convolution generated motion and generalized huygens principles for interface motion a physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry accordingly there is great interest in algorithms that generate such geometric interface motion in this paper we unify and generalize two simple algorithms for constant and mean curvature based interface motion the classical huygens principle and diffusion generated motion we show that the resulting generalization can be viewed both geometrically as a type of huygens principle and algebraically as a convolution generated motion using the geometric algebraic duality from the unification we construct specific convolution generated motion algorithms for a common class of anisotropic curvature dependent motionlaws we validate these algorithms with numerical experiments and show that they can be implemented accurately and efficiently with adaptive resolution and fast fourier transform techniques

internal diffusion limited aggregation parallel algorithms and complexity the computational complexity of internal diffusion limited aggregation dla is examined from both a theoretical and a practical point of view we show that for two or more dimensions the problem of predicting the cluster from a given set of paths is complete for the complexity class cc the subset of p characterized by circuits composed of comparator gates cc completeness is believed to imply that in the worst case growing a cluster of size n requires polynomial time in n even on a parallel computer a parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths and then corrects defects in the guessed cluster through a nonlocal annihilation process the parallel running time of the relaxation algorithm for two dimensional internal dla is studied by simulating it on a serial computer the numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n thus the computational resources needed to grow large clusters are significantly less on average than the worst case analysis would suggest for a parallel machine with k processors we show that random clusters in d dimensions can be generated in mathcal o n k logk n 2 d steps this is a significant speedup over explicit sequential simulation which takes mathcal o n 1 2 d time on average finally we show that in one dimension internal dla can be predicted in mathcal o logn parallel time and so is in the complexity class nc

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

limiting shapes for deterministic centrally seeded growth models we study the rotor router model and two deterministic sandpile models for the rotor router model in z d levine and peres proved that the limiting shape of the growth clus ter is a sphere for the other two models only bounds in dimension 2 are known a unified approach for these models with a new parameter h the initial number of particles at each site allows to prove a number of new limiting shape results in any dimension d 1 for the rotor router model the limiting shape is a sphere for all values of h for one of the sandpile models and h 2d 2 the maximal value the limiting shape is a cube for both sandpile models the limiting shape is a sphere in the limit h finally we prove that the rotor router shape contains a diamond

efficient algorithms for diffusion generated motion by mean curvature the problem of simulating the motion of evolving surfaces with junctions according to some curvature dependent speed arises in a number of applications by alternately diffusing and sharpening characteristic functions for each region a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums however the usual finite difference discretization of these methods is often excessively slow when accurate solutions are sought even in two dimensions we propose a new spectral discretization of these diffusion generated methods which obtains greatly improved efficiency over the usual finite difference approach these efficiency gains are obtained in part through the use of a quadrature based refinement technique by integrating fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients indeed numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization our findings are demonstrated on several examples

number of longest increasing subsequences we study the entropy s of longest increasing subsequences liss i e the logarithm of the number of distinct liss we consider two ensembles of sequences namely random permutations of integers and sequences drawn independent and identically distributed i i d from a limited number of distinct integers using sophisticated algorithms we are able to exactly count the number of liss for each given sequence furthermore we are not only measuring averages and variances for the considered ensembles of sequences but we sample very large parts of the probability distribution p s with very high precision especially we are able to observe the tails of extremely rare events which occur with probabilities smaller than 10 600 we show that the distribution of the entropy of the liss is approximately gaussian with deviations in the far tails which might vanish in the limit of long sequences further we propose a large deviation rate function which fits best to our observed data

large deviations of the length of the longest increasing subsequence of random permutations and random walks we study numerically the length distribution of the longest increasing subsequence lis for random permutations and one dimensional random walks using sophisticated large deviation algorithms we are able to obtain very large parts of the distribution especially also covering probabilities smaller than 10 1000 this enables us to verify for the length of the lis of random permutations the analytically known asymptotics of the rate function and even the whole tracy widom distribution we observe a rather fast convergence in the larger than typical part to this limiting distribution for the length l of lis of random walks no analytical results are known to us we test a proposed scaling law and observe convergence of the tails into a collapse for increasing system size further we obtain estimates for the leading order behavior of the rate functions in both tails

