asymptotics in asep with step initial condition in previous work the authors considered the asymmetric simple exclusion process on the integer lattice in the case of step initial condition particles beginning at the positive integers there it was shown that the probability distribution for the position of an individual particle is given by an integral whose integrand involves a fredholm determinant here we use this formula to obtain three asymptotic results for the positions of these particles in one an apparently new distribution function arises and in another the distribution function f2 arises the latter extends a result of johansson on tasep to asep and hence proves kpz universality for asep with step initial condition

monte carlo study of the point for collapsing trees we investigate the collapse transition of lattice trees with nearest neighbor attraction in two and three dimensions two methods are used 1 a stochastic optimization process of the robbins monro type which is designed solely to locate the maximum value of the specific heat and 2 umbrella sampling which is designed to sample data over a wide temperature range as well as to combat the quasiergodicity of metropolis algorithms in the collapsed phase we find good evidence that the transition is second order with a divergent specific heat and that the divergence of the specific heat coincides with the metric collapse

density scaling a new monte carlo technique in statistical mechanics abstract we demonstrate the feasibility of using umbrella sampling to do monte carlo markov sampling runs each covering a substantial range of density density scaling monte carlo or dsmc one can obtain in this way not only the usual canonical averages but also the relative free energy as a function of density to test this it has been applied to systems for which there are some previous reliable results the hard sphere system and the restricted primitive model of 1 1 and 2 2 electrolytes the method proves to be startlingly powerful in that very extensive results can be obtained with very few dsmc runs an important further motivation is the prospect of using the technique to study phase transition regions

kardar parisi zhang universality class and the anchored toom interface we revisit the anchored toom interface and use kardar parisi zhang scaling theory to argue that the interface fluctuations are governed by the airy1 process with the role of space and time interchanged the predictions which contain no free parameter are numerically well confirmed for space time statistics in the stationary state in particular the spatial fluctuations of the interface computed numerically agree well with those given by the goe edge distribution of tracy and widom

duality relations for asymmetric exclusion processes we derive duality relations for a class ofu q su 2 symmetric stochastic processes including among others the asymmetric exclusion process in one dimension like the known duality relations for symmetric hopping processes these relations express certainm point correlation functions inn particle systems n m in terms of sums of correlation functions of the same system but with onlym particles for the totally asymmetric case we obtain exact expressions for some boundary density correlation functions the dynamical exponent for these correlators isz 2 which is different from the dynamical exponent for bulk density correlations which is known to bez 3 2

exact solution for the stationary kardar parisi zhang equation we obtain the first exact solution for the stationary one dimensional kardar parisi zhang equation a formula for the distribution of the height is given in terms of a fredholm determinant which is valid for any finite time t the expression is explicit and compact enough so that it can be evaluated numerically furthermore by extending the same scheme we find an exact formula for the stationary two point correlation function

nonuniversality of front fluctuations for compact colonies of nonmotile bacteria the front of a compact bacterial colony growing on a petri dish is a paradigmatic instance of non equilibrium fluctuations in the celebrated eden or kardar parisi zhang kpz universality class while in many experiments the scaling exponents crucially differ from the expected kpz values the source of this disagreement has remained poorly understood we have performed growth experiments with b subtilis 168 and e coli atcc 25922 under conditions leading to compact colonies in the classically alleged eden regime where individual motility is suppressed non kpz scaling is indeed observed for all accessible times kpz asymptotics being ruled out for our experiments due to the monotonic increase of front branching with time simulations of an effective model suggest the occurrence of transient nonuniversal scaling due to diffusive morphological instabilities agreeing with expectations from detailed models of the relevant biological reaction diffusion processes

kardar parisi zhang equation in the weak noise limit pattern formation and upper critical dimension we extend the previously developed nonperturbative weak noise scheme applied to the noisy burgers equation in one dimension to the kardar parisi zhang equation for a growing interface in arbitrary dimensions by means of the cole hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes in the cole hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear schrodinger equation subsequently boosted to finite velocity by a galilei transformation we discuss the dynamics of the pattern formation and briefly the superimposed linear modes implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector in the dipole sector we obtain the hurst exponent h 3 d 4 d or dynamic exponent zdip 4 d 3 d for the random walk of growth modes below d 3 the dipole growth modes show anomalous diffusion above d 3 the dipole growth modes freeze finally applying derrick s theorem based on constrained minimization we show that the upper critical dimension is d 4 in the sense that growth modes cease to exist above this dimension

