random walks on hierarchical lattices at the percolation threshold random walks on hierarchical lattices are considered at the percolation threshold real space renormalisation methods are employed to obtain the exact fractal dimension of the incipient infinite cluster iic df and the fractal dimension of the effective one dimensional resistance length of the iic dr for some hierarchical lattices the recent conjecture of aharony and stauffer does not hold for diamond and wheatstone bridge hierarchical lattices

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

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

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

rates of clustering for some gaussian self similar processes the analogue of strassen s functional law of the iterated logarithm in known for many gaussian processes which have suitable scaling properties and here we establish rates at which this convergence takes place we provide a new proof of the best upper bound for the convergence tok by suitably normalized brownian motion and then continue with this method to get similar bounds for the brownian sheet and other self similar gaussian processes the previous method which produced these results for brownian motion in 1 was highly dependent on many special properties unavailable when dealing with other gaussian processes

levy area of wiener processes in banach spaces the goal of this paper is to construct canonical levy area processes for banach space valued brownian motions via dyadic approximations the significance of the existence of canonical levy area processes is that a stochastic integration theory can be established for such brownian motions in banach spaces existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of lyons and lyons and qian see e g system control and rough paths 2000 oxford univ press this investigation involves a careful analysis on the choice of tensor norms motivated by the applications to infinite dimensional stochastic differential equations

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

longest path in percolating hierarchical lattice we determine numerically the probability distribution for the longest self avoiding path lengths connecting two distant points on a diluted hierarchical lattice at the percolation threshold the evolution of this distribution with the system size is studied and the distribution is observed to approach a universal scale invariant form under proper rescaling of its argument the longest path length scales as p max and our estimate for max 1 816 0 013 is clearly different from the previously estimated min 1 531 0 002 for the shortest path lengths on the same hierarchical lattice this gives support to the multifractal behavior of saws on percolating clusters

flow of diffeomorphisms induced by a geometric multiplicative functional in this paper it is shown that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional consitutes a flow of local diffeomorphisms in the case where the driving geometric multiplicative functional is generated by a brownian motion the result in particular presents an answer to an open problem proposed in ikeda and watanabe 4

laplace s method for the laws of heat processes on loop spaces abstract in this paper we will consider laplace s method for a class of heat processes on loop spaces we will obtain the first term of the asymptotics under assumptions that the function under consideration attains its minimum at a unique point and that the hessian at the point is non degenerate this kind of process was first introduced by p malliavin in 1990 for the loop group case and then gradually generalized by various authors our tool is the rough path theory of t lyons this technique was pioneered by s aida for finite dimensional processes in his unpublished paper

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space abstract we study a semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space key results are semiboundedness theorem of the schrodinger operator laplace type asymptotic formula and ims localization formula we also make a remark on the semiclassical problem of a schrodinger operator on a path space over a riemannian manifold

spreading of percolation in three and four dimensions the authors present monte carlo simulations of the spreading of bond percolation on three and four dimensional simple hyper cubic lattices the algorithm is the same as that applied previously in two dimensions with the spreading proceeding from a hyperplane they find the spreading dimension to be d 1 82 or 0 02 three dimensional and d 1 88 or 0 03 four dimensional also they obtain values of the percolation probability and of the static exponents with errors at least comparable to the best values from the literature

stochastic analysis rough path analysis and fractional brownian motions in this paper we show by using dyadic approximations the existence of a geometric rough path associated with a fractional brownian motion with hurst parameter greater than 1 4 using the integral representation of fractional brownian motions we furthermore obtain a skohorod integral representation of the geometric rough path we constructed by the results in ly1 a stochastic integration theory may be established for fractional brownian motions and strong solutions and a wong zakai type limit theorem for stochastic differential equations driven by fractional brownian motions can be deduced accordingly the method can actually be applied to a larger class of gaussian processes with covariance functions satisfying a simple decay condition

stochastic area for brownian motion on the sierpinski gasket we construct a levy stochastic area for brownian motion on the sierpinski gasket the standard approach via ito integrals fails because this diffusion has sample paths which are far rougher than those of semimartingales we thus provide an example demonstrating the restrictions of the semimartingale framework as a consequence of the existence of the area one has a stochastic calculus and can solve stochastic differential equations driven by brownian motion on the sierpinski gasket

