on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

statistics of persistent events an exactly soluble model it was recently realized that the persistence exponent appearing in the dynamics of nonequilibrium systems is a special member of a continuously varying family of exponents describing generalized persistence properties we propose and solve a simple stochastic spin model where time intervals between spin flips are independent and distributed according to a l evy law both the limit distribution of the mean magnetization and the generalized persistence exponents are obtained exactly we discuss the relevance of this model for phase ordering spin glasses and random walks

residence time distribution for a class of gaussian markov processes we study the distribution of residence time or equivalently that of mean magnetization for a family of gaussian markov processes indexed by a positive parameter alpha the persistence exponent for these processes is simply given by theta alpha but the residence time distribution is nontrivial the shape of this distribution undergoes a qualitative change as theta increases indicating a sharp change in the ergodic properties of the process we develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary alpha for some special values of alpha we obtain closed form expressions of the distribution function s1063 651x 99 03306 1

self diffusion and visited surface in the droplet condensation problem breath figures abstract we study experimentally and theoretically new aspects of condensation growth and coalescence of droplets on a substrate we address in particular the dynamics of a marked droplet which undergoes coalescences with neighbouring droplets we find that the number of coalescences of the droplet grows as log t t is time and that the traveled distance increases as r t the average radius of the droplets at time t the fraction of the surface which was never covered by any droplet an important quantity for applications such as drug spreading or surface decontamination decays as t showing that completion occurs only slowly heuristic arguments accurate numerical simulations on simplified models which neglect polydispersity and an exact solution reported elsewhere a bray b derrida and c godreche europhys lett 27 1994 175 strongly support these findings and show that this power law decay is a generic feature common to many different situations finally the contour of the ensemble of dry sites appears fractal with dimension d 1 22 an experimental result not reproduced by the simplified models

large deviations and nontrivial exponents in coarsening systems we investigate the statistics of the mean magnetization of its large deviations and persistent large deviations in simple coarsening systems in particular we consider more specifically the case of the diffusion equation of the ising chain at zero temperature and of the two dimensional voter model for the diffusion equation at large times the mean magnetization has a limit law which is studied analytically using the independent interval approximation the probability of persistent large deviations defined as the probability that the mean magnetization was for all previous times greater than some level x decays algebraically at large times with an exponent continuously varying with x when x 1 is the usual persistence exponent similar behaviour is found for the glauber ising chain at zero temperature for the two dimensional voter model large deviations of the mean magnetization are algebraic while the probability of persistent large deviations seem to behave as the usual persistence probability

statistics of the occupation time for a random walk in the presence of a moving boundary we investigate the distribution of the time spent by a random walker to the right of a boundary moving with constant velocity v for the continuous time problem brownian motion we provide a simple alternative proof of newman s recent result newman t j 2001 j phys a math gen 34 l89 using a method developed by kac we then discuss the same problem for the case of a random walk in discrete time with an arbitrary distribution of steps taking advantage of the general set of results of sparre andersen for the binomial random walk we analyse the corrections to the continuum limit on the example of the mean occupation time the case of cauchy distributed steps is also studied

reaction diffusion models in one dimension with disorder we study a large class of one dimensional reaction diffusion models with quenched disorder using a real space renormalization group method rsrg which yields exact results at large time particles e g of several species undergo diffusion with random local bias sinai model and may react upon meeting we obtain a detailed description of the asymptotic states i e attractive fixed points of the rsrg such as the large time decay of the density of each specie their associated universal amplitudes and the spatial distribution of particles we also derive the spectrum of nontrivial exponents which characterize the convergence towards the asymptotic states for reactions which lead to several possible asymptotic states separated by unstable fixed points we analyze the dynamical phase diagram and obtain the critical exponents characterizing the transitions we also obtain a detailed characterization of the persistence properties for single particles as well as more complex patterns we compute the decay exponents for the probability of no crossing of a given point by respectively the single particle trajectories ensuremath theta or the thermally averaged packets overline ensuremath theta the generalized persistence exponents associated to n crossings are also obtained specifying to the process a stackrel ensuremath rightarrow a ensuremath emptyset or a with probabilities r 1 ensuremath r we compute exactly the exponents ensuremath delta r and ensuremath psi r characterizing the survival up to time t of a domain without any merging or with mergings respectively and the exponents ensuremath delta a r and ensuremath psi a r characterizing the survival up to time t of a particle a without any coalescence or with coalescences respectively overline ensuremath theta ensuremath psi and ensuremath delta obey hypergeometric equations and are numerically surprisingly close to pure system exponents though associated to a completely different diffusion length the effect of additional disorder in the reaction rates as well as some open questions are also discussed

diffusive persistence and the sign time distribution we present a new method for extracting the persistence exponent theta for the diffusion equation based on the distribution p of sign times with the aid of a numerically verified ansatz for p we derive an exact formula for theta in arbitrary spatial dimension d our results are in excellent agreement with previous numerical studies furthermore our results indicate a qualitative change in p above d 36 signalling the existence of a sharp change in the ergodic properties of the diffusion field

