violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices it is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity these lattices are special examples of sierpinski carpets their being of low lacunarity means that they are arbitrarily close to being translationally invariant the claim is substantiated for the ising model in d 1 ensuremath epsilon dimensions and for resistor network models with 1ldl2

multifractal diffusion entropy analysis on stock volatility in financial markets abstract this paper introduces a generalized diffusion entropy analysis method to analyze long range correlation then applies this method to stock volatility series the method uses the techniques of the diffusion process and renyi entropy to focus on the scaling behaviors of regular volatility and extreme volatility respectively in developed and emerging markets it successfully distinguishes their differences where regular volatility exhibits long range persistence while extreme volatility reveals anti persistence

lacunarity and multifractal analysis of the large dla mass distribution abstract we show the methodology used to analyze fractal and mass multifractal properties of very large diffusion limited aggregation dla clusters with a maximum of 10 9 particles for 2d aggregates and 10 8 particles for 3d clusters to support our main result the scaling behavior obtained by our experimental results corresponds to the expected performance of monofractal objects in order to estimate lacunarity measures for large dla clusters we develop a variant of the gliding box algorithm which reduces the computer time needed to obtain experimental results we show how our mass multifractal data have a tendency to present monofractal behavior for the mass distribution of the cases presented in this paper in the limit of very large clusters lacunarity analysis shows provided we study small clusters mass distributions data which might be interpreted as two different values of fractal dimensions while the cluster grows however this effect tends to vanish when the cluster size increases further in such a way that monofractality is achieved the outcomes of this paper lead us to conclude that the previously reported mass multifractality behavior vicsek et al 1990 13 detected for dla clusters is a consequence of finite size effects and floating point precision limitations and not an intrinsic feature of the phenomena since the scaling behavior of our dla clusters space corresponds to monofractal objects being this situation remarkably noticeable in the limit of very large clusters

local and cluster critical dynamics of the 3d random site ising model abstract we present the results of monte carlo simulations for the critical dynamics of the three dimensional site diluted quenched ising model three different dynamics are considered these correspond to the local update metropolis scheme as well as to the swendsen wang and wolff cluster algorithms the lattice sizes of l 10 96 are analysed by a finite size scaling technique the site dilution concentration p 0 85 was chosen to minimize the correction to scaling effects we calculate numerical values of the dynamical critical exponents for the integrated and exponential autocorrelation times for energy and magnetization as expected cluster algorithms are characterized by lower values of dynamical critical exponent than the local one also in the case of dilution critical slowing down is more pronounced for the metropolis algorithm however the striking feature of our estimates is that they suggest that dilution leads to decrease of the dynamical critical exponent for the cluster algorithms this phenomenon is quite opposite to the local dynamics where dilution enhances critical slowing down

dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

critical behavior of a three dimensional random bond ising model using finite time scaling with extensive monte carlo renormalization group method we have investigated the critical behavior of a three dimensional random bond ising model for a series of the disorder strength by a finite time scaling combining with monte carlo renormalization group method in the presence of a linearly varying temperature the method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures the static exponents obtained agree well with most existing results verify both the hyperscaling and the rushbrooke scaling laws and their combined scaling law which in turn validate their asymptotic nature and corroborate the universality of the relevant random fixed point with respect to the forms of disorder the dynamic critical exponent z is estimated to 2 114 51 which is compatible with those obtained from experiments and renormalization group analyses the exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point they also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover our results demonstrate the effectiveness of the finite time scaling method

factorization of j unitary matrix polynomials on the line and a schur algorithm for generalized nevanlinna functions we prove that a 2 2 matrix polynomial which is j unitary on the real line can be written as a product of normalized elementary j unitary factors and a j unitary constant in the second part we give an algorithm for this factorization using an analog of the schur transformation

dynamic critical behavior of the swendsen wang algorithm the two dimensional three state potts model revisited we have performed a high precision monte carlo study of the dynamic critical behavior of the sweden wang algorithm for the two dimensional three state potts model we find that the li sokal bound int const cn is almost but not quite sharp the ratio int cn seems to diverge either as a small power 0 08 or as a logarithm