on factorization of matrix polynomials abstract an interrelationship between the numerical range of matrix polynomials and its factorization with linear or nonlinear factors is investigated here also a bound on the number of components of numerical range is given

factorization of matrix polynomials with multiple roots abstract the concept of a multiple root of matrix polynomial l is introduced and associated spectral properties of l are investigated a statement concerning factorization of l is presented applications are made to factorizations of the matrix polynomial l for any positive integer

stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension the d dimensional fractional brownian motion fbm for short b t b 1 t b d t t r with hurst exponent 0 1 is a d dimensional centered self similar gaussian process with covariance e b i s b i t 1 2 i j lsl 2 ltl 2 t s 2 the long standing problem of defining a stochastic integration with respect to fbm and the related problem of solving stochastic differential equations driven by fbm has been addressed successfully by several different methods although in each case with a restriction on the range of either d or the case corresponds to the usual stochastic integration with respect to brownian motion while most computations become singular when gets under various threshhold values due to the growing irregularity of the trajectories as 0 we provide here a new method valid for any d and for 1 4 by constructing an approximation e t e 0 of fbm which allows to define iterated integrals and then applying the geometric rough path theory the approximation relies on the definition of an analytic process z on the cut plane z c r of which fbm appears to be a boundary value and allows to understand very precisely the well known see 5 but as yet a little mysterious divergence of levy s area for 1 4

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

family of growth fractals with continuously tunable chemical dimension we introduce a new class of statistical growth fractals which is of interest because the chemical dimension d is continuously tunable we also study other exponents charac terising these fractals there has been considerable recent interest in uncovering the fashion in which the familiar laws of physics are modified for fractal objects in part because of the large number of important realisations of fractals in nature l 81 it has only recently become appreciated that the physics of fractals is determined by more than just the fractal dimension df which describes how the cluster mass n scales with the cluster radius r

approximation metric entropy and small ball estimates for gaussian measures a precise link proved by kuelbs and li relates the small ball behavior of a gaussian measure on a banach space e with the metric entropy behavior of k the unit ball of the reproducing kernel hilbert space of in e we remove the main regularity assumption imposed on the unknown function in the link this enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete gaussian processes moreover we show that the small ball behavior of a gaussian process is also tightly connected with the speed of approximation by finite rank processes

a block bidiangonal form for block companion matrices abstract the problem of the linear factorization of a polynomial matrix is related with a similarity condition linking the block companion matrix and a block upper bidiagonal matrix constructed from a chain of solvents this result is the applied to the solution of differential and difference linear matrix equations

factorization of the matrix polynomial a λ a0λt a1λt 1 at 1λ at abstract this paper gives necessary and sufficient conditions for a linear factorization of a matrix polynomial a a 0 t a 1 t 1 a t in the form a 0 l t i n l t 1 i n l 1 in terms of the strict equivalence of two pencils associated to a

nonlinear resistor fractal networks topological distances singly connected bonds and fluctuations the authors consider a fractal network of nonlinear resistors with the voltage v behaving as a power of the current i mod v mod r mod i mod alpha the resistance between two points at a distance l is r l varies as lzeta alpha they prove that zeta 0 describes the scaling of the topological chemical distance while zeta infinity describes that of the number of singly connected red bonds for random resistors they also consider the width of the resistance distribution delta r varies as l zeta 2 alpha values for zeta and zeta 2 are explicitly derived for two model fractals and delta r r is found to grow with l for the sierpinski gasket and alpha 1 612 the relevance of the results to percolation clusters is discussed

precise asymptotics of certain wiener functionals abstract in this article we present a method for developing certain wiener integrals in an asymptotic expansion as the variance of the wiener goes to 0 the basic ideas are a combination of methods borrowed from the theory of large deviations together with techniques from the calculus of wiener space

asymptotic expansions for the laplace approximations for ito functionals of brownian rough paths abstract in this paper we establish asymptotic expansions for the laplace approximations for ito functionals of brownian rough paths under the condition that the phase function has finitely many non degenerate minima our main tool is the banach space valued rough path theory of t lyons we use a large deviation principle and the stochastic taylor expansion with respect to the topology of the space of geometric rough paths this is a continuation of a series of papers by inahama y inahama laplace s method for the laws of heat processes on loop spaces j funct anal 232 2006 148 194 and by inahama and kawabi y inahama h kawabi large deviations for heat kernel measures on loop spaces via rough paths j london math soc 73 3 2006 797 816 y inahama h kawabi on asymptotics of certain banach space valued ito functionals of brownian rough paths in proceedings of the abel symposium 2005 stochastic analysis and applications a symposium in honor of kiyosi ito springer berlin in press available at http www abelprisen no no abelprisen deltagere 2005 html

