rectifiability of measures with locally uniform cube density the conjecture that radon measures in euclidean space with positive finite density are rectifiable was a central problem in geometric measure theory for fifty years this conjecture was positively resolved by preiss in 1986 using methods entirely dependent on the symmetry of the euclidean unit ball since then due to reasons of isometric immersion of metric spaces into l infty and the uncommon nature of the sup norm even in finite dimensions a popular model problem for generalising this result to non euclidean spaces has been the study of 2 uniform measures in l 3 infty the rectifiability or otherwise of these measures has been a well known question in this paper the stronger result that locally 2 uniform measures in l 3 infty are rectifiable is proved this is the first result that proves rectifiability from an initial condition about densities for general radon measures of dimension greater than 1 outside euclidean space

survival in equilibrium step fluctuations we report the results of analytic and numerical investigations of the time scale of survival or non zero crossing probability s t in equilibrium step fluctuations described by langevin equations appropriate for attachment detachment and edge diffusion limited kinetics an exact relation between long time behaviors of the survival probability and the autocorrelation function is established and numerically verified s t is shown to exhibit a simple scaling behavior as a function of system size and sampling time our theoretical results are in agreement with those obtained from an analysis of experimental dynamical scanning tunneling microscopy data on step fluctuations on al si 111 and ag 111 surfaces

renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

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

instability intermittency and multiscaling in discrete growth models of kinetic roughening we show by numerical simulations that discretized versions of commonly studied continuum nonlinear growth equations such as the kardar parisi zhangequation and the lai das sarma villain equation and related atomistic models of epitaxial growth have a generic instability in which isolated pillars or grooves on an otherwise flat interface grow in time when their height or depth exceeds a critical value depending on the details of the model the instability found in the discretized version may or may not be present in the truly continuum growth equation indicating that the behavior of discretized nonlinear growth equations may be very different from that of their continuum counterparts this instability can be controlled either by the introduction of higher order nonlinear terms with appropriate coefficients or by restricting the growth of pillars or grooves by other means a number of such controlled instability models are studied by simulation for appropriate choice of the parameters used for controlling the instability these models exhibit intermittent behavior characterized by multiexponent scaling of height fluctuations over the time interval during which the instability is active the behavior found in this regime is very similar to the turbulent behavior observed in recent simulations of several one and two dimensional atomistic models of epitaxial growth

unrectifiable 1 sets with moderate essential flatness satisfy besicovitch s 12 conjecture abstract in this paper we show that for a wide class of totally unrectifiable 1 sets in the plane or even a hilbert space satisfying a mild measure theoretic flatness condition almost everywhere at sufficiently small scales the lower spherical density is bounded above by 1 2 at almost every point thereby affirming besicovitch s conjecture which states that for all totally unrectifiable 1 sets in the plane or possibly even in r n or a hilbert space the lower spherical density is bounded above by 1 2 at almost every point

width distribution for random walk interfaces roughening of a one dimensional interface is studied under the assumption that the interface configurations are continuous periodic random walks the distribution of the square of the width of interface mathit w 2 is found to scale as p mathit w 2 mathit w 2 mathrm mathrm ensuremath 1 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 we calculate the scaling function ensuremath phi x exactly and compare it both to exact enumerations for a discrete slope surface evolution model and to ensuremath phi s obtained in monte carlo simulations of equilibrium and driven interfaces of chemically reacting systems

partial regularity of almost minimizing rectifiable g chains in hilbert space we adapt to an infinite dimensional ambient space e r reifenberg s epiperimetric inequality and a quantitative version of d preiss second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable g chain in l is dense in its support whenever the group g of coefficients is so that g g g is discrete and closed

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

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

vectorized and parallel simulations of the kardar parisi zhang equation in 3 1 dimensions the kardar parisi zhang kpz equation describing kinetic roughening is solved numerically for 3 1 dimensional systems in the strong coupling phase by massive use of supercomputing tools we calculate for the first time the exponent beta 0 181 or 0 007 with an accuracy comparable with that of lattice growth model simulations a possible source of errors in parallel monte carlo calculations is pointed out

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

a marstrand theorem for measures with polytope density given s in 0 2 and any centrally symmetric convex polytope theta subset mathbb r n define theta r x r theta x we prove that if a radon measure has the property label df1 0 mathop lim limits r to 0 frac mu big theta r x big r s infty quad rm for mu textrm a e x then s is an integer for the case is the euclidean ball this result was first proved by marstrand in 1955 for hausdorff measure in the plane marstrand in proc lond math soc 3 4 257 302 1954 and later for general radon measures in mathbb r n marstrand in trans am math soc 205 369 392 1964