a sharper threshold for bootstrap percolation in two dimensions two dimensional bootstrap percolation is a cellular automaton in which sites become infected by contact with two or more already infected nearest neighbours we consider these dynamics which can be interpreted as a monotone version of the ising model on an n n square with sites initially infected independently with probability p the critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1 2 holroyd determined the sharp first order approximation pc 2 18 log n as n here we sharpen this result proving that the second term in the expansion is log n 3 2 o 1 and moreover determining it up to a poly log log n factor the exponent 3 2 corrects numerical predictions from the physics literature

score statistics of global sequence alignment from the energy distribution of a modified directed polymer and directed percolation problem sequence alignment is one of the most important bioinformatics tools for modern molecular biology the statistical characterization of gapped alignment scores has been a long standing problem in sequence alignment research using a variant of the directed path in random media model we investigate the score statistics of global sequence alignment taking into account in particular the compositional bias of the sequences compared such statistics are used to distinguish accidental similarity due to compositional similarity from biologically significant similarity to accommodate the compositional bias we introduce an extra parameter p indicating the probability for positive matching scores to occur when p is small a high scoring alignment obviously cannot come from compositional similarity when p is large the highest scoring point within a global alignment tends to be close to the end of both sequences in which case we say the system percolates by applying finite size scaling theory on percolating probability functions of various sizes sequence lengths the critical p at infinite size is obtained for alignment of length t the fact that the score fluctuation grows as chi t 1 3 is confirmed upon investigating the scaling form of the alignment score using the kolmogorov smirnov statistics test we show that the random variable if properly scaled follows the tracy widom distributions gaussian orthogonal ensemble for p slightly larger than pc and gaussian unitary ensemble for larger p although these results deepen our understanding of the distribution of alignment scores the use of these results in practical applications remains somewhat heuristic and needs to be further developed nevertheless the possibility of characterizing score statistics for modest system size sequence lengths via proper reparametrization of alignment scores is illustrated

empirical scaling of the length of the longest increasing subsequences of random walks we provide monte carlo estimates of the scaling of the length l n of the longest increasing subsequences of n step random walks for several different distributions of step lengths short and heavy tailed our simulations indicate that barring possible logarithmic corrections with the leading scaling exponent for the heavy tailed distributions of step lengths examined with values increasing as the distribution becomes more heavy tailed and for distributions of finite variance irrespective of the particular distribution the results are consistent with existing rigorous bounds for although in a somewhat surprising manner for random walks with step lengths of finite variance we conjecture that the correct asymptotic behavior of l n is given by and also propose the form for the subleading asymptotics the distribution of l n was found to follow a simple scaling form with scaling functions that vary with accordingly when the step lengths are of finite variance they seem to be universal the nature of this scaling remains unclear since we lack a working model microscopic or hydrodynamic for the behavior of the length of the longest increasing subsequences of random walks

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

applications of extreme value statistics in physics we present a descriptive review of physical problems dealing with extreme values in several fields of physics we consider different physical situations involving random variables that are correlated or not and study the statistics of extremal variables which is relevant for situations where height fluctuations catastrophic events such as material failure or power outrage occur we describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models in many cases however the random variables are correlated in interface problems for example and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution

longest increasing subsequence as expectation of a simple nonlinear stochastic partial differential equation with a low noise intensity we report some observations concerning the statistics of longest increasing subsequences lis we argue that the expectation of lis its variance and apparently the full distribution function appears in statistical analysis of some simple nonlinear stochastic partial differential equation in the limit of very low noise intensity

random walk generated by random permutations of 1 2 3 n 1 we study properties of a non markovian random walk x n l l 0 1 2 n evolving in discrete time l on a one dimensional lattice of integers whose moves to the right or to the left are prescribed by the rise and descent sequences characterizing random permutations of n 1 1 2 3 n 1 we determine exactly the probability of finding the end point xn x n n of the trajectory of such a permutation generated random walk pgrw at site x and show that in the limit n it converges to a normal distribution with a smaller compared to the conventional polya random walk diffusion coefficient we formulate as well an auxiliary stochastic process whose distribution is identical to the distribution of the intermediate points x n l l n which enables us to obtain the probability measure of different excursions and to define the asymptotic distribution of the number of turns of the pgrw trajectories

