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

identifying spatial clustering phenomena in forest fire sequences abstract spatial clustering in fire data recorded from 1997 to 2003 in an area of central italy has been deeply investigated using indices of different type i a topological index the voronoi polygon area ii a statistical index the morishita index iii a dimensional index the fractal dimension our findings reveal the presence of clustering in the spatial distribution of the fire set analysed the knowledge of such property can be very useful to estimate overdensity of fires in those areas where particular prevention and allocation of resources should be assessed

the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

periodic tasep with general initial conditions we consider the one dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain and obtain explicit formulas for the multi point distributions in the space time plane the formulas are given in terms of an integral involving a fredholm determinant we then evaluate the large time large period limit in the relaxation time scale which is the scale such that the system size starts to affect the height fluctuations the limit is obtained assuming certain conditions on the initial condition which we show that the step flat and step flat initial conditions satisfy hence we obtain the limit theorem for these three initial conditions in the periodic model extending the previous work on the step initial condition we also consider uniform random and uniform step random initial conditions

matrix li yau hamilton inequality for the cr heat equation in pseudohermitian 2n 1 manifolds in this paper we first derive the cr analogue of matrix li yau hamilton inequality for the positive solution to the cr heat equation in a closed pseudohermitian 2n 1 manifold with nonnegative bisectional curvature and bitorsional tensor we then obtain the cr li yau gradient estimate in the heisenberg group we apply this cr gradient estimate and extend the cr matrix li yau hamilton inequality to the case of the heisenberg group as a consequence we derive the hessian comparison property for the heisenberg group

rough paths and 1d sde with a time dependent distributional drift application to polymers motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions like the kardar parisi zhang kpz equation we reconsider the unique solvability of one dimensional stochastic differential equations the drift of which is a distribution by means of rough paths theory existence and uniqueness are established in the weak sense when the drift reads as the derivative of a holder continuous function regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed which makes the structure of the solutions more explicit than within the earlier framework of dirichlet processes

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

static fluctuations of a thick one dimensional interface in the 1 1 directed polymer formulation numerical study we study numerically the geometrical and free energy fluctuations of a static one dimensional 1d interface with a short range elasticity submitted to a quenched random bond gaussian disorder of finite correlation length ensuremath xi g0 and at finite temperature t using the exact mapping from the static 1d interface to the 1 1 directed polymer dp growing in a continuous space we focus our analysis on the disorder free energy of the dp end point a quantity which is strictly zero in the absence of disorder and whose sample to sample fluctuations at a fixed growing time t inherit the statistical translation invariance of the microscopic disorder explored by the dp constructing a new numerical scheme for the integration of the kardar parisi zhang evolution equation obeyed by the free energy we address numerically the time and temperature dependence of the disorder free energy fluctuations at fixed finite ensuremath xi we examine on one hand the amplitude stackrel ifmmode tilde else fi d t and effective correlation length stackrel ifmmode tilde else fi ensuremath xi t of the free energy fluctuations and on the other hand the imprint of the specific microscopic disorder correlator on the large time shape of the free energy two point correlator we observe numerically the crossover to a low temperature regime below a finite characteristic temperature t c ensuremath xi as previously predicted by gaussian variational method computations and scaling arguments and extensively investigated analytically in phys rev e 87 042406 2013 finally we address numerically the time and temperature dependence of the roughness b t which quantifies the dp end point transverse fluctuations and we show how the amplitude stackrel ifmmode tilde else fi d ensuremath infty t ensuremath xi controls the different regimes experienced by b t in agreement with the analytical predictions of a dp toy model approach

fluctuations for stationary q tasep we consider the q totally asymmetric simple exclusion process q tasep in the stationary regime and study the fluctuation of the position of a particle we first observe that the problem can be studied as a limiting case of an n particle q tasep with a random initial condition and with particle dependent hopping rate then we explain how this n particle q tasep can be encoded in a dynamics on a two sided gelfand tsetlin cone described by a two sided q whittaker process and present a fredholm determinant formula for the q laplace transform of the position of a particle two main ingredients in its derivation is the ramanujan s bilateral summation formula and the cauchy determinant identity for the theta function with an extra parameter based on this we establish that the position of a particle obeys the universal stationary kpz distribution the baik rains distribution in the long time limit

