on the estimation of the large deviations spectrum we propose an estimation algorithm for large deviations spectra of measures and functions the algorithm converges for natural examples of multifractals

self similarity of the branching structure in very large dla clusters and other branching fractals we study the branching structure of very large dla clusters of up to 100 million particles the horton strahler ordering of the branches in these clusters shows a relaxation towards a state with the stream numbers forming a geometric series this behaviour is compared with those of several self similar trees it indicates that dla clusters converge to topologically self similar objects

magnetic hierarchical deposition abstract we consider random deposition of debris or blocks on a line with block sizes following a rigorous hierarchy the linear size equals 1 n in generation n in terms of a rescaling factor without interactions between the blocks this model is described by a logarithmic fractal studied previously which is characterized by a constant increment of the length area or volume upon proliferation we study to what extent the logarithmic fractality survives if each block is equipped with an ising pseudo spin s 1 and the interactions between those spins are switched on ranging from antiferromagnetic to ferromagnetic it turns out that the dependence of the surface topology on the interaction sign and strength is not trivial for instance deep in the ferromagnetic regime our numerical experiments and analytical results reveal a sharp crossover from a euclidean transient consisting of aggregated domains of aligned spins to an asymptotic logarithmic fractal growth in contrast deep into the antiferromagnetic regime the surface roughness is important and is shown analytically to be controlled by vacancies induced by frustrated spins finally in the weak interaction regime we demonstrate that the non interacting model is extremal in the sense that the effect of the introduction of interactions is only quadratic in the magnetic coupling strength in all regimes we demonstrate the adequacy of a mean field approximation whenever vacancies are rare in sum the logarithmic fractal character is robust with respect to the introduction of spatial correlations in the hierarchical deposition process

growth of cayley and diluted cayley trees with two kinds of entities a kinetic growth model derived from the magnetic eden model is introduced in order to simulate the growth of hierarchical structures such as cayley trees we only consider the case where two kinds of entities are competing with each other and can be further subjected to an external field the very relevant case in which both kinds of entities have different coordination numbers is introduced here for the first time and is called the diluted cayley tree physical and geometrical properties of the finite and infinite trees are exactly found and simulated finite size effects are emphasized and illustrated on the global or local magnetization and on the chemical activity asymptotic limits are given in each case the generated patterns can be related to a correlated percolation problem briefly discussed in the appendix

magnetic diffusion limited aggregation an extra degree of freedom is introduced in the well known diffusion limited aggregation model the growth entities are spins taking e g two states that are coupled via a physically relevant interaction potential responsible for a competition process between the two components the presence of an external field favoring one spin species over the other is also considered this model leads to a wide variety of kinetic processes and morphologies distributed in a phase diagram of both growth control parameters i e the coupling energy and the field strength the brownian motion of the spins leads to fractal like structures with a fractal dimension varying from 1 68 plus minus 0 02 to 1 99 plus minus 0 01 depending upon the growth parameters a physical basis is presented to describe the new kinetic processes the spreading and geometry of the two components in fractal clusters have also been investigated the earlier stages of growth are driven by a dominating spin component this finite size process is found to imply a drastic change of the physical and geometrical properties of the cluster during growth history for large clusters the fractal branches can be divided into segments of the same spin species having a characteristic coherence length xi this length xi is found numericallymore to scale as a power law of the segment mass with a critical exponent mu 1 2 plus minus 0 1 less

uniform convergence of martingales in the branching random walk in a discrete time supercritical branching random walk let z n be the point process formed by the n th generation let m lambda be the laplace transform of the intensity measure of z 1 then w n lambda int e lambda x z n dx m lambda n which is the laplace transform of z n normalized by its expected value forms a martingale for any lambda with m lambda finite but nonzero the convergence of these martingales uniformly in lambda for lambda lying in a suitable set is the first main result of this paper this will imply that on that set the martingale limit w lambda is actually an analytic function of lambda the uniform convergence results are used to obtain extensions of known results on the growth of z n nc d with n for bounded intervals d and fixed c this forms the second part of the paper where local large deviation results for z n which are uniform in c are considered finally similar results both on martingale convergence and uniform local large deviations are also obtained for continuous time models including branching brownian motion

universality classes for spreading phenomena a new model with fixed static but continuously tunable kinetic exponents a new and quite tractable model for spreading phenomena is proposed which contains as a special case the eden model and a model for epidemics two exponents are defined one static and one kinetic growth the surprising feature is that the kinetic exponent can be continuously tuned while the static one does not change thus the dynamic universality classes are quite independent of the static one this is the first one cluster growth model showing dynamic universality classes unrelated to static ones and thereby yields insight into a generic feature for growth models