effect of a columnar defect on the shape of slow combustion fronts we report experimental results for the behavior of slow combustion fronts in the presence of a columnar defect with enhanced or reduced driving and compare them with those of mean field theory we also compare them with simulation results for an analogous problem of driven flow of particles with hard core repulsion asep and a single defect bond with a different hopping probability the difference in the shape of the front profiles for enhanced vs reduced driving in the defect clearly demonstrates the existence of a kardar parisi zhang type nonlinear term in the effective evolution equation for the slow combustion fronts we also find that slow combustion fronts display a faceted form for large enough enhanced driving and that there is a corresponding increase then in the average front speed this increase in the average front speed disappears at a nonzero enhanced driving in agreement with the simulated behavior of the asep 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

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

fixed degree for fuzzy mappings and a generalization of ky fan s theorem abstract in this paper the concept of some properties of fixed degree for fuzzy mappings are introduced and studied the results presented generalize and unify some recent results

caristi s fixed point theorem for fuzzy mappings and ekeland s variational principle abstract the purpose of this paper is to generalize caristi s fixed point theorem to fuzzy mappings and to obtain two formally stronger caristi type fixed point and common fixed point theorems for set valued mappings and sequences of set valued mappings the results presented in this paper generalize the corresponding results of caristi trans amer math soc 215 1976 241 251 and kasahara math seminar notes 35 2 1975 229 232 in addition by a simple method the equivalence between caristi s fixed point theorem for fuzzy mappings and ekeland s variational principle is shown this result improves and extends some important results of dancs et al kirk shi and sullivan

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

critical exponents of the kpz equation via multi surface coding numerical simulations we study the kpz equation in d 2 3 and 4 spatial dimensions by using a restricted solid on solid discretization of the surface we measure the critical exponents very precisely and we show that the rational guess is not appropriate and that d 4 is not the upper critical dimension we are also able to determine very precisely the exponent of the sub leading scaling corrections that turns out to be close to unity in all cases we introduce and use a multi surface coding technique that allows a gain of the order of 30 fold over usual numerical simulations

virial coefficients for hard discs and hard spheres the sixth seventh and eighth virial coefficients of hard discs and hard spheres are evaluated numerically monte carlo integration i improve on the best previous estimates for the seventh virial coefficients and the integration of the eighth virial coefficient is new the best estimates for these coefficients for hard discs are b7 b6 0 114 86 7 and b8 b7 0 065 14 8 and for hard spheres b7 b6 0 01307 7 and b8 b7 0 00432 10 b is the second virial coefficient in each case pade approximations to the excess pressure and the excess free energy are computed from these results and compared to data otherwise obtained

universality of fluctuations in the kardar parisi zhang class in high dimensions and its upper critical dimension we show that the theoretical machinery developed for the kardar parisi zhang kpz class in low dimensions is obeyed by the restricted solid on solid model for substrates with dimensions up to d 6 analyzing different restriction conditions we show that the height distributions of the interface are universal for all investigated dimensions it means that fluctuations are not negligible and consequently the system is still below the upper critical dimension at d 6 the extrapolation of the data to dimensions d 7 predicts that the upper critical dimension of the kpz class is infinite

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

polymers on disordered hierarchical lattices a nonlinear combination of random variables the problem of directed polymers on disordered hierarchical and hypercubic lattices is considered for the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way we present the results of numerical simulations of two hierarchical lattices finding evidence of a phase transition in one case for a limiting case we extend the perturbation theory developed by derrida and griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties overlaps of the model in this limit we obtain an interpolation formula allowing one to obtain the noninteger moments of the partition function from the integer moments we obtain bounds for the transition temperature for hierarchical and hypercubic lattices and some similarities between the problem on the two different types of lattice are discussed