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

differential harnack estimates for time dependent heat equations with potentials in a closed spherical cr 3 manifold abstract in this paper we derive differential harnack estimates for time dependent heat equations with potentials under the cr yamabe flow in a closed spherical cr 3 manifold with positive tanaka webster curvature and vanishing torsion it can be served as a generalization of li yau hamilton inequality for the cr yamabe flow the differential harnack estimate will play an important role in the study of the cr yamabe flow

forest fire model with immune trees abstract we present a generalization of the forest fire model of p bak et al by including the immunity g which is the probability that a tree is not ignited although one of its neighbors is burning when g reaches a critical value gc p which depends on the tree growth rate p the fire cannot survive any more i e a continuous phase transition takes place from a steady state with fire to a steady state without fire we present results of computer simulations and explain them by analytic calculations the fire spreading at the phase transition represents a new type of percolation which is called fluctuating site percolation

self organized critical forest fire model mean field theory and simulation results in 1 to 6 dimenisons we argue that the critical forest fire model ffm introduced by drossel and schwabl phys rev lett 69 1629 1992 is a critical branching process in the mean field approximation where the number s of trees burned in a forest fire is power law distributed with exponent s 5 2 the mean field model of the ffm in finite dimension is the percolation model and as in percolation the upper critical dimension is 6 simulations show that the ffm becomes increasingly percolationlike with increasing dimension d and is within error bars eully consistent with the percolation results when d 3

scaling laws and simulation results for the self organized critical forest fire model we discuss the properties of a self organized critical forest fire model which has been introduced recently we derive scaling laws and define critical exponents the values of these critical exponents are determined by computer simulations in 1 to 8 dimensions the simulations suggest a critical dimension d c 6 above which the critical exponents assume their mean field values changing the lattice symmetry and allowing trees to be immune against fire we show that the critical exponents are universal

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

fluctuations of tasep on a ring in relaxation time scale we consider the totally asymmetric simple exclusion process on a ring with flat and step initial conditions we assume that the size of the ring and the number of particles tend to infinity proportionally and evaluate the fluctuations of tagged particles and currents the crossover from the kpz dynamics to the equilibrium dynamics occurs when the time is proportional to the 3 2 power of the ring size we compute the limiting distributions in this relaxation time scale the analysis is based on an explicit formula of the finite time one point distribution obtained from the coordinate bethe ansatz method 2017 wiley periodicals inc

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

harnack inequality and pinching estimates for anisotropic curvature flow of hypersurfaces abstract we obtain a differential harnack inequality for anisotropic curvature flow of convex hypersurfaces in euclidean space with its speed given by a curvature function of homogeneity degree one in a certain class and restrictions depending only on the initial data and the anisotropic factor which reflects the influence of the ambient space moreover the pinching estimate for such flows is derived from the maximum principle for tensors

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 a self organized critical forest fire model the author studies a forest fire model introduced in a paper by drossel and schwabl 1992 by means of simulations with high statistics and very close to the critical point in this way he can correct some statements made in that paper in particular he finds that the critical exponents are not the classical ones proposed by drossel and schwabl in spite of scaling laws with anomalous exponents the typical states of the system are not critical in the sense of being marginally stable locally

heat kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

fractional derivatives of products of airy functions abstract fractional derivatives of the products of airy functions are investigated d ai 2 x and d ai x bi x where ai x and bi x are the airy functions of the first and second type respectively they turn out to be linear combinations of d ai x and d gi x where gi x is the scorer function it is also proved that the wronskian w x of the system of half integrals d 1 2 ai x d 1 2 gi x and its hilbert transform w x h w x can be considered special functions in their own right since they are expressed in terms of ai 2 x and ai x bi x respectively various integral relations are established integral representations for d ai x a ai x a and its hilbert transform h d ai x a ai x a are derived