persistence exponents for fluctuating interfaces numerical and analytic results for the exponent ensuremath theta describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear langevin equations the models are parametrized by the dynamic roughness exponent ensuremath beta with 0l ensuremath beta l1 for ensuremath beta frac 1 2 the time evolution is markovian using simulations of solid on solid models of the discretized continuum equations as well as of the associated zero dimensional stationary gaussian process we address two problems the return of an initially flat interface and the return to an initial state with fully developed steady state roughness the two problems are shown to be governed by different exponents for the steady state case we point out the equivalence to fractional brownian motion which has a return exponent ensuremath theta s 1 ensuremath ensuremath beta the exponent ensuremath theta 0 for the flat initial condition appears to be nontrivial we prove that ensuremath theta 0 ensuremath rightarrow ensuremath infty for ensuremath beta ensuremath rightarrow 0 ensuremath theta 0 g ensuremath theta s for ensuremath beta l frac 1 2 and ensuremath theta 0 l ensuremath theta s for ensuremath beta g frac 1 2 and calculate ensuremath theta 0 s perturbatively to first order in an expansion around the markovian case ensuremath beta frac 1 2 using the exact result ensuremath theta s 1 ensuremath ensuremath beta accurate upper and lower bounds on ensuremath theta 0 can be derived which show in particular that ensuremath theta 0 g 1 ensuremath ensuremath beta 2 ensuremath beta for small ensuremath beta

analytical results for generalized persistence properties of smooth processes we present a general scheme to calculate within the independent interval approximation generalized level dependent persistence properties for processes having a finite density of zero crossings our results are especially relevant for the diffusion equation evolving from random initial conditions one of the simplest coarsening systems exact results are obtained in certain limits and rely on a new method to deal with constrained multiplicative processes an excellent agreement of our analytical predictions with direct numerical simulations of the diffusion equation is found

a level crossing based scaling dimensionality transform applied to stationary gaussian processes the scaling dimensionality transform d sub a r theta of stochastic processes is introduced as a generalization of the fractal dimension concept over an infinite range of time scales it is based on the expected number of crossings of a constant level a and is a function of two variables the scaling factor r and the sampling time theta general properties of this transform are discussed whereby d sub a 1 theta emerges as the fundamental transform results for stationary gaussian processes calculable from rice s formula 1945 are applied to signals with asymptotic f sup beta spectra and to the problem of adjusting amplitude quantization to the sampling period in discrete signal representations

domain number distribution in the nonequilibrium ising model we study domain distributions in the one dimensional ising model subject to zero temperature glauber and kawasaki dynamics the survival probability of a domain s t t and an unreacted domain q1 t t are characterized by two independent nontrivial exponents we develop an independent interval approximation that provides close estimates for many characteristics of the domain length and number distributions including the scaling exponents

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

stationary definition of persistence for finite temperature phase ordering for the two dimensional kinetic ising model at finite temperature the local mean magnetization simply related to the fraction of time spent by a given spin in the positive direction has a limiting distribution singular at the onsager spontaneous magnetization the exponent of this singularity defines the persistence exponent we also study first passage exponents associated to persistent large deviations of and their temperature dependence

experimental measurement of the persistence exponent of the planar ising model using a twisted nematic liquid crystal system exhibiting planar ising model dynamics we have measured the scaling exponent ensuremath theta which characterizes the time evolution p t ensuremath sim t ensuremath ensuremath theta of the probability p t that the local order parameter has not switched its state by the time t for 0 4 sec 200 sec following the phase quench the system exhibits scaling behavior and measured over this interval ensuremath theta 0 19 ifmmode pm else textpm fi 0 03 in good agreement with theoretical analysis and numerical simulations

spin block persistence at finite temperature we explore a new definition of the persistence exponent measuring the probability that a spin never flips after a quench of an ising like model at a temperature while the usual definition only makes sense at t 0 this probability is now defined for spin blocks and a general scaling for it involving time and block linear size is introduced and illustrated by extensive simulations

exact exponent for the number of persistent spins in the zero temperature dynamics of the one dimensional potts model for the zero temperature glauber dynamics of theq state potts model the fractionr q t of spins which never flip up to timet decays like a power lawr q t t q when the initial condition is random by mapping the problem onto an exactly soluble one species coagulation model a a a or alternatively by transforming the problem into a free fermion model we obtain the exact expression of q for all values ofq the exponent q is in general irrational 3 0 53795082 4 0 63151575 with the exception ofq 2 andq for which 2 3 8 and 1

concatenated solutions of delay differential equations and their representation with time delay feedback systems stimulated by the monodromy operator approach to time delay systems tdss developed recently this article studies the conversion problems of delay differential equations ddes into the representation as time delay feedback systems we give two conversion methods including the conversion of initial conditions where we show that each of the two methods corresponds in general to one of the two different definitions for the solutions of ddes called pseudo concatenated solutions and continuous concatenated solution the study is actually closely related to the subtle behaviours of the solutions of ddes under discontinuous initial conditions and simple examples illustrating such subtleties as well as the validity of the conversion methods are also provided the results of this article suggest that the ability of representing tdss is higher in the representation as time delay feedback systems than in the representation as ddes