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

regularization of the indefinite stieltjes moment problem abstract a not regular indefinite stieltjes moment problem is studied in the generalized stieltjes class n k the new class of the sequence s s i i 0 l is defined which is called regular class regularization of the indefinite stieltjes moment problem is obtained the set of solutions of this problem is found in terms of the generalized stieltjes polynomials and s fractions with the shift

critical crossover phenomena in disordered ising systems the author has investigated the critical behaviour of site disordered ising systems over a large range of dilution 0 6 or p or 1 0 investing about 1015 single spin flips with conventional local dynamics critical temperatures have been determined with high precision using the cumulant method the finite size simulations lead to concentration dependent exponents which seem to violate universality the author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region the author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder

jl unitary factorization and the schur algorithm for nevanlinna functions in an indefinite setting abstract we introduce a schur transformation for generalized nevanlinna functions and show that it can be used in obtaining the unique minimal factorization of a class of rational j l unitary 2 2 matrix functions into elementary factors from the same class

crossover effects in dilute magnetic materials by finite time dynamics method abstract crossover phenomena are ubiquitous in disordered system however the crossover effects from the competition between different fixed points make it hard to identify the asymptotic scaling regime controlled by the random fixed point therefore the analyses on the crossover effects are of great importance to understand phase transition in dilute magnetic materials in this paper we derive the scaling function characterizing the crossovers to probe critical behavior according to these we show an alternative method to locate the asymptotic critical region in the quenched random systems in the finite time dynamics frame for the three dimensional bond diluted ising model the asymptotic critical exponents are identified with 0 686 1 0 356 6 0 057 3 1 345 20 z 2 170 11 at middle bond concentration p 0 7 from the effective ones describing the approach to the asymptotic regime for weak dilution p 0 9 and strong dilution p 0 5 the dynamic critical exponent from our non equilibrium simulation supports distinctly z 2 18 as suggested previously as opposed to the existing larger values

geometrical multifractality of growing structures the authors study the mass distribution of fractal objects growing by subsequent addition of units to the structure it is demonstrated that in the limit of very large sizes such structures generally exhibit multifractal properties without defining a singular measure on the fractal support using the mass as a measure on the fractal for a few simple deterministic processes they give the spectrum of corresponding fractal dimensions exactly according to their results such geometrical multifractals should be very common and consist of parts which have a varying local fractal scaling of mass different from that of the whole object they demonstrate the variation of multifractal properties as a function of the increasing size in this way establishing for the calculation of multifractality an approach common in the studies of fractal properties of growing clusters

inverse problems for periodic generalized jacobi matrices abstract some inverse problems for semi infinite periodic generalized jacobi matrices are considered in particular a generalization of the abel criterion is presented the approach is based on the fact that the solvability of the pell abel equation is equivalent to the existence of a certainly normalized j unitary 2 2 matrix polynomial the monodromy matrix

scale invariant distribution and multifractality of volatility multipliers in stock markets abstract the statistical properties of the multipliers of the absolute returns are investigated using 1 min high frequency data of financial time series the multiplier distribution is found to be independent of the box size s when s is larger than some crossover scale providing direct evidence of the existence of scale invariance in financial data the multipliers with base a 2 are well approximated by a normal distribution and the most probable multiplier scales as a power law with respect to the base a we unravel that the volatility multipliers possess multifractal nature which is independent of construction of the multipliers that is the values of s and a

precise determination of the bond percolation thresholds and finite size scaling corrections for the sc fcc and bcc lattices extensive monte carlo simulations were performed to study bond percolation on the simple cubic s c face centered cubic f c c and body centered cubic b c c lattices using an epidemic kind of approach these simulations provide very precise values of the critical thresholds for each of the lattices pc s c 0 2488126 0 0000005 pc f c c 0 1201635 0 0000010 and pc b c c 0 1802875 0 0000010 for p close to pc the results follow the expected finite size and scaling behavior with values for the fisher exponent 2 189 0 002 the finite size correction exponent 0 64 0 02 and the scaling function exponent 0 445 0 01 confirmed to be universal

numerical estimates of the generalized dimensions of the henon attractor for negative q the usual fixed size box counting algorithms are inefficient for computing generalized fractal dimensions in the range of q 0 in this letter we describe a new numerical algorithm specifically devised to estimate generalized dimensions for large negative q providing evidence of its better performance we compute the complete spectrum of the henon attractor and interpret our results in terms of a phase transition between different multiplicative laws

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

universality in the off equilibrium critical dynamics of the three dimensional diluted ising model we study the off equilibrium critical dynamics of the three dimensional diluted ising model we compute the dynamical critical exponent z and we show that it is independent of the dilution only when we take into account the scaling corrections to the dynamics finally we will compare our results with the experimental data