the asep and determinantal point processes we introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line with correlation kernels defined via spectral projections for the associated jacobi matrices for classical weights we show how such ensembles arise as limits of various hypergeometric orthogonal polynomial ensembles we then prove that the q laplace transform of the height function of the asep with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete laguerre ensemble a member of the new family this allows us to obtain the large time asymptotics of the asep in three limit regimes a for finitely many rightmost particles b gue tracy widom asymptotics of the height function c kpz asymptotics of the height function for the asep with weak asymmetry we also give similar results for two instances of the stochastic six vertex model in a quadrant the proofs are based on limit transitions for the corresponding determinantal point processes

blocks in the asymmetric simple exclusion process in earlier work the authors obtained formulas for the probability in the asymmetric simple exclusion process that the mth particle from the left is at site x at time t they were expressed in general as sums of multiple integrals and for the case of step initial condition as an integral involving a fredholm determinant in the present work these results are generalized to the case where the mth particle is the left most one in a contiguous block of l particles the earlier work depended in a crucial way on two combinatorial identities and the present work begins with a generalization of these identities to general l

the li yau hamilton estimate and the yang mills heat equation on manifolds with boundary the paper pursues two connected goals firstly we establish the li yau hamilton estimate for the heat equation on a manifold m with nonempty boundary results of this kind are typically used to prove monotonicity formulas related to geometric flows secondly we establish bounds for a solution t of the yang mills heat equation in a vector bundle over m the li yau hamilton estimate is utilized in the proofs our results imply that the curvature of t does not blow up if the dimension of m is less than 4 or if the initial energy of t is sufficiently small

expanding solitons with non negative curvature operator coming out of cones we consider ricci flow of complete riemannian manifolds which have bounded non negative curvature operator non zero asymptotic volume ratio and no boundary we prove scale invariant estimates for these solutions using these estimates we show that there is a limit solution obtained by scaling down this solution at a fixed point in space this limit solution is an expanding soliton coming out of the asymptotic cone at infinity

exact results for the one dimensional self organized critical forest fire model we present the analytic solution of the self organized critical soc forest fire model in one dimension proving soc in systems without conservation laws by analytic means under the condition that the system is in the steady state and very close to the critical point we calculate the probability that a string of ital n neighboring sites is occupied by a given configuration of trees the critical exponent describing the size distribution of forest clusters is exactly tau 2 and does not change under certain changes of the model rules computer simulations confirm the analytic results

large gap asymptotics for airy kernel determinants with discontinuities we obtain large gap asymptotics for airy kernel fredholm determinants with any number m of discontinuities these m point determinants are generating functions for the airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles our main result is that the m point determinants can be expressed asymptotically as the product of m 1 point determinants multiplied by an explicit constant pre factor which can be interpreted in terms of the covariance of the counting function of the process

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

formation of space time structure in a forest fire model we present a general stochastic forest fire model which shows a variety of different structures depending on the parameter values the model contains three possible states per site tree burning tree empty site and three parameters tree growth probability p lightning probability f and immunity g we review analytic and computer simulation results for a quasideterministic state with spiral shaped fire fronts for a percolation like phase transition and a self organized critical state possible applications to excitable systems are discussed

a multi layer extension of the stochastic heat equation motivated by recent developments on solvable directed polymer models we define a multi layer extension of the stochastic heat equation involving non intersecting brownian motions by developing a connection with darboux transformations and the two dimensional toda equations we conjecture a markovian evolution in time for this multi layer process as a first step in this direction we establish an analogue of the karlin mcgregor formula for the stochastic heat equation and use it to prove a special case of this conjecture

spreading analysis and finite size scaling study of the critical behavior of a forest fire model with immune trees abstract in a forest fire model with immune trees ffmit recently proposed by drossel and schwabl physica a 199 1993 183 the sites of the lattice are either empty a tree or a burning tree and at each time step the system is updated in parallel according to the following rules i a burning tree becomes an empty site ii trees grow with probability p from empty sites and iii a green tree becomes a burning tree with probability 1 g if at least one next neighbor is burning fixing an arbitrary grow probability p and starting with a small immunity g increments of g causes the fire density of the steady state to decrease until the fire becomes irreversibly extinguished at a certain critical point of coordinates pc gc the set of critical points defines a critical curve gc p which in the l limit divides the p g plane in two regions a steady state with fire fronts for g

