markov processes of infinitely many nonintersecting random walks consider an n dimensional markov chain obtained from n one dimensional random walks by doob h transform with the q vandermonde determinant we prove that as n becomes large these markov chains converge to an infinite dimensional feller markov process the dynamical correlation functions of the limit process are determinantal with an explicit correlation kernel the key idea is to identify random point processes on z with q gibbs measures on gelfand tsetlin schemes and construct markov processes on the latter space independently we analyze the large time behavior of pushasep with finitely many particles and particle dependent jump rates it arises as a marginal of our dynamics on gelfand tsetlin schemes the asymptotics is given by a product of a marginal of the gue minor process and geometric distributions

dirichlet form approach to infinite dimensional wiener processes with singular interactions we construct infinite dimensional wiener processes with interactions by constructing specific quasi regular dirichlet forms our assumptions are very mild accordingly our results can be applied to singular interactions such as hard core potentials lennard jones type potentials and dyson s model we construct nonequilibrium dynamics

discrete approximations of determinantal point processes on continuous spaces tree representations and tail triviality we prove tail triviality of determinantal point processes mu on continuous spaces tail triviality has been proved for such processes only on discrete spaces and hence we have generalized the result to continuous spaces to do this we construct tree representations that is discrete approximations of determinantal point processes enjoying a determinantal structure there are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic gaussian analytic function with bergman kernel and the thermodynamic limit of eigenvalues of gaussian random matrices for hbox sine 2 hbox airy 2 hbox bessel 2 and ginibre point processes our main theorem proves all these point processes are tail trivial

conformal welding problem flow line problem and multiple schramm loewner evolution a quantum surface qs is an equivalence class of pairs d h of simply connected domains d c and random distributions h on d induced by the conformal equivalence for random metric spaces this distribution valued random field is extended to a qs with n 1 marked boundary points mbps with n z 0 we propose the conformal welding problem for it in the case of n z 1 if n 1 it is reduced to the problem introduced by sheffield who solved it by coupling the qs with the schramm loewner evolution sle when n 3 there naturally appears room for making the configuration of mbps random and hence a new problem arises how to determine the probability law of the configuration we report that the multiple sle in h driven by the dyson model on r helps us to fix the problems and makes them solvable for any n 3 we also propose the flow line problem for an imaginary surface with boundary condition changing points bccps in the case when the number of bccps is two this problem was solved by miller and sheffield we address the general case with an arbitrary number of bccps in a similar manner to the conformal welding problem we again find that the multiple sle driven by the dyson model plays a key role to solve the flow line problem

continuum limits of random matrices and the brownian carousel we show that at any location away from the spectral edge the eigenvalues of the gaussian unitary ensemble and its general siblings converge to sine a translation invariant point process this process has a geometric description in term of the brownian carousel a deterministic function of brownian motion in the hyperbolic plane

interacting brownian motions in infinite dimensions with logarithmic interaction potentials we investigate the construction of diffusions consisting of infinitely numerous brownian particles moving in mathbb r d and interacting via logarithmic functions two dimensional coulomb potentials these potentials are very strong and act over a long range in nature the associated equilibrium states are no longer gibbs measures we present general results for the construction of such diffusions and as applications thereof construct two typical interacting brownian motions with logarithmic interaction potentials namely the dyson model in infinite dimensions and ginibre interacting brownian motions the former is a particle system in mathbb r while the latter is in mathbb r 2 both models are translation and rotation invariant in space and as such are prototypes of dimensions d 1 2 respectively the equilibrium states of the former diffusion model are determinantal or pfaffian random point fields with sine kernels they appear in the thermodynamical limits of the spectrum of the ensembles of gaussian random matrices such as goe gue and gse the equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non hermitian gaussian random matrices known as the ginibre ensemble

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

correction to list decoding of generalized reed solomon codes over commutative rings in the above titled paper ibid vol 51 no 1 pp 411 419 jan 05 lemma 6 is incorrect due to an error on the proof consequently proposition 2 is restated here