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

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

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

airy distribution function from the area under a brownian excursion to the maximal height of fluctuating interfaces the airy distribution function describes the probability distribution of the area under a brownian excursion over a unit interval surprisingly this function has appeared in a number of seemingly unrelated problems mostly in computer science and graph theory in this paper we show that this distribution function also appears in a rather well studied physical system namely the fluctuating interfaces we present an exact solution for the distribution p h m l of the maximal height h m measured with respect to the average spatial height in the steady state of a fluctuating interface in a one dimensional system of size l with both periodic and free boundary conditions for the periodic case we show that p h m l l 1 2f h m l 1 2 for all l 0 where the function f x is the airy distribution function this result is valid for both the edwards wilkinson ew and the kardar parisi zhang interfaces for the free boundary case the same scaling holds p h m l l 1 2f h m l 1 2 but the scaling function f x is different from that of the periodic case we compute this scaling function explicitly for the ew interface and call it the f airy distribution function numerical simulations are in excellent agreement with our analytical results our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables some of these results were announced in a recent letter s n majumdar and a comtet phys rev lett 92 225501 2004

statistics of extremal intensities for gaussian interfaces the extremal fourier intensities are studied for stationary edwards wilkinson type gaussian interfaces with power law dispersion we calculate the probability distribution of the maximal intensity and find that generically it does not coincide with the distribution of the integrated power spectrum i e roughness of the surface nor does it obey any of the known extreme statistics limit distributions the fisher tippett gumbel limit distribution is however recovered in three cases i in the nondispersive white noise limit ii for high dimensions and iii when only short wavelength modes are kept in the last two cases the limit distribution emerges in nonconventional scenarios

extreme value statistics of hierarchically correlated variables deviation from gumbel statistics and anomalous persistence we study analytically the distribution of the minimum of a set of hierarchically correlated random variables e1 e2 ellipsis e n where e i represents the energy of the ith path of a directed polymer on a cayley tree if the variables were uncorrelated the minimum energy would have an asymptotic gumbel distribution we show that due to the hierarchical correlations the forward tail of the distribution of the minimum energy becomes highly nonuniversal depends explicitly on the distribution of the bond energies epsilon and is generically different from the superexponential forward tail of the gumbel distribution the consequence of these results to the persistence of hierarchically correlated random variables is discussed and the persistence is also shown to be generically anomalous

on the area under a continuous time brownian motion till its first passage time the area swept out under a one dimensional brownian motion till its first passage time is analysed using a fokker planck technique we obtain an exact expression for the area distribution for the zero drift case and provide various asymptotic results for the nonzero drift case emphasizing the critical nature of the behaviour in the limit of vanishing drift the results offer important insights into the asymptotic behaviour of a number of discrete models we also provide a succinct derivation for the distribution of the maximum displacement observed during a first passage

the distribution of the area under a bessel excursion and its moments a bessel excursion is a bessel process that begins at the origin and first returns there at some given time t we study the distribution of the area under such an excursion which recently found application in the context of laser cooling the area a scales with the time as a sim t 3 2 independent of the dimension d but the functional form of the distribution does depend on d we demonstrate that for d 1 the distribution reduces as expected to the distribution for the area under a brownian excursion known as the airy distribution deriving a new expression for the airy distribution in the process we show that the distribution is symmetric in d 2 with nonanalytic behavior at d 2 we calculate the first and second moments of the distribution as well as a particular fractional moment we also analyze the analytic continuation from d 2 in the limit where d rightarrow 4 from below this analytically continued distribution is described by a one sided levy alpha stable distribution with index 2 3 and a scale factor proportional to 4 d t 3 2

universal large deviation function of the kardar parisi zhang equation in one dimension using the bethe ansatz we calculate the whole large deviation function of the displacement of particles in the asymmetric simple exclusion process asep on a ring when the size of the ring is large the central part of this large deviation function takes a scaling form independent of the density of particles we suggest that this scaling function found for the asep is universal and should be characteristic of all the systems described by the kardar parisi zhang equation in 1 1 dimension simulations done on two simple growth models are in reasonable agreement with this conjecture