endpoint distribution of directed polymers in 1 1 dimensions we give an explicit formula for the joint density of the max and argmax of the airy2 process minus a parabola the argmax has a universal distribution which governs the rescaled endpoint for large time or temperature of directed polymers in 1 1 dimensions

differential harnack inequalities on riemannian manifolds i linear heat equation abstract in the first part of this paper we get new li yau type gradient estimates for positive solutions of heat equation on riemannian manifolds with ricci m k k r as applications several parabolic harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with ricci curvature bounded from below in the second part we establish a perelman type li yau hamilton differential harnack inequality for heat kernels on manifolds with ricci m k which generalizes a result of l ni 2004 2006 20 21 as applications we obtain new harnack inequalities and heat kernel estimates on general manifolds we also obtain various entropy monotonicity formulas for all compact riemannian manifolds

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

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

length scales and power laws in the two dimensional forest fire model abstract we re examine a two dimensional forest fire model via monte carlo simulations and show the existence of two length scales with different critical exponents associated with clusters and with the usual two point correlation function of trees we check resp improve previously obtained values for other critical exponents and perform a first investigation of the critical behaviour of the slowest relaxational mode we also investigate the possibility of describing the critical point in terms of a distribution of the global density we find that some qualitative features such as a temporal oscillation and a power law of the cluster size distribution can nicely be obtained from such a model that discards the spatial structure

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

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

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

interpolating between constrained li yau and chow hamilton harnack inequalities for a nonlinear parabolic equation abstract we establish a one parameter family of harnack inequalities connecting the constrained trace li yau differential harnack inequality for a nonlinear parabolic equation to the constrained trace chow hamilton harnack inequality for this nonlinear equation with respect to evolving metrics related to the ricci flow on a 2 dimensional closed manifold this result can be regarded as a nonlinear version of the previous work of y zheng and the author j y wu y zheng interpolating between constrained li yau and chow hamilton harnack inequalities on a surface arch math 94 2010 591 600

plurisubharmonic functions and the kahler ricci flow in this paper a sharp linear trace li yau hamilton inequality for kahler ricci flow is proved the new inequality extends the previous trace harnack inequality obtained by h d cao we also establish sharp gradient estimates for the positive solution of the time dependent heat equation for some cases finally we apply this new linear trace li yau hamilton inequality to study the liouville properties of the plurisubharmonic functions on complete kahler manifolds with bounded nonnegative holomorphic bisectional curvature

a fredholm determinant representation in asep in previous work tracy and widom in commun math phys 279 815 844 2008 the authors found integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice the dynamics are uniquely determined once the initial state is specified in this note we restrict our attention to the case of step initial condition with particles at the positive integers and consider the distribution function for the mth particle from the left in tracy and widom commun math phys 279 815 844 2008 an infinite series of multiple integrals was derived for the distribution in this note we show that the series can be summed to give a single integral whose integrand involves a fredholm determinant we use this determinant representation to derive non rigorously at this writing a scaling limit

on a forest fire model with supposed self organized criticality we study a stochastic forest fire model introduced by p baket al as a model showing self organized criticality this model involves a growth parameterp and the criticality is supposed to show up in the limitp 0 by simulating the model on much larger lattices and with much smaller values ofp we find that the correlations with longest range do not show a nontrivial critical phenomenon in this limit though we cannot rule out percolation like critical behavior on a smaller but still divergent length scale in contrast the model shows nontrivialdeterministic evolution over time scales 1 p in the limitp 0

on asymmetric simple exclusion process with periodic step bernoulli initial condition we consider the asymmetric simple exclusion process asep on the integers in which the initial density at a site the probability that it is occupied is given by a periodic function on the positive integers when the function is constant this is the step bernoulli initial condition starting with a result in earlier work we find a formula for the probability distribution for a given particle at a given time which is a sum over positive integers k of integrals of order k