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

fixed point theorems for fuzzy mappings in complete metric spaces in this paper we are concerned with mathematical properties of the previous contractive types of set valued maps or fuzzy mappings on a metric space see for instance nadler pacific j math 30 1969 475 488 heilpern j math anal appl 83 1981 566 569 papageorgiou nonlinear anal theory methods appl 7 1983 763 770 bose and sahani fuzzy sets and systems 21 1987 53 58 som and mukherjee fuzzy sets and systems 33 1989 213 219 park and jeong fuzzy sets and systems 59 1993 231 235 87 1997 111 116 lee et al fuzzy sets and systems 101 1999 143 152 and we prove some crucial theorems relating fixed points of fuzzy mappings on complete metric spaces first we prove a theorem by using the concept of w distance kada et al math japonica 44 1996 381 391 which generalizes the results of amemiya and takahashi fuzzy sets and systems 114 2000 469 476 and kada et al 1996 next we prove two theorems for fuzzy mappings which are connected with fixed point theorems for set valued maps then we shall observe that our theorems can be used to obtain almost all results proved so far concerning fixed points of fuzzy mappings or set valued maps on complete metric spaces and therefore that our results imply the existence of fixed points of various contractive types of fuzzy mappings or set valued maps on complete metric spaces

upper critical dimension of the kardar parisi zhang equation numerical results for the directed polymer model in 1 4 dimensions in various types of disorder are presented the results are obtained for a system size that is considerably larger than considered previously for the extreme strong disorder case min max system associated with the directed percolation model the expected value of the meandering exponent 0 5 is clearly revealed with very weak finite size effects for the weak disorder case associated with the kardar parisi zhang equation finite size effects are stronger but the value of is clearly seen in the vicinity of 0 57 in systems with strong disorder it is expected that the system will cross over sharply from min max behavior at short chains to weak disorder behavior at long chains our numerical results agree with that expectation to complete the picture we obtain the energy fluctuation exponent for weak disorder and we find that the value of is in the vicinity of 0 14 thus the meandering exponent and the energy fluctuation exponent obey the strong coupling scaling relation 2 1 our results indicate that 1 4 is not the upper critical dimension in the weak disorder case and thus 4 1 does not seem to be the upper critical dimension for the kardar parisi zhang equation

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

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

rate of convergence for polymers in a weak disorder we consider directed polymers in random environment on the lattice z d at small inverse temperature and dimension d 3 then the normalized partition function w n is a regular martingale with limit w we prove that n d 2 4 w n w w n converges in distribution to a gaussian law both the polynomial rate of convergence and the scaling with the martingale w n are different from those for polymers on trees

universal aspects of curved flat and stationary state kardar parisi zhang statistics motivated by the recent exact solution of the stationary state kardar parisi zhang kpz statistics by imamura and sasamoto phys rev lett 108 190603 2012 as well as a precursor experimental signature unearthed by takeuchi phys rev lett 110 210604 2013 we establish here the universality of these phenomena examining scaling behaviors of directed polymers in a random medium the stochastic heat equation with multiplicative noise and kinetically roughened kpz growth models we emphasize the value of cross kpz class universalities revealing crossover effects of experimental relevance finally we illustrate the great utility of kpz scaling theory by an optimized numerical analysis of the ulam problem of random permutations

diffusion of directed polymers in a random environment we consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time in spatial dimensionsd 2 we establish that the behavior is diffusive with probability one the diffusion constant is not renormalized by the interaction

directed polymers in a random medium abstract the problem of a polymer in a random medium can be formulated as follows consider a regular lattice with on each bond ij a random energy ij the energy e w of a self avoiding walk w on this lattice is the sum of the energies of all the bonds visited by the walk then the problem is to compute the partition function z w exp e w t several recent results will be presented 1 the mean field version of this problem can be solved exactly and one observes a low temperature phase similar to the spin glass phase 2 for finite dimensional lattices one can derive bounds on the transition temperature in high enough dimension 3 for hierarchical lattices the problem can be reduced to finding stable laws when one combines random variables in a non linear way for example z z 1 z 2 z 3 z 4

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

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

transversal fluctuations of the asep stochastic six vertex model and hall littlewood gibbsian line ensembles we consider the asep and the stochastic six vertex model started with step initial data after a long time t it is known that the one point height function fluctuations for these systems are of order t1 3 we prove the kpz prediction of t2 3 scaling in space namely we prove tightness and brownian absolute continuity of all subsequential limits as t goes to infinity of the height function with spatial coordinate scaled by t2 3 and fluctuations scaled by t1 3 the starting point for proving these results is a connection discovered recently by borodin bufetov wheeler between the stochastic six vertex height function and the hall littlewood process a certain measure on plane partitions interpreting this process as a line ensemble with a gibbsian resampling invariance we show that the one point tightness of the top curve can be propagated to the tightness of the entire curve