mean field renormalisation group for the spin 1 2 anisotropic heisenberg model the mean field renormalisation group approach is applied to the spin 1 2 anisotropic heisenberg model the critical temperature and the critical exponent are obtained as a function of the anisotropy for the two dimensional model on square triangular and hexagonal lattices the three dimensional model on a simple cubic lattice is also analysed

power laws in surface physics the deep the shallow and the useful abstract the growth and dynamics of solid surfaces displays a multitude of power law relationships which are often associated with geometric self similarity in many cases the mechanisms behind these power laws are comparatively trivial and require little more than dimensional analysis for their derivation the information of interest to surface physicists then resides in the prefactors this point will be illustrated by recent experimental and theoretical work on the growth induced roughening of thin films and step fluctuations on vicinal surfaces the conventional distinction between trivial and nontrivial power laws will be critically examined in general and specifically in the context of persistence of step fluctuations

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

the true self avoiding walk with bond repulsion on z limit theorems the true self avoiding walk with bond repulsion is a nearest neighbor random walk on z for which the probability of jumping along a bond of the lattice is proportional to exp g number of previous jumps along that bond first we prove a limit theorem for the distribution of the local time process of this walk using this result later we prove a local limit theorem as a for the distribution of a 2 3 x s a where s a is a random time distributed geometrically with mean e s a 1 e s a 1 a s 0 1 as a by product we also obtain an apparently new identity related to brownian excursions and bessel bridges

extreme value statistics and traveling fronts various applications abstract an intriguing connection between extreme value statistics and traveling fronts has been found recently in a number of diverse problems in this short review we outline a few such problems and consider their various applications

maximum of n independent brownian walkers till the first exit from the half space we consider the one dimensional target search process that involves an immobile target located at the origin and n searchers performing independent brownian motions starting at the initial positions all on the positive half space the process stops when the target is first found by one of the searchers we compute the probability distribution of the maximum distance m visited by the searchers till the stopping time and show that it has a power law tail for large m thus all moments of m up to the order n 1 are finite while the higher moments diverge the prefactor bn increases with n faster than exponentially our solution gives the exit probability of a set of n particles from a box 0 l through the left boundary incidentally it also provides an exact solution of laplace s equation in an n dimensional hypercube with some prescribed boundary conditions the analytical results are in excellent agreement with monte carlo simulations

phase separation in random cluster models ii the droplet at equilibrium and local deviation lower bounds we study the droplet that results from conditioning the planar subcritical fortuin kasteleyn random cluster model on the presence of an open circuit 0 encircling the origin and enclosing an area of at least or exactly n2 we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary conv 0 of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length namely the length of the longest line segment of which the polygon conv 0 is formed we prove that there exists a constant c 0 such that the conditional probability that the normalised quantity n 1 3 log n 2 3 mlr 0 exceeds c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mfl 0 to obtain these bounds we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a markov chain that resamples sections of the circuit boundary we analyse the chain at equilibrium to prove the local roughness lower bounds alongside complementary upper bounds provided in 14 the fluctuations mlr 0 and mfl 0 are determined up to a constant factor

concentrated nearly monotonic epiperimetric measures in euclidean space we characterize holder continuously differentiable m dimensional submanifolds of euclidean space among m rectifiable sets s in terms of growth conditions on the m density ratios of the hausdorff measure h m l s

scaling function and universal amplitude combinations for self avoiding polygons we analyse new data for self avoiding polygons saps on the square and triangular lattices enumerated by both perimeter and area providing evidence that the scaling function is the logarithm of an airy function the results imply universal amplitude combinations for all area moments and suggest that rooted saps may satisfy a q algebraic functional equation

statistics of the first passage time of brownian motion conditioned by maximum value or area we derive the moments of the first passage time for brownian motion conditioned by either the maximum value or the area swept out by the motion these quantities are the natural counterparts to the moments of the maximum value and area of brownian excursions of fixed duration which we also derive for completeness within the same mathematical framework various applications are indicated

mean field renormalization group for the spin 1 2 xxz model the mean field renormalization group mfrg method is applied to the spin 1 2 xxz model the phase diagram is obtained for various lattice structures in two and three dimensions as well as estimates for the critical exponents the results of the original mfrg method are quite good but the more sophisticated approach that also includes surface exponents is not suitable for application to the small clusters used here