high temperature limits for 1 1 dimensional directed polymer with heavy tailed disorder the directed polymer model at intermediate disorder regime was introduced by alberts khanin quastel ann probab 42 2014 1212 1256 it was proved that at inverse temperature n n with 1 4 1 4 the partition function centered appropriately converges in distribution and the limit is given in terms of the solution of the stochastic heat equation this result was obtained under the assumption that the disorder variables posses exponential moments but its universality was also conjectured under the assumption of six moments we show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments we also explain the behavior of the scaling exponent for the log partition function under different moment assumptions and values of

self organized criticality in forest fire models we review properties of the self organized critical soc forest fire model ffm self organized critical systems drive themselves into a critical state without fine tuning of parameters after an introduction the rules of the model and the conditions for spiral shaped and soc large scale structures are given for the soc state critical exponents and scaling relations are introduced the existence of an upper critical dimension and the universal behavior of the model are discussed the relations and differences between ffm and percolation systems are outlined considering an extension of the ffm into the regions beyond the critical point phase transitions and the various structures found in these regions are illustrated

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

weakly asymmetric non simple exclusion process and the kardar parisi zhang equation we analyze a class of non simple exclusion processes and the corresponding growth models by generalizing the discrete cole hopf transformation of gartner stoch process appl 27 233 260 1987 we identify the main non linearity and eliminate it by imposing a gradient type condition for hopping range at most 3 using the generalized transformation we prove the convergence of the exclusion process toward the kardar parisi zhang kpz equation this is the first universality result under the weak asymmetry concerning interacting particle systems while this class of exclusion processes are not explicitly solvable under the weak asymmetry we obtain the exact one point limiting distribution for the step initial condition by using the previous result of amir et al commun pure appl math 64 4 466 537 2011 and our convergence result

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

differential harnack estimates for backward heat equations with potentials under the ricci flow abstract in this paper we derive a general evolution formula for possible harnack quantities as a consequence we prove several differential harnack inequalities for positive solutions of backward heat type equations with potentials including the conjugate heat equation under the ricci flow we shall also derive perelman s harnack inequality for the fundamental solution of the conjugate heat equation under the ricci flow

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

a monotonicity formula on complete kahler manifolds with nonnegative bisectional curvature in y yau proposed to study the uniformization of complete kahler manifolds with nonnegative curvature in particular one wishes to determine whether or not a complete kahler manifold m with positive bisectional curvature is biholomorphic to cm see also gw si for this sake it was further asked in y whether or not the ring of the holomorphic functions with polynomial growth which we denote by op m is finitely generated and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on cm this paper addresses the latter questions we denote by 0 m the space of holomorphic functions of polynomial growth with degree d see section 3 for the precise definition then op m jd 0 od m in this paper we show that

a central limit theorem for the kpz equation we consider the kpz equation in one space dimension driven by a stationary centred space time random field which is sufficiently integrable and mixing but not necessarily gaussian we show that in the weakly asymmetric regime the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the hopf cole solution to the kpz equation driven by space time gaussian white noise while the limiting process depends only on the integrated variance of the driving field the diverging constants appearing in the definition of the reference frame also depend on higher order moments

limit processes for tasep with shocks and rarefaction fans we consider the totally asymmetric simple exclusion process tasep with two sided bernoulli initial condition i e with left density and right density we study the associated height function whose discrete gradient is given by the particle occurrences macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of we characterize the large time scaling limit of the multipoint fluctuations as a function of the densities and of the different macroscopic regions moreover using a slow decorrelation phenomena the results are extended from fixed time to the whole space time except along the some directions the characteristic solutions of the related burgers equation where the problem is still open

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

scaling limits of wick ordered kpz equation consider the kpz equation x d where w t x is a space time white noise this paper investigates the question of whether for some exponents and z k u k z t kx converges in some sense as k to infty and if so what are the values of these exponents the non linear term in the kpz equation is interpreted as a wick product and the equation is solved in a suitable space of stochastic distributions the main tools for establishing the scaling properties of the solution are those of white noise analysis in particular the wiener chaos expansion a notion of convergence in law in the sense of wiener chaos is formulated and convergence in this sense of k u k z t kx as k is established for various values of and z depending on the dimension d