hermite predictor corrector scheme for regular volterra integral equations and for some integro differential equations of turbulent diffusion abstract in this paper we present a hermite predictor corrector scheme for the nonlinear regular volterra equations of the second kind and some integro differential equations of turbulent diffusion the method developed here closely parallels the corresponding theory for ordinary differential equations

a computation of fermi dirac integrals by asymptotics and the hermite corrector formula we present a computation of fermi dirac integrals based on asymptotic expansions and the hermite corrector formula

a shape theorem for the spread of an infection in ksb we studied the following model for the spread of a rumor or infection there is a gas of so called a particles each of which performs a continuous time simple random walk on z d with jump rate da we assume that just before the start the number of a particles at x na x 0 has a mean a poisson distribution and that the na x 0 x z d are independent in addition there are b particles which perform continuous time simple random walks with jump rate db we start with a finite number of b particles in the system at time 0 the positions of these initial b particles are arbitrary but they are nonrandom the b particles move independently of each other the only interaction occurs when a b particle and an a particle coincide the latter instantaneously turns into a b particle ksb gave some basic estimates for the growth of the set b t x z d a b particle visits x during 0 t

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

convolution generated motion as a link between cellular automata and continuum pattern dynamics cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical biological and ecological systems however for patterns in which sharp interfaces form and propagate automata simulations can exhibit undesirable properties including spurious anisotropy and poor representation of interface curvature effects these simulations are also prohibitively slow when high accuracy is required even in two dimensions also the highly discrete nature of automata models makes theoretical analysis difficult in this paper we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata but based on continuous convolutions rather than discrete sums these convolution generated motions naturally achieve the fine grid limit of the corresponding automata and they are also well suited to numerical and theoretical analysis because of this the desired pattern dynamics can be computed accurately and efficiently using adaptive resolution and fast fourier transform techniques and for a large class of convolutions the limiting interface motion laws can be derived analytically thus convolution generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics this is useful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics in this latter role it also has a number of benefits over the traditional reaction diffusion ginzburg landau continuum pde models of pattern formation which yield true moving interfaces only as singular limits we illustrate the power of this approach with convolution generated motion models for pattern dynamics in developmental biology and excitable media

convergence of nonlocal threshold dynamics approximations to front propagation in this note we prove that appropriately scaled threshold dynamics type algorithms corresponding to the fractional laplacian of order 0 2 converge to moving fronts when 1 the resulting interface moves by weighted mean curvature while for 1 the normal velocity is nonlocal of fractional type the results easily extend to general nonlocal anisotropic threshold dynamics schemes

threshold growth dynamics we study the asymptotic shape of the occupied region for monotone deterministic dynamics in d dimensional euclidean space parametrized by a threshold 0 and a borel set n r d with positive and finite lebesgue measure if a n denotes the oocupied set of the dynamics at integer time n then a n 1 is obtained by adjoining any point x for which the volume of overlap between x n and a n exceeds except in some degenerate cases we prove that n 1 a n converges to a unique limiting shape l starting from any bounded initial region a 0 that is suitably large moreover l is computed as the polar transform for 1 w where w is an explicit width function that depends on n and

exact solution for the kardar parisi zhang equation with flat initial conditions we provide the first exact calculation of the height distribution at arbitrary time t of the continuum kardar parisi zhang kpz growth equation in one dimension with flat initial conditions we use the mapping onto a directed polymer with one end fixed one free and the bethe ansatz for the replicated attractive boson model we obtain the generating function of the moments of the directed polymer partition sum as a fredholm pfaffian our formula valid for all times exhibits convergence of the free energy i e kpz height distribution to the gaussian orthogonal ensemble tracy widom distribution at large time

logarithmic fluctuations for internal dla let each of n particles starting at the origin in z 2 perform simple random walk until reaching a site with no other particles lawler bramson and griffeath proved that the resulting random set a n of n occupied sites is with high probability close to a disk br of radius r p n we show that the discrepancy between a n and the disk is at most logarithmic in the radius i e there is an absolute constant c such that with probability 1 br c log r a r 2 br c log r for all suciently large r