on correlations between certain random variables associated with first passage brownian motion we analyse how the area swept out by a brownian motion up to its first passage time correlates with the first passage time itself obtaining several exact results in the process additionally we analyse the relationship between the time average of a brownian motion during a first passage and the maximum value attained the results which find various applications are in excellent agreement with simulations

length scaling in mfrg the mean field renormalisation group mfrg has so far given poor estimates for critical indices it is argued that this can be traced to an unsatisfactory definition of the length scaling factor a new definition that improves the estimates in most cases is proposed

improved discretization of the kardar parisi zhang equation pected but quite surprisingly n 1 14fin 0 the parameters d and n were computed at a short time limit to avoid any renormalization these values imply a scaling amplitude a 5d n 0 877 which agrees nicely with independent estimates of a 0 879 and 0 876 taken respectively from saturated surface width and correlation function measurements the discrepancy between n and n 0 cannot be rectified by decreasing dt or dx it has thus been concluded in ref 9 that the conventional discretization is not a genuine approximation of the continuum kpz equation although universality does imply many common properties the problem has been attributed to microscopic roughness in the surfaces which leads to inaccuracy in the finite difference expressions in eq 3 this work aims at a better understanding of the relationship between the kpz equation and its discretizations this is made possible by studying a spatial discretization of the kpz equation in 111 dimensions unlike conventional schemes these discrete equations can be studied analytically and their properties can thus be compared with those of the continuum counterpart by calculating analytically the values of three correlators we found that the effective parameters l n and d all agree with the nominal values we emphasize that we have only investigated three correlators out of infinitely many possible ones and have not shown that effective parameters extracted from other correlators all give consistent values

controlled instability and multiscaling in models of epitaxial growth we show that discretized versions of commonly studied nonlinear growth equations have a generic instability in which isolated pillars or grooves on an otherwise flat interface grow in time when their height or depth exceeds a critical value controlling this instability by the introduction of higher order nonlinear terms leads io intermittent behavior characterized by multiexponent scaling of height fluctuations similar to the turbulent behavior found in recent simulations of one dimensional atomistic models of epitaxial growth

anomaly in numerical integrations of the kardar parisi zhang equation we demonstrate that conventional finite difference schemes for direct numerical integration do not approximate the continuum kardar parisi zhang equation the effective diffusion coefficient is found to be inconsistent with the nominal one this is explained by the existence of microscopic roughness in the resulting surfaces

universal asymptotic statistics of maximal relative height in one dimensional solid on solid models we study the probability density function p h m l of the maximum relative height h m in a wide class of one dimensional solid on solid models of finite size l for all these lattice models in the large l limit a central limit argument shows that for periodic boundary conditions p h m l takes a universal scaling form p h m l ensuremath sim sqrt 12 w l ensuremath 1 f mathbf h m sqrt 12 w l mathbf with w l the width of the fluctuating interface and f x the airy distribution function for one instance of these models corresponding to the extremely anisotropic ising model in two dimensions this result is obtained by an exact computation using the transfer matrix technique valid for any lg0 these arguments and exact analytical calculations are supported by numerical simulations which show in addition that the subleading scaling function is also universal up to a nonuniversal amplitude and simply given by the derivative of the airy distribution function f ensuremath x

strong coupling behaviour in discrete kardar parisi zhang equations we present a systematic discretization scheme for the kardar parisi zhang kpz equation which correctly captures the strong coupling properties of the continuum model in particular we show that the scheme contains no finite time singularities in contrast to conventional schemes the implications of these results to i previous numerical integration of the kpz equation and ii the non trivial diversity of universality classes for discrete models of kpz type are examined the new scheme makes the strong coupling physics of the kpz equation more transparent than the original continuum version and allows the possibility of building new continuum models which may be easier to analyse in the strong coupling regime

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

numerical solution of the kardar parisi zhang equation in one two and three dimensions abstract on a course grained level a family of microscopic growth processes may be described by a stochastic differential equation which is solved numerically for surface dimensions d 1 2 and 3 dimensional analysis shows that the spatial discretization parameter has the meaning of an effective coupling constant the numerical stability of the euler integration scheme is discussed for the strong coupling exponents defined by surface width time the following effective values were obtained d 1 0 330 0 004 and d 2 0 24 0 005 considering the width and its ensemble fluctuations at constant dimensionless time the transition between strong and weak coupling phases is located in d 3 for the largest coupling for which reliable data are available we obtain an effective exponent close to the best estimates on discrete models d 3 0 17