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

uniqueness and ergodicity of stationary directed polymers on mathbb z 2 we study the ergodic theory of stationary directed nearest neighbor polymer models on mathbb z 2 with i i d weights such models are equivalent to specifying a stationary distribution on the space of weights and correctors that satisfy certain consistency conditions we show that for a prescribed weight distribution and corrector mean there is at most one stationary polymer distribution which is ergodic under the e 1 or e 2 shift further if the weights have more than two moments and the corrector mean vector is an extreme point of the superdifferential of the limiting free energy then the corrector distribution is ergodic under each of the e 1 and e 2 shifts

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

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

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

directed polymers and the quantum toda lattice we characterize the law of the partition function of a brownian directed polymer model in terms of a diffusion process associated with the quantum toda lattice the proof is via a multidimensional generalization of a theorem of matsumoto and yor concerning exponential functionals of brownian motion it is based on a mapping which can be regarded as a geometric variant of the rsk correspondence

from duality to determinants for q tasep and asep we prove duality relations for two interacting particle systems the q deformed totally asymmetric simple exclusion process q tasep and the asymmetric simple exclusion process asep expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents respectively duality implies that they solve systems of odes these systems are integrable and for particular step and half stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems solutions and hence also the moments we form laplace transform like generating functions of these moments and via residue calculus we compute two different types of fredholm determinant formulas for such generating functions for asep the first type of formula is new and readily lends itself to asymptotic analysis as necessary to reprove gue tracy widom distribution fluctuations for asep while the second type of formula is recognizable as closely related to tracy and widom s asep formula comm math phys 279 2008 815 844 j stat phys 132 2008 291 300 comm math phys 290 2009 129 154 j stat phys 140 2010 619 634 for q tasep both formulas coincide with those computed via borodin and corwin s macdonald processes probab theory related fields 2014 158 225 400 both q tasep and asep have limit transitions to the free energy of the continuum directed polymer the logarithm of the solution of the stochastic heat equation or the hopf cole solution to the kardar parisi zhang equation thus q tasep and asep are integrable discretizations of these continuum objects the systems of odes associated to their dualities are deformed discrete quantum delta bose gases and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics

mode coupling and renormalization group results for the noisy burgers equation we investigate the noisy burgers equation kardar parisi zhang equation in 1 1 dimensions using the dynamical renormalization group to two loop order and mode coupling techniques the roughness and dynamic exponent are fixed by galilean invariance and a fluctuation dissipation theorem the fact that there are no singular two loop contributions to the two point vertex functions supports the mode coupling approach which can be understood as a self consistent one loop theory where vertex corrections are neglected therefore the numerical solution of the mode coupling equations yields very accurate results for the scaling functions in addition finite size effects can be studied furthermore the results from exact ward identities as well as from second order perturbation theory permit the quantitative evaluation of the vertex corrections and thus provide a quantitative test for the mode coupling approach it is found that the vertex corrections themselves are of the order 1 surprisingly however their effect on the correlation function is substantially smaller

generalization of shadows and fixed point theorems for fuzzy sets in this paper we define a fuzzy set concerning shadows without assuming the existence of the metric projection and extend a well known result in a hilbert space to that in a normed linear space we shall observe that the fuzzy set presented here is in essence a generalization of shadows further we study fuzzy mappings in the framework of fixed point theory we first prove two theorems extending nadler s fixed point theorem lots of fixed point theorems for fuzzy mappings have been proved however the method used here is completely different from those in the previous results next we prove a theorem by applying fan s fixed point theorem and conversely we show that this is a generalization of this theorem

brownian bridges for late time asymptotics of kpz fluctuations in finite volume height fluctuations are studied in the one dimensional totally asymmetric simple exclusion process with periodic boundaries with a focus on how late time relaxation towards the non equilibrium steady state depends on the initial condition using a reformulation of the matrix product representation for the dominant eigenstate the statistics of the height at large scales is expressed for arbitrary initial conditions in terms of extremal values of independent standard brownian bridges comparison with earlier exact bethe ansatz asymptotics leads to explicit conjectures for some conditional probabilities of non intersecting brownian bridges with exponentially distributed distances between the endpoints

maximal height statistics for 1 f alpha signals numerical and analytical results are presented for the maximal relative height distribution of stationary periodic gaussian signals one dimensional interfaces displaying a 1 f alpha power spectrum for 0 1 regime of strong correlations and a highly accurate picture gallery of distribution functions can be constructed numerically analytical results can be obtained in the limit alpha infinity and for large alpha by perturbation expansion furthermore using path integral techniques we derive a trace formula for the distribution function valid for alpha 2n even integer from the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha 1 is found to be in agreement with simulations comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions

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

steady state scaling function of the 1 1 dimensional single step model the steady state height height correlation function for the 1 1 dimensional single step model is calculated in a large scale monte carlo simulation analysis of the data yields a universal ratio of scaling amplitudes which differs from the value obtained recently from a mode coupling calculation an empirical form for a universal scaling function is also presented

continuum model for radial interface growth abstract a stochastic partial differential equation along the lines of the kardar parisi zhang equation is introduced for the evolution of a growing interface in a radial geometry regular polygon solutions as well as radially symmetric solutions are identified in the deterministic limit the polygon solutions of relevance to on lattice eden growth from a seed in the zero noise limit are unstable in the continuum in favour of the symmetric solutions the asymptotic surface width scaling for stochastic radial interface growth is investigated through numerical simulations and found to be characterized by the same scaling exponent as that for stochastic growth on a substrate

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

spectral gaps of open tasep in the maximal current phase we study spectral gaps of the one dimensional totally asymmetric simple exclusion process tasep with open boundaries in the maximal current phase earlier results for the model with periodic boundaries suggest that the gaps contributing to the universal kpz regime may be understood as points on an infinite genus riemann surface built from a parametric representation of the cumulant generating function of the current we perform explicit analytic continuations from the known large deviations of the current for open tasep and confirm the results for the gaps by an exact bethe ansatz calculation with additional checks using high precision extrapolation numerics

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

universal fluctuations in kardar parisi zhang growth on one dimensional flat substrates we present a numerical study of the evolution of height distributions hds obtained in interface growth models belonging to the kardar parisi zhang kpz universality class the growth is done on an initially flat substrate the hds obtained for all investigated models are very well fitted by the theoretically predicted gaussian orthogonal ensemble goe distribution the first cumulant has a shift that vanishes as t 1 3 while the cumulants of order 2 n 4 converge to goe as t 2 3 or faster behaviors previously observed in other kpz systems these results yield evidences for the universality of the goe distribution in kpz growth on flat substrates finally we further show that the surfaces are described by the airy 1 process

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

power accretion in social systems we consider a model of power distribution in a social system where a set of agents plays a simple game on a graph the probability of winning each round is proportional to the agent s current power and the winner gets more power as a result we show that when the agents are distributed on simple one dimensional and two dimensional networks inequality grows naturally up to a certain stationary value characterized by a clear division between a higher and a lower class of agents high class agents are separated by one or several lower class agents which serve as a geometrical barrier preventing further flow of power between them moreover we consider the effect of redistributive mechanisms such as proportional nonprogressive taxation sufficient taxation will induce a sharp transition towards a more equal society and we argue that the critical taxation level is uniquely determined by the system geometry interestingly we find that the roughness and shannon entropy of the power distributions are a very useful complement to the standard measures of inequality such as the gini index and the lorenz curve

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

kardar parisi zhang universality class in 2 1 dimensions universal geometry dependent distributions and finite time corrections the dynamical regimes of models belonging to the kardar parisi zhang kpz universality class are investigated in d 2 1 by extensive simulations considering flat and curved geometries geometry dependent universal distributions different from their tracy widom counterpart in one dimension were found distributions exhibit finite time corrections hallmarked by a shift in the mean decaying as t where is the growth exponent our results support a generalization of the ansatz h v t t t to higher dimensions where v and are nonuniversal quantities whereas and are universal and the last one depends on the surface geometry generalized gumbel distributions provide very good fits of the distributions in at least four orders of magnitude around the peak which can be used for comparisons with experiments our numerical results call for analytical approaches and experimental realizations of the kpz class in two dimensional systems