open asep in the weakly asymmetric regime we consider asep on a bounded interval and on a half line with sources and sinks on the full line bertini and giacomin in 1997 proved convergence under weakly asymmetric scaling of the height function to the solution of the kpz equation we prove here that under similar weakly asymmetric scaling of the sources and sinks as well the bounded interval asep height function converges to the kpz equation on the unit interval with neumann boundary conditions on both sides different parameter for each side and likewise for the half line asep to kpz on a half line this result can be interpreted as showing that the kpz equation arises at the triple critical point maximal current high density low density of the open asep 2018 wiley periodicals inc

formulas for joint probabilities for the asymmetric simple exclusion process in earlier work the authors tracy c a and widom h integral formulas for the asymmetric simple exclusion process commun math phys 279 815 2008 obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process here formulas are obtained for joint probabilities for several particles in the case of a single particle the derivation here is simpler than the one in the earlier work for one of its main results

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

the kpz limit of asep with boundary it was recently proved in corwin and shen cpam cs16 that under weakly asymmetric scaling the height functions for asep with sources and sinks converges to the hopf cole solution of the kpz equation with inhomogeneous neumann boundary conditions in their assumptions cs16 chose positive values for the neumann boundary condition and they assumed initial data which is close to stationarity by developing more extensive heat kernel estimates we clarify and extend their results to negative values of the neumann boundary parameters and we also show how to generalize their results to empty initial data which is very far from stationarity combining our result with barraquand et al duke math j bbcw17 we obtain the laplace transform of the one point distribution for half line kpz and use this to confirm t1 3 scale goe tracy widom long time fluctuations at the origin

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

constrained matrix li yau hamilton estimates on kahler manifolds we derive a family of constrained matrix li yau hamilton estimates on kahler manifolds as a result we first get a constrained matrix li yau hamilton estimate for heat equation on a kahler manifold with fixed kahler metric secondly we get such estimate for forward conjugate heat equation on kahler manifolds with time dependent kahler metrics evolving under the kahler ricci flow

integral formulas for the asymmetric simple exclusion process in this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process asep on the integer lattice mathbb z with nearest neighbor hopping rates p to the right and q 1 p to the left for the most part we consider an n particle system but for certain of these formulas we can take the n to infty limit first we obtain for the n particle system a formula for the probability of a configuration at time t given the initial configuration for this we use bethe ansatz ideas to solve the master equation extending a result of schutz for the case n 2 the main results of the paper derived from this are integral formulas for the probability for given initial configuration that the mth left most particle is at x at time t in one of these formulas we can take the n to infty limit and it gives the probability for an infinite system where the initial configuration is bounded on one side for the special case of the totally asymmetric simple exclusion process tasep our formulas reduce to the known ones

complete riemannian manifolds with pointwise pinched curvature an analogous bonnet myers theorem is obtained for a complete and positively curved n dimensional n 3 riemannian manifold m n we prove that if n 4 and the curvature operator of m n is pointwise pinched or if n 3 and the ricci curvature of m 3 is pointwise pinched then m n is compact

stochastic higher spin six vertex model and macdonald measures we prove an identity that relates the q laplace transform of the height function of a higher spin inhomogeneous stochastic six vertex model in a quadrant on one side and a multiplicative functional of a macdonald measure on the other the identity is used to prove the gue tracy widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding schur measures

on the chaotic character of the stochastic heat equation ii consider the stochastic heat equation partial t u varkappa 2 delta u sigma u dot f where the solution u u t x is indexed by t x in 0 infty times bf r d and dot f is a centered gaussian noise that is white in time and has spatially correlated coordinates we analyze the large x fixed t behavior of the solution u in different regimes thereby study the effect of noise on the solution in various cases among other things we show that if the spatial correlation function f of the noise is of riesz type that is f x propto x alpha then the fluctuation exponents of the solution are psi for the spatial variable and 2 psi 1 for the time variable where psi 2 4 alpha moreover these exponent relations hold as long as alpha in 0 d wedge 2 that is precisely when dalang s theory dalang electron j probab 4 paper no 6 29 1999 implies the existence of a solution to our stochastic pde these findings bolster earlier physical predictions kardar et al phys rev lett 58 20 889 892 1985 kardar and zhang phys rev lett 58 20 2087 2090 1987

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn