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

the fractal dimension of cluster perimeters generated by a kinetic walk the fractal dimension dg of cluster perimeters generated by a recently proposed butterfly growth walk is considered in the long range limit of the walk on a percolation cluster dg appears to be equal to the fractal dimension of the singly connected bonds dg 1 nu the new relation for chemical dimension dl is proposed dl df df dg in the short range limit the butterfly walk on a euclidean lattice appears to be in the same universality class as a random walk the dynamic aspect of the growth walk is discussed and the continuously tunable spectral dimension is obtained both short and long range limits of this diffusion process are different from a random diffusion on percolation

a connection between linear and nonlinear resistor networks the authors explore the connection between the higher moments of the current or voltage distribution in a random linear resistor network and the resistance of a nonlinear random resistor network they find that the two problems are very similar and that an infinite set of exponents are required to fully characterise each problem these exponent sets are shown to be identical on a particular hierarchical lattice a simple model which accurately describes the geometrical properties of the backbone of the infinite cluster at the percolation threshold and also the voltage distribution on this structure

a self avoiding walk on random strips to describe a polymer in a random medium the author considers a self avoiding walk saw on a lattice whose bonds are randomly favourable with probability 1 p or unfavourable with probability p the size of a saw of n steps is calculated when the lattice is a strip by performing products of random matrices when the weight of unfavourable bonds tends to 0 there exist critical concentrations pc where the size of the saw undergoes first order transitions the origin of these transitions is explained as well as the fact that the limits p to 0 or p to 1 are discontinuous possible consequences for higher dimensional systems are discussed

the resistive susceptibility of random diode insulator networks the authors consider regular crystal lattices in which the bonds are either diodes probability p or insulators the mean number of backbone bonds lbb p when averaged over lattice points is found to be related to the mean size of clusters s p by lbb p zps p where z is the number of bonds directed away from any lattice site thus lbb diverges at pc with the mean size exponent gamma the resistive susceptibility chi r p of harris and fisch 1977 1978 is expanded graphically as a power series in p and it is found that term by term chi r p psi r p s p 2 where psi r p is obtained from chi r p by ignoring contributions from nodal graphs the above results are valid for bond and site dilution in any dimension for bond dilution on the square lattice the authors have determined psi r p to p17 and pade analysis of the resulting series for chi r p shows that it diverges with exponent gamma r 3 654 or 0 017 using a scaling relation the exponent t for the conductivity of the infinite cluster is estimated to be t 0 75 or 0 02

a transfer matrix for the backbone exponent of two dimensional percolation rephrasing the backbone of two dimensional percolation as a monochromatic path crossing problem we investigate the latter by a transfer matrix approach conformal invariance links the backbone dimension db to the highest eigenvalue of the transfer matrix t and we obtain the result db 1 6431 0 0006 for a strip of width l t is roughly of size 23l but we manage to reduce it to l we find that the value of db is stable with respect to inclusion of additional blobs tangent to the backbone in a finite number of points

control of chaos by oscillating feedback parametric feedback control of chaos relies on detailed knowledge of the locations of unstable periodic orbits we show that unstable periodic orbits of dynamical systems with unknown locations but known periodicity ensuremath tau can be stabilized by an oscillating feedback term proportional to ensuremath varepsilon t x ensuremath rightarrow t ensuremath x ensuremath rightarrow t ensuremath ensuremath tau where x ensuremath rightarrow t is the location of the trajectory at time t and ensuremath varepsilon t is periodic in time periodic feedback overcomes the limitations of giona s theorem nonlinearity 4 911 1991 which states that constant feedback i e a time independent ensuremath varepsilon can stabilize an unstable periodic orbit only if the stability matrix has no positive eigenvalues greater than unity as an application of oscillating feedback we use it to stabilize the memory patterns in an associative memory hopfield proc natl acad sci usa 79 2554 1982 81 3088 1984 network thereby enhancing the total capacity of the memory device we extend our method to high dimensional systems described by differential equations in this framework it is possible to stabilize the spatiotemporal chaos generated by the kuramoto sivashinsky equation g j sivashinsky and d m michelson prog theor phys 63 2122 1980

scaling theory for the statistics of self avoiding walks on random lattices the authors study self avoiding walks saw on randomly diluted quenched lattices with direct configurational averaging over the moments of the saw distribution function a scaling function representation of rn the average end to end distance of n step walks is studied here both for saw on a the infinite percolation cluster and b any cluster they have shown that at the percolation threshold nu p nu p 1 beta p 2 nu p where beta p and nu p are the percolation order parameter and correlation length exponents respectively the authors also propose a scaling function representation for the total number of n step saw configuration gn approximately mu nngamma 1 for infinite cluster averaging which gives gamma p gamma d nu p nu for all cluster averaging gamma will remain unchanged

maximal lyapunov exponents for random matrix products in this article we study the lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions this leads to new explicit formulae for the lyapunov exponents and to an efficient method for their computation

invariant radon measures for horocycle flows on abelian covers we classify the ergodic invariant radon measures for horocycle flows on d covers of compact riemannian surfaces of negative curvature thus proving a conjecture of m babillot and f ledrappier an important tool is a result in the ergodic theory of equivalence relations concerning the reduction of the range of a cocycle by the addition of a coboundary