exact occupation time distribution in a non markovian sequence and its relation to spin glass models we compute exactly the distribution of the occupation time in a discrete non markovian toy sequence that appears in various physical contexts such as the diffusion processes and ising spin glass chains the non markovian property makes the results nontrivial even for this toy sequence the distribution is shown to have non gaussian tails characterized by a nontrivial large deviation function that is computed explicitly an exact mapping of this sequence to an ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model at a given temperature what is the distribution over disorder of the thermally averaged number of spins that are aligned to their local fields we show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the sherrington kirkpatrick model with gaussian disorder

domain statistics in coarsening systems we study the domain number and size distributions in the one dimensional ising and q state potts models subject to zero temperature glauber dynamics the survival probability of a domain s t t 2c and an unreacted domain q1 t t 2d are characterized by two independent nontrivial exponents for the ising case we find c50 126 and d51 27 using numerical simulations we develop an independent interval approximation that predicts the qualitative behavior of the domain distribution and provides good estimates for the exponents exact results for the domain distribution are also obtained in several solvable cases s1063 651x 97 00110 4

analytical results for random walk persistence in this paper we present a detailed calculation of the persistence exponent straight theta for a nearly markovian gaussian process x t a problem initially introduced elsewhere in phys rev lett 77 1420 1996 describing the probability that the walker never crosses the origin resummed perturbative and nonperturbative expressions for straight theta are derived which suggest a connection with the result of the alternative independent interval approximation the perturbation theory is extended to the calculation of straight theta for non gaussian processes by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem finally we give perturbative and nonperturbative expressions for the persistence exponent straight theta x0 describing the probability that the process remains larger than x 0 sqrt

statistics of persistent events in the binomial random walk will the drunken sailor hit the sober man the statistics of persistent events recently introduced in the context of phase ordering dynamics is investigated in the case of the one dimensional lattice random walk in discrete time we determine the survival probability of the random walker in the presence of an obstacle moving ballistically with velocity v i e the probability that the random walker remains up to time n on the left of the obstacle three regimes are to be considered for the long time behavior of this probability according to the sign of the difference between v and the drift velocity v of the random walker in one of these regimes v v the survival probability has a nontrivial limit at long times which is discontinuous at all rational values of v an algebraic approach allows us to compute these discontinuities as well as several related quantities the mathematical structure underlying the solvability of this model combines elementary number theory algebraic functions and algebraic curves defined over the rationals

are there any julia sets for the laguerre iteration function abstract for polynomials some of whose zeros are complex little is known about the overall convergence properties of the laguerre s method the existence of free critical points of the laguerre function as applied to one parameter families of quadratic and cubic polynomials is examined with the help of microcomputer plots we investigate the julia sets of the laguerre function in the special case wherein this function is constructed to converge to the nth roots of unity

statistical properties of randomly broken objects and of multivalley structures in disordered systems the statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common for all these problems if w denotes the weight of the sth piece we show that the probability distributions pi w of the largest piece w p2 w of the second largest piece and n y of y 1 wi always have singularities at w l n w2 l n and y ijn n 1 2 3

coarsening and persistence in the voter model the voter model is a simple model for coarsening with a nonconserved scalar order parameter we investigate coarsening and persistence in the voter model by introducing the quantity pn t defined as the fraction of voters who changed their opinion n times up to time t we show that pn t exhibits scaling behavior that strongly depends on the dimension as well as on the initial opinion concentrations exact results are obtained for the average number of opinion changes n and the autocorrelation function a t 21 n pn t 2d 2 in arbitrary dimension d these exact results are complemented by a mean field theory heuristic arguments and numerical simulations for dimensions d 2 the system does not coarsen and the opinion changes follow a nearly poissonian distribution in agreement with mean field theory for dimensions d 2 the distribution is given by a different scaling form which is characterized by nontrivial scaling exponents for unequal opinion concentrations an unusual situation occurs where different scaling functions correspond to the majority and the minority as well as for even and odd n

non trivial exponents in the zero temperature dynamics of the 1d ising and potts models we consider the glauber dynamics of the q state potts model in one dimension at zero temperature starting with a random initial configuration we measure the density r t of spins which have never flipped from the beginning of the simulation until time t we find that for large t the density r t has a power law decay left r t sim t theta right where the exponent theta varies with q our simulations lead to theta simeq 37 for q 2 theta simeq 53 for q 3 and theta longrightarrow 1 as q longrightarrow infty

non markovian persistence and nonequilibrium critical dynamics the persistence exponent ensuremath theta for the global order parameter m t of a system quenched from the disordered phase to its critical point describes the probability p t ensuremath sim t ensuremath ensuremath theta that m t does not change sign in the time interval t following the quench we calculate ensuremath theta to o ensuremath epsilon 2 for model a of hohenberg and halperin rev mod phys 49 435 1977 and to order ensuremath epsilon for model c and show that at this order m t is a non markov process consequently to our knowledge ensuremath theta is a new exponent the calculation is performed by expanding around a markov process using a simplified version of the perturbation theory recently introduced by majumdar and sire phys rev lett 77 1420 1996