solutions to complex smoothing equations we consider smoothing equations of the form begin aligned x mathop limits text law sum j ge 1 t j x j c end aligned where c t 1 t 2 ldots is a given sequence of random variables and x 1 x 2 ldots are independent copies of x and independent of the sequence c t 1 t 2 ldots the focus is on complex smoothing equations i e the case where the random variables x c t 1 t 2 ldots are complex valued but also more general multivariate smoothing equations are considered in which the t j are similarity matrices under mild assumptions on c t 1 t 2 ldots we describe the laws of all random variables x solving the above smoothing equation these are the distributions of randomly shifted and stopped levy processes satisfying a certain invariance property called u alpha stability which is related to operator semi stability the results are applied to various examples from applied probability and statistical physics

the screening of species in a darwinistic tree like model of evolution abstract numerical and theoretical investigations of a tree like model of self organized critical evolution are presented the screening of species from the evolution process is numerically studied critical exponents for i the avalanche distribution ii the correlation function iii the order parameter and iv the fraction of unscreened species are given a mean field approximation is developed this theory is quite accurate for describing the critical state but is not able to describe the transient regimes i e the self organized slowing down of both order parameter and fraction of unscreened species towards the critical state

on explosions in heavy tailed branching random walks consider a branching random walk on r with offspring distribution z and nonnegative displacement distribution w we say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin in this paper we investigate this phenomenon when the offspring distribution z is heavy tailed under an appropriate condition we are able to characterize the pairs z w for which explosion occurs by demonstrating the equivalence of explosion with a seemingly much weaker event that the sum over generations of the minimum displacement in each generation is finite furthermore we demonstrate that our condition on the tail is best possible for this equivalence to occur we also investigate under additional smoothness assumptions the behavior of mn the position of the particle in generation n closest to the origin when explosion does not occur and hence limn mn

on l convergence of the biggins martingale with complex parameter abstract we prove necessary and sufficient conditions for the l p convergence p 1 of the biggins martingale with complex parameter in the supercritical branching random walk the results and their proofs are much more involved especially in the case p 1 2 than those for the biggins martingale with real parameter our conditions are ultimate in the case p 2 only

hausdorff and packing spectra large deviations and free energy for branching random walks in mathbb r d consider an mathbb r d valued branching random walk brw on a supercritical galton watson tree without any assumption on the distribution of this brw we compute almost surely and simultaneously the hausdorff and packing dimensions of the level sets e k of infinite branches in the boundary of the tree endowed with its standard metric along which the averages of the brw have a given closed connected set of limit points k this goes beyond multifractal analysis which only considers those level sets when k ranges in the set of singletons alpha alpha in mathbb r d we also give a 0 law for the hausdorff and packing measures of the level sets e and compute the free energy of the associated logarithmically correlated random energy model in full generality moreover our results complete the previous works on multifractal analysis by including the levels which do not belong to the range of the gradient of the free energy this covers in particular a situation that was until now badly understood namely the case where a first order phase transition occurs as a consequence of our study we can also describe the whole singularity spectrum of mandelbrot measures as well as the associated free energy function or lq spectrum when a first order phase transition occurs

exact inference for categorical data recent advances and continuing controversies methods for exact small sample analyses with categorical data have been increasingly well developed in recent years a variety of exact methods exist primarily using the approach that eliminates unknown parameters by conditioning on their sufficient statistics in addition a variety of algorithms now exist for implementing the methods this paper briefly summarizes the exact approaches and describes recent developments controversy continues about the appropriateness of some exact methods primarily relating to their conservative nature because of discreteness this issue is examined for two simple problems in which discreteness can be severe interval estimation of a proportion and the odds ratio in general adjusted exact methods based on the mid p value seem a reasonable way of reducing the severity of this problem

spines skeletons and the strong law of large numbers for superdiffusions consider a supercritical superdiffusion x t t 0 on a domain d r d with branching mechanism x z x z x z 2 0 e zy 1 zy x dy the skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion we use this decomposition to derive the strong law of large numbers slln for a wide class of superdiffusions from the corresponding result for branching particle diffusions that is we show that for suitable test functions f and starting measures f x t p f x t w p almost surely as t where w is a finite non deterministic random variable characterized as a martingale limit our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the slln for superdiffusions the result covers many of the key examples of interest and in particular proves a conjecture by fleischmann and swart stochastic process appl 106 2003 141 165 for the super wright fisher diffusion

diffusion and reaction in random media and models of evolution processes a diffusion equation including source terms representing randomly distributed sources and sinks is considered for quasilinear growth rates the eigenvalue problem is equivalent to that of the quantum mechanical motion of electrons in random fields correspondingly there exist localized and extended density distributions dependent on the statistics of the random field and on the dimension of the space besides applications in physics nonequilibrium processes in pumped disordered solid materials a new evolution model is discussed which considers evolution as hill climbing in a random landscape

analyticity of the partition function of the random energy model we consider the random energy model with complex temperature for beta lying in a suitable set the function zn beta ezn beta is at the thermodynamic limit an analytic function of beta analytic properties of the high temperature free energy follow directly

seneta heyde norming in the branching random walk malization of the general crump mode jagers branching process is obtained in this case the convergence holds almost surely the results rely heavily on a detailed study of the functional equation that the laplace transform of the limit must satisfy 1 introduction this paper considers the usual supercritical branching random walk thus ignoring the spatial element the population grows like a supercritical galton watson process the initial ancestor is at the origin of the real line and the positions of her children are given by a point process z each of these children has children in the same way in that the positions of each family relative to the parent are given by an independent copy of z and so on individuals are labelled by their line of descent so if u i i 1 n then u is the i th child of the i th child of the i th child of the initial nn 11 ancestor now let u be the generation in which u is born and write v u if v is a strict ancestor of u let t be the set of all people ever born which can be viewed as a tree with the population members as the nodes sums products sets and so on defined with an index ranging over individual s labels will be restricted to those actually born without this being made 4

scaling properties of the surface of the eden model in d 2 3 4 the surface of the eden model is investigated numerically by finite size scaling using a strip geometry three different versions are studied and it shows that the one mostly used previously exhibits strong finite size corrections in two dimensions it is found that the surface thickness simply grows as the square root of the width of the strip for infinitely long strips this result as well as the results in d 3 and d 4 suggest that the surface of the eden shares some properties with the equilibrium models used to describe the roughening transition moreover it is found that the surface thickness scales differently with the height of the cluster for infinitely large strips

cluster structure of the eden model and directed polymers in a random potential the cluster structures of the eden model and directed polymers in a random potential are described the heights hs and widths ws of each cluster grows as hs approximately sv and ws approximately sv perpendicular to with v sub z d 1 z and vperpendicular to 1 d 1 z in d dimensions where s is the size of the cluster and z is the dynamic critical exponent generalization to include the cluster size distribution is discussed

power and sample size calculations for exact conditional tests with ordered categorical data we develop an algorithm for computing sample sizes equal or unequal for categorical data we illustrate its use in the two sample setting using the wilcoxon rank sum statistic but the algorithm accommodates the entire class of linear rank statistics and can be extended to include nonlinear rank statistics as well the sample size determinations can be based on either exact power or on a very precise monte carlo estimate of it to reduce the computations further power can be computed as a function of asymptotic critical values when the number of categories is not too small for the wilcoxon statistic we show that this approximation works well if there are more than five response categories

complex random energy model zeros and fluctuations the partition function of the random energy model at inverse temperature beta is a sum of random exponentials mathcal z n beta sum k 1 n exp beta sqrt n x k where x 1 x 2 ldots are independent real standard normal random variables random energies and n log n we study the large n limit of the partition function viewed as an analytic function of the complex variable beta we identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature we prove limit theorems for the random partition function at complex beta both on the logarithmic scale and on the level of limiting distributions our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent

travelling waves for the fkpp equation via probabilistic arguments we outline a completely probabilistic study of travelling wave solutions of the fkpp reaction diffusion equation that are monotone and connect 0 to 1 the necessary asymptotics of such travelling waves are proved using martingale and brownian motion techniques recalling the connection between the fkpp equation and branching brownian motion through the work of mckean and neveu we show how the necessary asymptotics and results about branching brownian motion combine to give the existence and uniqueness of travelling waves of all speeds greater than or equal to the critical speed

the number of open paths in an oriented ρ percolation model we study the asymptotic properties of the number of open paths of length n in an oriented percolation model we show that this number is en 1 o 1 as n the exponent is deterministic it can be expressed in terms of the free energy of a polymer model and it can be explicitly computed in some range of the parameters moreover in a restricted range of the parameters we even show that the number of such paths is n 1 2wen 1 o 1 for some nondegenerate random variable w we build on connections with the model of directed polymers in random environment and we use techniques and results developed in this context

a simple approximation for calculating sample sizes for detecting linear trend in proportions a simple approximate formula for sample sizes for detecting a linear trend in proportions is derived the formulas for both the uncorrected and corrected cochran armitage test are given for two binomial proportions these reduce to those given by casagrande pike and smith 1978 biometrics 34 483 486 some numerical results of a power study for small sample sizes show that the nominal power corresponding to the approximate sample size is a reasonably good approximation to the actual power

irreversible growth of binary mixtures on small world networks binary mixtures growing on small world networks under far from equilibrium conditions are studied by means of extensive monte carlo simulations for any positive value of the shortcut fraction of the network p 0 the system undergoes a continuous order disorder phase transition while it is noncritical in the regular lattice limit p 0 using finite size scaling relations the phase diagram is obtained in the thermodynamic limit and the critical exponents are evaluated the small world networks are thus shown to trigger criticality a phenomenon analogous to similar observations reported recently in the investigation of equilibrium systems