distribution of the time at which n vicious walkers reach their maximal height we study the extreme statistics of n nonintersecting brownian motions vicious walkers over a unit time interval in one dimension using path integral techniques we compute exactly the joint distribution of the maximum m and of the time m at which this maximum is reached we focus in particular on nonintersecting brownian bridges watermelons without wall and nonintersecting brownian excursions watermelons with a wall we discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium dprm with one free end point on the other hand we also check our results using numerical simulations of dyson s brownian motion and confront them with numerical simulations of the polynuclear growth model png and of a model of dprm on a discrete lattice some of the results presented here were announced in a recent letter j rambeau and g schehr europhys lett 91 60006 2010

exact asymptotic results for the bernoulli matching model of sequence alignment finding analytically the statistics of the longest common subsequence lcs of a pair of random sequences drawn from c alphabets is a challenging problem in computational evolutionary biology we present exact asymptotic results for the distribution of the lcs in a simpler yet nontrivial variant of the original model called the bernoulli matching bm model we show that in the bm model for all c the distribution of the asymptotic length of the lcs suitably scaled is identical to the tracy widom distribution of the largest eigenvalue of a random matrix whose entries are drawn from a gaussian unitary ensemble

bethe ansatz in the bernoulli matching model of random sequence alignment for the bernoulli matching model of the sequence alignment problem we apply the bethe ansatz technique via an exact mapping to the five vertex model on a square lattice considering the terracelike representation of the sequence alignment problem we reproduce by the bethe ansatz the results for the averaged length of the longest common subsequence in the bernoulli approximation in addition we compute the average number of nucleation centers of the terraces

statistics of noncoding rnas alignment and secondary structure prediction a new statistical approach to alignment finding the longest common subsequence of two random rna type sequences is proposed we have constructed a generalized dynamic programming algorithm for finding the extreme value of the free energy of two noncoding rnas in our procedure we take into account the binding free energy of two random heteropolymer chains which are capable of forming the cloverleaf like spatial structures typical for rna molecules the algorithm is based on two observations i the standard alignment problem can be considered as a zero temperature limit of a more general statistical problem of binding of two associating heteropolymer chains ii this last problem can be generalized naturally to consider sequences with hierarchical cloverleaf like structures i e of rna type the approach also permits us to perform a secondary structure recovery namely we can predict the optimal secondary structures of interacting rnas in a zero temperature limit knowing only their primary sequences

mutually avoiding paths in random media and largest eigenvalues of random matrices recently it was shown that the probability distribution function pdf of the free energy of a single continuum directed polymer dp in a random potential equivalently to the height of a growing interface described by the kardar parisi zhang kpz equation converges at large scale to the tracy widom distribution the latter describes the fluctuations of the largest eigenvalue of a random matrix drawn from the gaussian unitary ensemble gue and the result holds for a dp with fixed end points i e for the kpz equation with droplet initial conditions a more general conjecture can be put forward relating the free energies of n 1 noncrossing continuum dp in a random potential to the sum of the nth largest eigenvalues of the gue here using replica methods we provide an important test of this conjecture by calculating exactly the right tails of both pdfs and showing that they coincide for arbitrary n

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

internal dla and the stefan problem generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as poisson sources of particles which then perform symmetric random walks with an attractive zero range interaction until they reach the first site which has been visited by fewer than alpha particles at which point they stop sites on which particles are frozen constitute the occupied set we prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one phase stefan problem this is then used to study the asymptotic behavior of the occupied set in two dimensions when the walks are independent with one source at the origin and alpha 1 we obtain in particular that the occupied set is asymptotically a disc of radius k sqrt t where k is the solution of exp k 2 4 pi k 2 settling a conjecture of lawler bramson and griffeath

anisotropic ballistic deposition model with links to the ulam problem and the tracy widom distribution we compute exactly the asymptotic distribution of scaled height in a 1 1 dimensional anisotropic ballistic deposition model by mapping it to the ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers using the known results for the ulam problem we show that the scaled height in our model has the tracy widom distribution appearing in the theory of random matrices near the edges of the spectrum our result supports the hypothesis that various growth models in 1 1 dimensions that belong to the kardar parisi zhang universality class perhaps all share the same universal tracy widom distribution for the suitably scaled height variables