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

probability distribution function and multiscaling properties in the korean stock market abstract we consider the probability distribution function pdf and the multiscaling properties of the index and the traded volume in the korean stock market we observed the power law of the pdf at the fat tail region for the return volatility the traded volume and changes of the traded volume we also investigate the multifractality in the korean stock market we consider the multifractality by the detrended fluctuation analysis mfdfa we observed the multiscaling behaviors for index return traded volume and the changes of the traded volume we apply mfdfa method for the randomly shuffled time series to observe the effects of the autocorrelations the multifractality is strongly originated from the long time correlations of the time series

the schur algorithm for generalized schur functions iii j unitary matrix polynomials on the circle the main result is that for j 1 0 0 1 every j unitary 2 x 2 matrix polynomial on the unit circle is an essentially unique product of elementary j unitary 2 x 2 matrix polynomials which are either of degree 1 or 2k this is shown by means of the generalized schur transformation introduced in ann inst fourier 8 1958 211 ann acad sci fenn ser a i 250 9 1958 1 7 and studied in pisot and salem numbers birkhauser verlag basel 1992 philips j res 41 1 1986 1 54 and also in the first two parts operator theory adv appl 129 birkhauser verlag basel 2000 p 1 monatshefte fur mathematik in press of this series the essential tool in this paper are the reproducing kernel pontryagin spaces associated with generalized schur functions c 2003 elsevier science inc all fights reserved

multifractality in stock indexes fact or fiction abstract multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes including hsi szsc s p 500 and nasdaq to check whether the indexes instead of the returns possess multifractality we find that the mass exponent q is linear and the singularity q is close to 1 for all trading days and all indexes furthermore we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of shuffled time series hence the so called multifractality in the intraday stock market indexes is merely an illusion

finite size effects in diffusion limited aggregation this paper discusses a large variety of numerical results on diffusion limited aggregation dla to support the view that asymptotically large dla is self similar and the scaling of the geometry can be speci6ed by the fractal dimension alone deviations from simple scaling observed in many simulations are due to 6nite size effects i explain the relationship between the 6nite size effects in various measurements and how they can arise due to a crossover of the noise magnitude in the growth process complex scaling hypotheses including anomalous scaling of the width of the growing region multiscaling of the cluster radial density infinite drift of the e neighborhood filling ratio nonmultifractal scaling of the growth probability measure and geometrical multifractality are shown to lead to physically unacceptable predictions

star graph expansions for bond diluted potts models we derive high temperature series expansions for the free energy and the susceptibility of random bond q state potts models on hypercubic lattices using a star graph expansion technique this method enables the exact calculation of quenched disorder averages for arbitrary uncorrelated coupling distributions moreover we can keep the disorder strength p as well as the dimension d as symbolic parameters by applying several series analysis techniques to the new series expansions one can scan large regions of the p d parameter space for any value of q for the bond diluted four state potts model in three dimensions which exhibits a rather strong first order phase transition in the undiluted case we present results for the transition temperature and the effective critical exponent gamma as a function of p as obtained from the analysis of susceptibility series up to order 18 a comparison with recent monte carlo data chatelain et al phys rev e 64 036120 2001 shows signals for the softening to a second order transition at finite disorder strength

single cluster monte carlo dynamics for the ising model we present an extensive study of a new monte carlo acceleration algorithm introduced by wolff for the ising model it differs from the swendsen wang algorithm by growing and flipping single clusters at a random seed in general it is more efficient than swendsen wang dynamics ford 2 giving zero critical slowing down in the upper critical dimension monte carlo simulations give dynamical critical exponentszw 0 33 0 05 and 0 44 0 10 ind 2 and 3 respectively and numbers consistent withzw 0 ind 4 and mean field theory we present scaling arguments which indicate that the wolff mechanism for decorrelation differs substantially from swendsen wang despite the apparent similarities of the two methods

an efficient implementation of the gliding box lacunarity algorithm lacunarity is a measure of how data fills space it complements fractal dimension which measures how much space is filled many researchers use the gliding box algorithm to calculate lacunarity this paper introduces a fast algorithm for making this calculation this algorithm is akin to fast box counting algorithms used by some researchers to estimate fractal dimension a simplified gliding box measure equation along with key pseudo code implementations for the algorithm are presented applications for the gliding box lacunarity measure have ranged from biological community modelling to target detection