geometry dependence in linear interface growth the effect of geometry in the statistics of nonlinear universality classes for interface growth has been widely investigated in recent years and it is well known to yield a split of them into subclasses in this work we investigate this for the linear classes of edwards wilkinson and of mullins herring in one and two dimensions from comparison of analytical results with extensive numerical simulations of several discrete models belonging to these classes as well as numerical integrations of the growth equations on substrates of fixed size flat geometry or expanding linearly in time radial geometry we verify that the height distributions hds and the spatial and the temporal covariances are universal but geometry dependent in fact the hds are always gaussian and when defined in terms of the so called kpz ansatz h v t t their probability density functions p have mean null so that all their cumulants are null except by their variances which assume different values in the flat and radial cases the shape of the rescaled covariance curves is analyzed in detail and compared with some existing analytical results for them overall these results demonstrate that the splitting of such university classes is quite general being not restricted to the nonlinear ones

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

two loop renormalization group analysis of the burgers kardar parisi zhang equation a systematic analysis of the burgers kardar parisi zhang equation in d 1 dimensions by dynamic renormalization group theory is described the fixed points and exponents are calculated to two loop order we use the dimensional regularization scheme carefully keeping the full d dependence originating from the angular parts of the loop integrals for dimensions less than mathit d mathit c 2 we find a strong coupling fixed point which diverges at d 2 indicating that there is nonperturbative strong coupling behavior for all d ensuremath ge 2 at d 1 our method yields the identical fixed point as in the one loop approximation and the two loop contributions to the scaling functions are nonsingular for dg2 dimensions there is no finite strong coupling fixed point in the framework of a 2 ensuremath epsilon expansion we find the dynamic exponent corresponding to the unstable fixed point which described the nonequilibrium roughening transition to be z 2 o mathrm ensuremath epsilon 3 in agreement with a recent scaling argument by doty and kosterlitz phys rev lett 69 1979 1992 similarly our result for the correlation length exponent at the transition is 1 ensuremath nu ensuremath epsilon o mathrm ensuremath epsilon 3 for the smooth phase some aspects of the crossover from gaussian to critical behavior are discussed

minimal path on the hierarchical diamond lattice we consider the minimal paths on a hierarchical diamond lattice where bonds are assigned a random weight depending on the initial distribution of weights we find all possible asymptotic scaling properties the different cases found are the small disorder case the analog of levy s distributions with a power law decay at and finally a limit of large disorder which can be identified as a percolation problem the asymptotic shape of the stable distributions of weights of the minimal path are obtained as well as their scaling properties as a side result we obtain the asymptotic form of the distribution of effective percolation thresholds for finite size hierarchical lattices

possible origin for the experimental scarcity of kpz scaling in non conserved surface growth abstract the kardar parisi zhang kpz equation is generically expected to describe the scaling properties of rough surfaces growing in the absence of conservation laws however very few experimental realizations are known of this universality class here we focus on the role of instabilities whether of diffusional origin or other as physical mechanisms hindering the observation of kpz scaling examples are drawn from various growth processes such as electrochemical deposition ecd chemical vapor deposition cvd and erosion by ion beam sputtering we moreover consider an interface equation which starting from the corresponding constitutive equations can be derived to describe growth by either ecd or cvd depending on the interpretation of the quantities appearing this approach makes contact with phenomenological parameters and suggests that a more generic description of non conserved growth may be provided by the kuramoto sivashinsky equation and generalizations thereof in this case the experimental scarcity of kpz scaling would be due to exceedingly long transients determined by the instabilities that occur

multisurface coding simulations of the restricted solid on solid model in four dimensions we study the restricted solid on solid model for surface growth in spatial dimension d 4 by means of a multisurface coding technique that allows us to analyze samples of size up to 256 4 in the steady state regime for such large systems we are able to achieve a controlled asymptotic regime where the typical scale of the fluctuations are larger than the lattice spacing used in the simulations a careful finite size scaling analysis of the critical exponents clearly indicate that d 4 is not the upper critical dimension of the model

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

field theoretic thermodynamic uncertainty relation general formulation exemplified with the kardar parisi zhang equation we propose a field theoretic thermodynamic uncertainty relation as an extension of the one derived so far for a markovian dynamics on a discrete set of states and for overdamped langevin equations we first formulate a framework which describes quantities like current entropy production and diffusivity in the case of a generic field theory we will then apply this general setting to the one dimensional kardar parisi zhang equation a paradigmatic example of a non linear field theoretic langevin equation in particular we will treat the dimensionless kardar parisi zhang equation with an effective coupling parameter measuring the strength of the non linearity it will be shown that a field theoretic thermodynamic uncertainty relation holds up to second order in a perturbation expansion with respect to a small effective coupling constant the calculations show that the field theoretic variant of the thermodynamic uncertainty relation is not saturated for the case of the kardar parisi zhang equation due to an excess term stemming from its non linearity

scaling for a one dimensional directed polymer with boundary conditions we study a 1 1 dimensional directed polymer in a random environment on the integer lattice with log gamma distributed weights among directed polymers this model is special in the same way as the last passage percolation model with exponential or geometric weights is special among growth models namely both permit explicit calculations with appropriate boundary conditions the polymer with log gamma weights satisfies an analogue of burke s theorem for queues building on this we prove the conjectured values for the fluctuation exponents of the free energy and the polymer path in the case where the boundary conditions are present and both endpoints of the polymer path are fixed for the polymer without boundary conditions and with either fixed or free endpoint we get the expected upper bounds on the exponents

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

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

circular kardar parisi zhang equation as an inflating self avoiding ring polymer we consider the kardar parisi zhang equation for a circular interface in two dimensions unconstrained by the standard small slope and no overhang approximations numerical simulations using an adaptive scheme allow us to elucidate the complete time evolution as a crossover between a short time regime with the interface fluctuations of a self avoiding ring or two dimensional vesicle and a long time regime governed by the tracy widom distribution expected for this geometry for small noise amplitudes scaling behavior is only of the latter type large noise is also seen to renormalize the bare physical parameters of the ring akin to analogous parameter renormalization for equilibrium three dimensional membranes our results bear particular importance on the relation between relevant universality classes of scale invariant systems in two dimensions

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

multiparticle biased diffusion limited aggregation with surface diffusion a comprehensive model of electrodeposition we present a complete study of the multiparticle biased diffusion limited aggregation mbdla model supplemented with surface diffusion sd focusing on the relevance and effects of the latter transport mechanism by comparing different algorithms we show that mbdla 1 sd is a very good qualitative model for electrodeposition in essentially the whole range of current intensities provided one introduces sd in the model in the proper fashion we have found that the correct procedure involves simultaneous bulk diffusion and sd introducing a time scale arising from the ratio of the rates of the two processes we discuss in detail the different morphologies obtained and compare them to the available experimental data with very satisfactory results we also characterize the aggregates thus obtained by means of the dynamic scaling exponents of the interface height allowing us to distinguish several regimes in the mentioned interface growth our asymptotic scaling exponents are again in good agreement with recent experiments we conclude by discussing a global picture of the influence and consequences of sd in electrodeposition

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

exact solution for a random walk in a time dependent 1d random environment the point to point beta polymer we consider the beta polymer an exactly solvable model of directed polymer on the square lattice introduced by barraquand and corwin bc 2016 probab theory relat fields 1 16 we study the statistical properties of its point to point partition sum the problem is equivalent to a model of a random walk in a time dependent and in general biased 1d random environment in this formulation we study the sample to sample fluctuations of the transition probability distribution function pdf of the random walk using the bethe ansatz we obtain exact formulas for the integer moments and fredholm determinant formulas for the laplace transform of the directed polymer partition sum random walk transition probability the asymptotic analysis of these formulas at large time t is performed both i in a diffusive vicinity of the optimal direction in space time chosen by the random walk where the fluctuations of the pdf are found to be gamma distributed ii in the large deviations regime of the random walk where the fluctuations of the logarithm of the pdf are found to grow with time as t 1 3 and to be distributed according to the tracy widom gue distribution our exact results complement those of bc for the cumulative distribution function of the random walk in regime ii and in regime i they unveil a novel fluctuation behavior we also discuss the crossover regime between i and ii identified as our results are confronted to extensive numerical simulations of the model