geometrical properties of single connected bonds in percolation clusters numerical simulation of the two dimensional bond percolation at the percolation threshold on an l l lattice shows that the radius of gyration of singly connected bonds scb scales as l1 this result confirms an assumption used to derive a lower bound on the critical exponent of elasticity a persistent anisotropy in the distribution of scb is found and used to estimate the ratio between the bulk and shear moduli of a random percolating system local anisotropy in the direction of scb disappears for large l

the levy area process for the free brownian motion abstract in this paper we construct a levy area process for the free brownian motion and in this way a typical geometric rough path in the sense of t lyons 1998 rev mat iberoamer 14 215 310 lying above the free brownian path thus the general results of lyons on differential equations driven by rough paths may be applied to the free brownian motion case

topological properties of diffusion limited aggregation and cluster cluster aggregation the detailed topological or connectivity properties of the clusters formed in diffusion limited aggregation dla and cluster cluster aggregation cca are considered for spatial dimensions d 2 3 and 4 specifically for both aggregation phenomena the authors calculate the fractal dimension dmin nu 1 defined by l approximately rd min where l is the shortest path between two points separated by a pythagorean distance r for cca they find that dmin increases monotonically with d presumably tending toward a limiting value dmin 2 at the upper critical dimensionality dc as found previously for lattice animals and percolation for dla on the other hand they find that dmin 1 within the accuracy of the calculations for d 2 3 and 4 suggesting the absence of an upper critical dimension they also discuss some of the subtle features encountered in calculating dmin for dla

surface and edge exponents for the spreading of 3d percolation monte carlo results are presented for the spreading of 3d percolation where the seed consists of an infinite straight line this line can either be in the bulk of the material into which spreading is possible on a planar surface of the material or on a rectangular edge in the last two cases spreading occurs only into angular regions with 180 degrees resp 90 degrees while the mean distance grows at p pc in all three cases with the same exponent the average number of growth sites grows resp vanishes in all three cases with different exponents in particular this implies that at p pc the sites on the surface of a big cube belonging to the maximal cluster within that cube have fractal dimension 1

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

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

position space renormalisation group study of shortest paths in percolation clusters the average length of the shortest paths along the backbone of the incipient infinite cluster is expected to diverge near the percolation threshold pc as mod p pc mod nu s with nu s different from the correlation length exponent nu the author applies the position space renormalisation group psrg technique to obtain nu s both for square and cubic lattices and for a triangular lattice in two dimensions the results obtained nu s 1 49 for d 2 and 1 21 for d 3 are in very good agreement with the existing results obtained by different methods

ritz and pseudo ritz values using matrix polynomials abstract in this paper we present a unified analysis of two block krylov subspace methods based on the block arnoldi procedure on a real nonsymmetric matrix a we show that matrix polynomials can be the appropriate tool for analyzing the convergence of jordan ritz pairs when a is not diagonalizable we also prove that analogous arguments can be used for studying the convergence properties of a linear system solver that minimizes the residual norm on the generated krylov subspace bgmres known results for single krylov subspace methods can be naturally generalized to the block setting explicitly demonstrating how the presence of blocks can influence the convergence of the method at hand

application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media we consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media it is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions in addition to the fractal dimensiond f 1 d the fractal dimension of the backbone arises in connection with electric current flow 2 d red the fractal dimension of the singly connected bonds in the backbone arises in connection with its equivalence to the thermal scaling power 3 d e the fractal dimension of the of the elastic backbone 4 d u the fractal dimension of the unscreened perimeter arises in connection with the viscosity singularity at the gelation threshold 5 d min the fractal dimension of the minimum path or chemical distance between two sites arises in co nnection with the aharony stauffer conjecture 6 dw the fractal dimension of a random walk 7 d g the fractal dimension of growth sites that arise as a random walk creates a cluster relations among these fractal dimensions are discussed some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases