correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

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

tropical combinatorics and whittaker functions we establish a fundamental connection between the geometric robinson schensted knuth rsk correspondence and gl n r whittaker functions analogous to the well known relationship between the rsk correspondence and schur functions this gives rise to a natural family of measures associated with gl n r whittaker functions which are the analogues in this setting of the schur measures on integer partitions the corresponding analogue of the cauchy littlewood identity can be seen as a generalization of an integral identity for gl n r whittaker functions due to bump and stade as an application we obtain an explicit integral formula for the laplace transform of the law of the partition function associated with a 1 dimensional directed polymer model with log gamma weights recently introduced by one of the authors

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

geometric rsk correspondence whittaker functions and symmetrized random polymers we show that the geometric lifting of the rsk correspondence introduced by a n kirillov physics and combinatorics proc nagoya 2000 2nd internat workshop pp 82 150 2001 is volume preserving with respect to a natural product measure on its domain and that the integrand in givental s integral formula for mathit gl n mathbb r whittaker functions arises naturally in this context apart from providing further evidence that whittaker functions are the natural analogue of schur polynomials in this setting our results also provide a new combinatorial framework for the study of random polymers when the input matrix consists of random inverse gamma distributed weights the probability distribution of a polymer partition function constructed from these weights can be written down explicitly in terms of whittaker functions next we restrict the geometric rsk mapping to symmetric matrices and show that the volume preserving property continues to hold we determine the probability law of the polymer partition function with inverse gamma weights that are constrained to be symmetric about the main diagonal with an additional factor on the main diagonal the third combinatorial mapping studied is a variant of the geometric rsk mapping for triangular arrays which is again showed to be volume preserving this leads to a formula for the probability distribution of a polymer model whose paths are constrained to stay below the diagonal we also show that the analogues of the cauchy littlewood identity in the setting of this paper are equivalent to a collection of whittaker integral identities conjectured by bump number theory trace formulas and discrete groups pp 49 109 1989 and bump and friedberg festschrift in honor of piatetski shapiro part ii pp 47 65 1990 and proved by stade am j math 123 121 161 2001 israel j math 127 201 219 2002 our approach leads to new combinatorial proofs and generalizations of these identities with some restrictions on the parameters

fluctuation properties of the tasep with periodic initial configuration we consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process tasep they are expressed as fredholm determinants with a kernel defining a signed determinantal point process we then consider certain periodic initial conditions and determine the kernel in the scaling limit this result has been announced first in a letter by one of us sasamoto in j phys a 38 l549 l556 2005 and here we provide a self contained derivation connections to last passage directed percolation and random matrices are also briefly discussed

conjectures of alperin and broue for 2 blocks with elementary abelian defect groups of order 8 using the classification of finite simple groups we prove alperin s weight conjecture and the character theoretic version of broue s abelian defect group conjecture for 2 blocks of finite groups with an elementary abelian defect group of order 8

log gamma polymer free energy fluctuations via a fredholm determinant identity we prove that under n1 3 scaling the limiting distribution as n of the free energy of seppalainen s log gamma discrete directed polymer is gue tracy widom the main technical innovation we provide is a general identity between a class of n fold contour integrals and a class of fredholm determinants applying this identity to the integral formula proved in corwin et al tropical combinatorics and whittaker functions http arxiv org abs 1110 3489v3 math pr 2012 for the laplace transform of the log gamma polymer partition function we arrive at a fredholm determinant which lends itself to asymptotic analysis and thus yields the free energy limit theorem the fredholm determinant was anticipated in borodin and corwin macdonald processes http arxiv org abs 1111 4408v3 math pr 2012 via the formalism of macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach

determinantal structures in the o connell yor directed random polymer model we study the semi discrete directed random polymer model introduced by o connell and yor we obtain a representation for the moment generating function of the polymer partition function in terms of a determinantal measure this measure is an extension of the probability measure of the eigenvalues for the gaussian unitary ensemble in random matrix theory to establish the relation we introduce another determinantal measure on larger degrees of freedom and consider its few properties from which the representation above follows immediately

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

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

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

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

current distribution and random matrix ensembles for an integrable asymmetric fragmentation process we calculate the time evolution of a discrete time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size dependent rates in the continuous time limit the process turns into the totally asymmetric simple exclusion process only pieces of size 1 break off a given cluster we express the exact solution of the master equation for the process in terms of a determinant which can be derived using the bethe ansatz from this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the gaussian unitary ensemble of random matrices this result confirms universality of the scaling form of the current distribution in the kpz universality class and suggests that there is a link between integrable particle systems and random matrix ensembles

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

o connell s process as a vicious brownian motion vicious brownian motion is a diffusion scaling limit of fisher s vicious walk model which is a system of brownian particles in one dimension such that if two motions meet they kill each other we consider the vicious brownian motions conditioned never to collide with each other and call it noncolliding brownian motion this conditional diffusion process is equivalent to the eigenvalue process of the hermitian matrix valued brownian motion studied by dyson j math phys 3 1191 1962 recently o connell ann probab to be published introduced a generalization of the noncolliding brownian motion by using the eigenfunctions the whittaker functions of the quantum toda lattice in order to analyze a directed polymer model in 1 1 dimensions we consider a system of one dimensional brownian motions with a long ranged killing term as a generalization of the vicious brownian motion and construct the o connell process as a conditional process of the killing brownian motions to survive forever

system of complex brownian motions associated with the o connell process the o connell process is a softened version a geometric lifting with a parameter a 0 of the noncolliding brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length a this process is not determinantal under a special entrance law however borodin and corwin gave a fredholm determinant expression for the expectation of an observable which is a softening of an indicator of a particle position we rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show if the number of particles is n the rank of the matrix of the fredholm determinant is n then we give a representation for the quantity by using an n particle system of complex brownian motions cbms the complex function which gives the determinantal expression to the weight of cbm paths is not entire but in the combinatorial limit a 0 it becomes an entire function providing conformal martingales and the cbm representation for the noncolliding brownian motion is recovered

transition between airy1 and airy2 processes and tasep fluctuations we consider the totally asymmetric simple exclusion process a model in the kpz universality class we focus on the fluctuations of particle positions starting with certain deterministic initial conditions for large time t one has regions with constant and linearly decreasing density the fluctuations on these two regions are given by the airy1 and airy2 processes whose one point distributions are the goe and gue tracy widom distributions of random matrix theory in this paper we analyze the transition region between these two regimes and obtain the transition process its one point distribution is a new interpolation between goe and gue edge distributions 2007 wiley periodicals inc

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

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

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

stationary correlations for the 1d kpz equation we study exact stationary properties of the one dimensional kardar parisi zhang kpz equation by using the replica approach the stationary state for the kpz equation is realized by setting the initial condition the two sided brownian motion bm with respect to the space variable developing techniques for dealing with this initial condition in the replica analysis we elucidate some exact nature of the height fluctuation for the kpz equation in particular we obtain an explicit representation of the probability distribution of the height in terms of the fredholm determinants furthermore from this expression we also get the exact expression of the space time two point correlation function

large time asymptotics of growth models on space like paths ii png and parallel tasep we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints the joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process the long time scaling limit of the surface height is shown to coincide with the airy1 process this result holds more generally for the observation points located along any space like path in the space time plane we also obtain the corresponding results for the discrete time tasep totally asymmetric simple exclusion process with parallel update

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

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

chattering analysis a formal mathematical definition of chattering is proposed chattering phenomena are classified into three types in particular the first type is harmless and cannot be avoided chattering properties of various control approaches are considered the dangerous second and third types of chattering phenomena are proved to be removable by proper use of high order sliding modes hosm fast stable actuators and sensors only generate the first type of chattering in hosm systems and practically never affect the sliding motion computer simulation confirms the theoretical results