random growth models with polygonal shapes we consider discrete time random perturbations of monotone cellular automata ca in two dimensions under general conditions we prove the existence of half space velocities and then establish the validity of the wulff construction for asymptotic shapes arising from finite initial seeds such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases in many cases exact stability is observed that is for small perturbations the shapes of the deterministic and random processes agree exactly we give a complete characterization of such cases and show that they are prevalent among threshold growth ca with box neighborhood we also design a nontrivial family of ca in which the shape is exactly computable for all values of its probability parameter

a problem in one dimensional diffusion limited aggregation dla and positive recurrence of markov chains we consider the following problem in one dimensional diffusion limited aggregation dla at time we have an aggregate consisting of z 0 r t with r t a positive integer we also have n i t particles at i i r t all these particles perform independent continuous time symmetric simple random walks until the first time t t at which some particle tries to jump from r t 1 to r t the aggregate is then increased to the integers in 0 r t 0 r t 1 so that r t r t 1 and all particles which were at r t 1 at time t are removed from the system the problem is to determine how fast r t grows as a function of t if we start at time 0 with r 0 0 and the n i 0 i i d poisson variables with mean 0 it is shown that if 1 then r t is of order t in a sense which is made precise it is conjectured that r t will grow linearly in t if is large enough

how long does it take for internal dla to forget its initial profile internal dla is a discrete model of a moving interface on the cylinder graph mathbb z n times mathbb z a particle starts uniformly on mathbb z n times 0 and performs simple random walk on the cylinder until reaching an unoccupied site in mathbb z n times mathbb z ge 0 which it occupies forever this operation defines a markov chain on subsets of the cylinder we first show that a typical subset is rectangular with at most logarithmic fluctuations we use this to prove that two internal dla chains started from different typical subsets can be coupled with high probability by adding order n 2 log n particles for a lower bound we show that at least order n 2 particles are required to forget which of two independent typical subsets the process started from

a kpz cocktail shaken not stirred the stochastic partial differential equation proposed nearly three decades ago by kardar parisi and zhang kpz continues to inspire intrigue and confound its many admirers here we i pay debts to heroic predecessors ii highlight additional experimentally relevant aspects of the recently solved 1 1 kpz problem iii use an expanding substrates formalism to gain access to the 3d radial kpz equation and lastly iv examining extremal paths on disordered hierarchical lattices set our gaze upon the fate of d infty kpz clearly there remains ample unexplored territory within the realm of kpz and for the hearty much work to be done especially in higher dimensions where numerical and renormalization group methods are providing a deeper understanding of this iconic equation

approximation schemes for propagation of fronts with nonlocal velocities and neumann boundary conditions the convergence of schemes for propagation of fronts in a bounded domain moving with normal velocities is studied the velocities considered depend on the principal curvatures the normal direction the location as well as some nonlocal properties of the front most of the schemes considered are in essence threshold dynamics type approximation schemes modified for neumann boundary conditions and nonlocal terms the existence and uniqueness of appropriately defined viscosity solutions of the level set equations describing the nonlocal motions is also shown

statistics of layered zigzags a two dimensional generalization of tasep a novel discrete growth model in 2 1 dimensions is presented in three equivalent formulations i directed motion of zigzags on a cylinder ii interacting interlaced tasep layers and iii growing heap over 2d substrate with a restricted minimal local height gradient we demonstrate that the coarsegrained behavior of this model is described by the two dimensional kardar parisi zhang equation the coefficients of different terms in this hydrodynamic equation can be derived from the steady state flow density curve the so called fundamental diagram a conjecture concerning the analytical form of this flow density curve is presented and is verified numerically

the numerical analysis of implicit runge kutta methods for a certain nonlinear integro differential equation the mathematical theory and numerical methods for the nonlinear integro differential equation u t p t u t 0 t k t s u t s u s ds q t 0 t t u 0 u 0 it is proved that the implicit runge kutta methods with m stages are of optimal approximation order p 2m