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

critical dynamics a field theoretical approach we review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of halperin and hohenberg we unify the presentation of the theoretical background by restricting ourselves to the field theoretic renormalization group method the main results obtained in the different universality classes are presented this contains the critical dynamics near the gas liquid transition in pure fluids model h the plait point and consolute point in mixtures model h the superfluid transition in 4 he model f and 4 he 3 he mixtures model f the curie point model j and neel point model g in heisenberg magnets and the superconducting transition in comparison with experimental results it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non universal effective behaviour either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics e g at the superfluid transition or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas liquid or demixing transition the calculation of the critical dynamics is adequate in most cases only in two loop order we review these results and present the solution to unreasonable features found for some models thus we consider model c where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model model a have not been understood in general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role their fixed point values and the approach to the fixed point are changed considerably in two loop order compared to their values in one loop order these considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium model f and the isotropic antiferromagnet model g as far as possible the comparison of results obtained by the renormalization group theory with numerical simulations has been made

the dynamic critical exponent of dilute and pure ising systems a massive dynamic field theory is employed to calculate the critical exponent z of the three dimensional dilute ising model to two loop order without expansion in square root 4 d with result z approximately 2 191 applying the same method to the pure ising model at three loop order we find z 2 022 for d 3 and z 2 124 for d 2

multifractal detrended fluctuation analysis of derivative and spot markets abstract we investigate the multifractal properties of price increments in the cases of derivative and spot markets through the multifractal detrended fluctuation analysis we estimate the generalized hurst and the renyi exponents for price fluctuations by deriving the singularity spectrum from the above exponents we quantify the multifractality of a financial time series and compare the multifractal properties of two different markets the different behavior of each agent group in transactions is also discussed in order to identify the nature of the underlying multifractality we apply the method of surrogate data to both sets of financial data it is shown that multifractality due to a fat tailed distribution is significant

probing surface characteristics of diffusion limited aggregation clusters with particles of variable size we develop a technique for probing the harmonic measure of a diffusion limited aggregation dla cluster surface with variable size particles and generate 1000 clusters with 50 x 10 6 particles using an original off lattice killing free algorithm taking in sequence the limit of the vanishing size of the probing particles and then sending the growing cluster size to infinity we achieve unprecedented accuracy in determining the fractal dimension d 1 7100 2 crucial to the characterization of the geometric properties of dla clusters

an improved multifractal formalism and self similar measures abstract to characterize the geometry of a measure its generalized dimensions dq have been introduced recently the mathematically precise definition given by falconer fractal geometry 1990 turns out to be unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates in the negative q range a new definition is introduced which is also based on box counting but which carries relevant information about for negative q also in particular the legendre connection between generalized dimensions and the multifractal spectrum is established using large deviations and rigorous proofs are provided for the implicit formula giving the generalized dimensions of selfsimilar measures which was until now known only for positive q

universal aging properties at a disordered critical point we investigate analytically near the dimension d uc 4 and numerically in d 3 the nonequilibrium relaxational dynamics of the randomly diluted ising model at criticality using the exact renormalization group method to one loop we compute the two times t phantom rule 0 2em 0ex t w correlation function and fluctuation dissipation ratio fdr for any fourier mode of the order parameter of finite wave vector q in the large time separation limit the fdr is found to reach a nontrivial value x ensuremath infty independently of small q and coincide with the fdr associated to the total magnetization obtained previously explicit calculations in real space show that the fdr associated to the local magnetization converges in the asymptotic limit to this same value x ensuremath infty through a monte carlo simulation we compute the autocorrelation function in three dimensions for different values of the dilution fraction p at t c p taking properly into account the corrections to scaling we find according to the renormalization group predictions that the autocorrelation exponent ensuremath lambda c is independent of p the analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state

generic multifractality in exponentials of long memory processes we find that multifractal scaling is a robust property of a large class of continuous stochastic processes constructed as exponentials of long memory processes the long memory is characterized by a power law kernel with tail exponent ensuremath varphi 1 2 where ensuremath varphi g0 this generalizes previous studies performed only with ensuremath varphi 0 with a truncation at an integral scale by showing that multifractality holds over a remarkably large range of dimensionless scales for ensuremath varphi g0 the intermittency multifractal coefficient can be tuned continuously as a function of the deviation ensuremath varphi from 1 2 and of another parameter ensuremath sigma 2 embodying information on the short range amplitude of the memory kernel the ultraviolet cutoff viscous scale and the variance of the white noise innovations in these processes both a viscous scale and an integral scale naturally appear bracketing the inertial scaling regime we exhibit a surprisingly good collapse of the multifractal spectra ensuremath zeta q on a universal scaling function which enables us to derive high order multifractal exponents from the small order values and also obtain a given multifractal spectrum ensuremath zeta q by different combinations of ensuremath varphi and ensuremath sigma 2

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation