error correcting codes in projective space the projective space of order n over the finite field bbfq denoted here as pq n is the set of all subspaces of the vector space bbfqn the projective space can be endowed with the distance function d u v dimu dimv 2 dim u v which turns pq n into a metric space with this an n m d code bbc in projective space is a subset of pq n of size m such that the distance between any two codewords subspaces is at least d koetter and kschischang recently showed that codes in projective space are precisely what is needed for error correction in networks an n m d code can correct t packet errors and packet erasures introduced adversarially anywhere in the network as long as 2t 2 d this motivates our interest in such codes in this paper we investigate certain basic aspects of coding theory in projective space first we present several new bounds on the size of codes in pq n which may be thought of as counterparts of the classical bounds in coding theory due to johnson delsarte and gilbert varshamov some of these are stronger than all the previously known bounds at least for certain code parameters we also present several specific constructions of codes and code families in pq n finally we prove that nontrivial perfect codes in pq n do not exist

on the capacity of multiplicative finite field matrix channels this paper deals with the multiplicative finite field matrix channel a discrete memoryless channel whose input and output are matrices over a finite field related by a multiplicative transfer matrix the model considered here assumes that all transfer matrices with the same rank are equiprobable so that the channel is completely characterized by the rank distribution of the transfer matrix this model is seen to be more flexible than previously proposed ones in describing random linear network coding systems subject to link erasures while still being sufficiently simple to allow tractability the model is also conservative in the sense that its capacity provides a lower bound on the capacity of any channel with the same rank distribution a main contribution is to express the channel capacity as the solution of a convex optimization problem which can be easily solved by numerical computation for the special case of constant rank input a closed form expression for the capacity is obtained the behavior of the channel for asymptotically large field size or packet length is studied and it is shown that constant rank input suffices in this case finally it is proved that the well known approach of treating inputs and outputs as subspaces is information lossless even in this more general model

extremely large scale simulation of a kardar parisi zhang model using graphics cards the octahedron model introduced recently has been implemented onto graphics cards which permits extremely large scale simulations via binary lattice gases and bit coded algorithms we confirm scaling behavior belonging to the two dimensional kardar parisi zhang universality class and find a surface growth exponent 0 2415 15 on 2 17 2 17 systems ruling out 1 4 suggested by field theory the maximum speedup with respect to a single cpu is 240 the steady state has been analyzed by finite size scaling and a growth exponent 0 393 4 is found correction to scaling exponent are computed and the power spectrum density of the steady state is determined we calculate the universal scaling functions and cumulants and show that the limit distribution can be obtained by the sizes considered we provide numerical fitting for the small and large tail behavior of the steady state scaling function of the interface width

the crossover phenomena in surface growth models with height dependent noise abstract in this paper we consider several known growth processes with height dependent noise this type of noise is interesting from a theoretical standpoint for example it paves the way to the derivation of the exact height distribution of the kpz equation through the hopf cole transformation in addition it may have implications for experimental growth processes using numerical methods we observe that adding such a noise to different growth processes can change their universality class or ruin the scaling laws in the case of mullins herring equation a two fold cross over is observed

exact formulas for the form factors of local operators in the lieb liniger model we present exact formulas for the form factors of local operators in the repulsive lieb liniger model at finite size these are essential ingredients for both numerical and analytical calculations from the theory of algebraic bethe ansatz it is known that the form factors of local operators satisfy a particular type of recursive relations we show that in some cases these relations can be used directly to derive practical expressions in terms of the determinant of a matrix whose dimension scales linearly with the system size our main results are determinant formulas for the form factors of the operators and r 0 for arbitrary integer r where are the usual field operators from these expressions we also derive the infinite size limit of the form factors of these local operators in the attractive regime

nonperturbative renormalization group for the stationary kardar parisi zhang equation scaling functions and amplitude ratios in 1 1 2 1 and 3 1 dimensions we investigate the strong coupling regime of the stationary kardar parisi zhang equation for interfaces growing on a substrate of dimension d 1 2 and 3 using a nonperturbative renormalization group nprg approach we compute critical exponents correlation and response functions extract the related scaling functions and calculate universal amplitude ratios we work with a simplified implementation of the second order in the response field approximation proposed in a previous work phys rev e 84 061150 2011 and phys rev e 86 019904 e 2012 which greatly simplifies the frequency sector of the nprg flow equations while keeping a nontrivial frequency dependence for the two point functions the one dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second order nprg equations and with the exact scaling function furthermore the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d 2 and d 3 we argue that our ansatz is reliable up to d symbol see text 3 5

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

iterative coding for network coding we consider communication over a noisy network under randomized linear network coding possible error mechanisms include node or link failures byzantine behavior of nodes or an overestimate of the network min cut building on the work of kotter and kschischang we introduce a systematic oblivious random channel model within this model codewords contain a header this is the systematic part the header effectively records the coefficients of the linear encoding functions thus simplifying the decoding task under this constraint errors are modeled as random low rank perturbations of the transmitted codeword we compute the capacity of this channel and we define an error correction scheme based on random sparse graphs and a low complexity decoding algorithm by optimizing over the code degree profile we show that this construction achieves the channel capacity in complexity which is jointly quadratic in the number of coded information bits and sublogarithmic in the error probability

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

exact scaling functions for one dimensional stationary kpz growth we determine the stationary two point correlation function of the one dimensional kpz equation through the scaling limit of a solvable microscopic model the polynuclear growth model the equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a riemann hilbert problem related to the painleve ii equation we solve these equations numerically with very high precision and compare our up to numerical rounding exact result with the prediction of colaiori and moore 1 obtained from the mode coupling approximation

efficient encoding of low density parity check codes low density parity check ldpc codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy constrained random code ensembles and iterative decoding algorithms we consider the encoding problem for ldpc codes more generally we consider the encoding problem for codes specified by sparse parity check matrices we show how to exploit the sparseness of the parity check matrix to obtain efficient encoders for the 3 6 regular ldpc code for example the complexity of encoding is essentially quadratic in the block length however we show that the associated coefficient can be made quite small so that encoding codes even of length n spl sime 100000 is still quite practical more importantly we show that optimized codes actually admit linear time encoding

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

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

spatial correlations of the 1d kpz surface on a flat substrate we study the spatial correlations of the one dimensional kpz surface for the flat initial condition it is shown that the multi point joint distribution for the height is given by a fredholm determinant with its kernel in the scaling limit explicitly obtained this may also describe the dynamics of the largest eigenvalue in the goe dyson s brownian motion model our analysis is based on a reformulation of the determinantal green s function for the totally asep in terms of a vicious walk problem

scaling limit for the space time covariance of the stationary totally asymmetric simple exclusion process the totally asymmetric simple exclusion process tasep on the one dimensional lattice with the bernoulli measure as initial conditions 0 1 is stationary in space and time let nt j be the number of particles which have crossed the bond from j to j 1 during the time span 0 t for open image in new window we prove that the fluctuations of nt j for large t are of order t1 3 and we determine the limiting distribution function open image in new window which is a generalization of the gue tracy widom distribution the family open image in new window of distribution functions have been obtained before by baik and rains in the context of the png model with boundary sources which requires the asymptotics of a riemann hilbert problem in our work we arrive at open image in new window through the asymptotics of a fredholm determinant open image in new window is simply related to the scaling function for the space time covariance of the stationary tasep equivalently to the asymptotic transition probability of a single second class particle

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

probability distribution of the free energy of a directed polymer in a random medium we calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder by using the fact that the nth moment of the partition function is given by the ground state energy of a quantum problem of n interacting particles on a ring of length l we write an integral equation allowing to expand these moments in powers of the strength of the disorder gamma or in powers of n for n small and n approximately lgamma 1 2 the moments take a scaling form which allows us to describe all the fluctuations of order 1 l of the free energy per unit length of the directed polymer the distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process indicating that it is characteristic of all the systems described by the kardar parisi zhang equation in 1 1 dimensions

two time distribution function in one dimensional random directed polymers the explicit expression for the two time free energy distribution function in one dimensional random directed polymers is derived in terms of the bethe ansatz replica technique it is shown that such type of the distribution function can be represented in the form of a generalized two dimensional fredholm determinant

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

maximum relative height of elastic interfaces in random media the distribution of the maximal relative height mrh of self affine one dimensional elastic interfaces in a random potential is studied we analyze the ground state configuration at zero driving force and the critical configuration exactly at the depinning threshold both for the random manifold and random periodic universality classes these configurations are sampled by exact numerical methods and their mrh distributions are compared with those with the same roughness exponent and boundary conditions but produced by independent fourier modes with normally distributed amplitudes using pickands theorem we derive an exact analytical description for the right tail of the latter after properly rescaling the mrh distributions we find that corrections from the gaussian independent modes approximation are in general small as previously found for the average width distribution of depinning configurations in the large size limit all corrections are finite except for the ground state in the random periodic class whose mrh distribution becomes for periodic boundary conditions indistinguishable from the airy distribution we find that the mrh distributions are in general sensitive to changes of boundary conditions

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

communication over finite field matrix channels this paper is motivated by the problem of error control in network coding when errors are introduced in a random fashion rather than chosen by an adversary an additive multiplicative matrix channel is considered as a model for random network coding the model assumes that n packets of length m are transmitted over the network and up to t erroneous packets are randomly chosen and injected into the network upper and lower bounds on capacity are obtained for any channel parameters and asymptotic expressions are provided in the limit of large field or matrix size a simple coding scheme is presented that achieves capacity in both limiting cases the scheme has decoding complexity o n 2 m and a probability of error that decreases exponentially both in the packet length and in the field size in bits extensions of these results for coherent network coding are also presented

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

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

the one dimensional kpz equation and its universality class our understanding of the one dimensional kpz equation alias noisy burgers equation has advanced substantially over the past 5 years we provide a non technical review where we limit ourselves to the stochastic pde and lattice type models approximating it

finite field matrix channels for network coding in 2010 silva et al studied certain classes of finite field matrix channels in order to model random linear network coding where exactly t random errors are introduced in this paper we consider a generalization of these matrix channels where the number of errors is not required to be constant indeed the number of errors may follow any distribution we show that a capacity achieving input distribution can always be taken to have a very restricted form the distribution should be uniform given the rank of the input matrix this result complements and is inspired by a paper of nobrega et al which establishes a similar result for a class of matrix channels that model network coding with link erasures our result shows that the capacity of our channels can be expressed as maximization over probability distributions on the set of possible ranks of input matrices a set of linear rather than exponential size

a rank metric approach to error control in random network coding the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of rotter and kschischang a large class of constant dimension subspace codes is investigated it is shown that codes in this class can be easily constructed from rank metric codes while preserving their distance properties moreover it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank metric codes where partial information about the error is available this partial information may be in the form of erasures knowledge of an error location but not its value and deviations knowledge of an error value but not its location taking erasures and deviations into account when they occur strictly increases the error correction capability of a code if mu erasures and delta deviations occur then errors of rank t can always be corrected provided that 2t les d 1 mu delta where d is the minimum rank distance of the code for gabidulin codes an important family of maximum rank distance codes an efficient decoding algorithm is proposed that can properly exploit erasures and deviations in a network coding application where n packets of length m over f q are transmitted the complexity of the decoding algorithm is given by o dm operations in an extension field f qn

ground state energy of a non integer number of particles with δ attractive interactions we show how to dene and calculate the ground state energy of a system of quantum particles with attractive interactions when the number of particles n is non integer the question is relevant to obtain the probability distribution of the free energy of a directed polymer in a random medium when one expands the ground state energy in powers of the interaction all the coecients of the perturbation series are polynomials in n allowing to dene the perturbation theory for non integer n we develop a procedure to calculate all the cumulants of the free energy of the directed polymer and we give explicit although complicated expressions of the rst three cumulants c 2000 elsevier science b v all rights reserved

integrability of zero dimensional replica field theories at β 1 building on insights from the theory of integrable lattices the integrability is claimed for nonlinear replica sigma models derived in the context of real symmetric random matrices specifically the fermionic and the bosonic replica partition functions are proven to form a single supersymmetric pfaff kp hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density density eigenvalue correlation function in the infinite dimensional gaussian orthogonal ensemble implications of the formalism outlined are briefly discussed

prediction of wall pressure fluctuation in turbulent flows with an immersed boundary method the objective of this paper is to assess the accuracy and efficiency of the immersed boundary ib method to predict the wall pressure fluctuations in turbulent flows where the flow dynamics in the near wall region is fundamental to correctly predict the overall flow the present approach achieves sufficient accuracy at the immersed boundary and overcomes deficiencies in previous ib methods by introducing additional constraints a compatibility for the interpolated velocity boundary condition related to mass conservation and the formal decoupling of the pressure on this surfaces the immersed boundary approximated domain method ib adm developed in the present study satisfies these conditions with an inexpensive computational overhead the ib adm correctly predicts the near wall velocity pressure and scalar fields in several example problems including flows around a very thin solid object for which incorrect results were obtained with previous ib methods in order to have sufficient near wall mesh resolution for les and dns computations the present approach uses local mesh refinement the present method has been also successfully applied to computation of the wall pressure space time correlation in dns of turbulent channel flow on grids not aligned with the boundaries when applied to a turbulent flow around an airfoil the computed flow statistics the mean rms flow field and power spectra of the wall pressure are in good agreement with experiment

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

a determinantal formula for the goe tracy widom distribution investigating the long time asymptotics of the totally asymmetric simple exclusion process sasamoto obtains rather indirectly a formula for the tracy widom distribution for the gaussian orthogonal ensemble we establish that his novel formula indeed agrees with more standard expressions

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process