power comparisons for tests of trend in dose response studies the cochran armitage test for trend is a popular statistical procedure for detecting increasing or decreasing probabilities of response when a categorical exposure is ordered such associations may arise in a variety of biomedical research settings particularly in dose response designs such as carcinogenicity experiments previously computing limitations mandated the use of the asymptotic trend test but with the availability of new algorithms increased computing power and appropriate software the exact trend test is now a practical option nevertheless the exact test is sometimes criticized on the grounds that it is conservative in this paper we investigate the implications of this conservatism by comparing the true type i error and power of three alternative tests of trend the asymptotic test the exact test and an admissible exact test proposed by cohen and sackrowitz the computations are performed by an extension to the network algorithm of mehta et al this allows us to make precise power comparisons between the tests under any given design without resorting to simulation we show how this tool can guide investigators in choosing the most appropriate test by considering the design of two year carcinogenicity studies carried out by the national toxicology program we additionally compare the tests for various other combinations of sample sizes and number of groups or levels of exposure we conclude that the asymptotic test while more powerful where it is valid generally does not preserve the type i error this violation of the a priori testing level can be greatly affected by imbalance in the data or unequal spacing of dose levels

criticality in simple models of evolution we consider two apparently similar models of biological evolution that have been claimed to exhibit self organized critical behavior a careful reanalysis of these models including several analytic results for one of them suggests that they are qualitatively different we demonstrate the limitations of the mean field description of these systems we argue that a more precise definition of self organized criticality is desirable and establish several criteria in this connection

phyllotaxis an interdisciplinary phenomenon abstract regular phyllotaxis is largely non uniform in the sense that 1 2 and the golden mean are the most frequently observed values of the characteristic parameter divergence the different approaches to this enigmatic phenomenon form a fascinating chapter in the history of science more recently different models have been independently suggested in an attempt to recreate the phenomenon these are dynamic systems in discrete time founded on a simple inhibiting hypothesis ih we analyse the divergences selected by these models and consequently we investigate the ih capability to account for regular phyllotaxis this capability is conditional it depends on the convex concave shape of the elementary inhibitions and involves considering the ih from a statistical standpoint our conclusions are essentially new and they partly conflict with previous predictions because of its generality our analysis applies to botanical as well as to physical systems such as vortex networks

multifractal analysis of complex random cascades we achieve the multifractal analysis of a class of complex valued statistically self similar continuous functions for we use multifractal formalisms associated with pointwise oscillation exponents of all orders our study exhibits new phenomena in multifractal analysis of continuous functions in particular we find examples of statistically self similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval 0

exact power and sample size computations for the cochran armitage trend test the cochran armitage test of trend is commonly used to determine if a dose response relationship exists in a wide variety of biomedical settings including clinical trials carcinogenicity studies and toxicological risk assessment for small sparse or unbalanced data sets one generally adopts the exact version of the cochran armitage test of trend for which numerical algorithms and software are readily available no corresponding algorithms or software exist for the exact power and samplesize computations that are needed at design time prior to gathering the data this paper develops a network algorithm for computing the exact power of the cochran armitage test of trend and applies it to several examples thereby demonstrating that the corresponding asymptotic power computations can be rather misleading

martingales in self similar growth fragmentations and their connections with random planar maps the purpose of the present work is twofold first we develop the theory of general self similar growth fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework in particular we establish many to one formulas for growth fragmentations and define the notion of intrinsic area of a growth fragmentation second we identify a distinguished family of growth fragmentations closely related to stable levy processes which are then shown to arise as the scaling limit of the perimeter process in markovian explorations of certain random planar maps with large degrees which are roughly speaking the dual maps of the stable maps of le gall and miermont in ann probab 39 1 69 2011 as a consequence of this result we are able to identify the law of the intrinsic area of these distinguished growth fragmentations this generalizes a geometric connection between large boltzmann triangulations and a certain growth fragmentation process which was established in bertoin et al ann probab accepted

self organized critical model of biological evolution abstract a punctuated equilibrium model of biological evolution with relative fitness between different species being the fundamental driving force of evolution is introduced mutation is modeled as a fitness updating cellular automation process where the change in fitness after mutation follows a gaussian distribution with mean x 0 and standard deviation scaling behaviors are observed in our numerical simulation indicating that the model is self organized critical besides the numerical experiment suggests that models with different x and belongs to the same universality class

mean field theory for a simple model of evolution a simple dynamical model for darwinian evolution on its slowest time scale is analyzed its mean field theory is formulated and solved a random neighbor version of the model is simulated as is a one dimensional version in one dimension the dynamics can be described in terms of a repetitious random walker and anomalous diffusion with exponent 0 4 in all cases the model self organizes to a robust critical attractor