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

brownian gibbs property for airy line ensembles consider a collection of n brownian bridges b i n n to mathbb r b i n b i n 0 1 i n conditioned not to intersect the edge scaling limit of this system is obtained by taking a weak limit as n of the collection of curves scaled so that the point 0 21 2 n is fixed and space is squeezed horizontally by a factor of n 2 3 and vertically by n 1 3 if a parabola is added to each of the curves of this scaling limit an x translation invariant process sometimes called the multi line airy process is obtained we prove the existence of a version of this process which we call the airy line ensemble in which the curves are almost surely everywhere continuous and non intersecting this process naturally arises in the study of growth processes and random matrix ensembles as do related processes with wanderers and outliers we formulate our results to treat these relatives as well note that the law of the finite collection of brownian bridges above has the property called the brownian gibbs property of being invariant under the following action select an index 1 k n and erase b k on a fixed time interval a b n n then replace this erased curve with a new curve on a b according to the law of a brownian bridge between the two existing endpoints a b k a and b b k b conditioned to intersect neither the curve above nor the one below we show that this property is preserved under the edge scaling limit and thus establish that the airy line ensemble has the brownian gibbs property an immediate consequence of the brownian gibbs property is a confirmation of the prediction of m prahofer and h spohn that each line of the airy line ensemble is locally absolutely continuous with respect to brownian motion we also obtain a proof of the long standing conjecture of k johansson that the top line of the airy line ensemble minus a parabola attains its maximum at a unique point this establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi line airy process and readily yields several other interesting properties of this process

a twisted generalization of linear poisson brackets of hydrodynamic type we study a twisted generalization of linear poisson brackets of hydrodynamic type in which the lie bracket is replaced by a hom lie bracket with some natural additional conditions such structures correspond to the hom novikov algebras introduced by yau which is the twisted version of balinsky novikov s approach of constructing a lie algebra from a novikov algebra certain central extensions of this twisted generalization of linear poisson brackets of hydrodynamic type are obtained from the bilinear forms on their corresponding hom novikov algebras satisfying some invariance conditions finally we give some examples of the infinite dimensional hom lie algebras constructed from the hom novikov algebras in particular there is an interesting twisted generalization of the virasoro algebra

positivity of the self diffusion matrix of interacting brownian particles with hard core we prove the positivity of the self diffusion matrix of interacting brownian particles with hard core when the dimension of the space is greater than or equal to 2 here the self diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle we will do this for all activities z 0 of gibbs measures in particular for large z the case of high density particles a typical example of such a particle system is an infinite amount of hard core brownian balls

efficient decoding of z sub p k linear codes we present a method for lifting a decoding scheme for a linear code over z sub p to a decoding scheme for a linear code over z sub p k and characterize the class of codes for which this kind of lifting works its interest lies in its great generality and efficiency

infinite dimensional stochastic differential equations and tail σ fields we present general theorems solving the long standing problem of the existence and pathwise uniqueness of strong solutions of infinite dimensional stochastic differential equations isdes called interacting brownian motions these isdes describe the dynamics of infinitely many brownian particles moving in mathbb r d with free potential varphi and mutual interaction potential varpsi we apply the theorems to essentially all interaction potentials of ruelle s class such as the lennard jones 6 12 potential and riesz potentials and to logarithmic potentials appearing in random matrix theory we solve isdes of the ginibre interacting brownian motion and the sine beta interacting brownian motion with beta 1 2 4 we also use the theorems in separate papers for the airy and bessel interacting brownian motions one of the critical points for proving the general theorems is to establish a new formulation of solutions of isdes in terms of tail sigma fields of labeled path spaces consisting of trajectories of infinitely many particles these formulations are equivalent to the original notions of solutions of isdes and more feasible to treat in infinite dimensions

a system of infinitely many mutually reflecting brownian balls in ℝd an infinite system of skorohod type equations is studied the unique solution of the system is obtained from a finite case by passing to the limit it is a diffusion process describing a system of infinitely many brownian hard balls and has a gibbs state associated with the hard core pair potential as a reversible measure

infinite dimensional stochastic differential equations related to random matrices we solve infinite dimensional stochastic differential equations isdes describing an infinite number of brownian particles interacting via two dimensional coulomb potentials the equilibrium states of the associated unlabeled stochastic dynamics are the ginibre random point field and dyson s measures which appear in random matrix theory to solve the isdes we establish an integration by parts formula for these measures because the long range effect of two dimensional coulomb potentials is quite strong the properties of brownian particles interacting with two dimensional coulomb potentials are remarkably different from those of brownian particles interacting with ruelle s class interaction potentials as an example we prove that the interacting brownian particles associated with the ginibre random point field satisfy plural isdes

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

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

hom novikov algebras we study a twisted generalization of novikov algebras called hom novikov algebras in which the two defining identities are twisted by a linear map it is shown that hom novikov algebras can be obtained from novikov algebras by twisting along any algebra endomorphism all algebra endomorphisms on complex novikov algebras of dimensions 2 or 3 are computed and their associated hom novikov algebras are described explicitly another class of hom novikov algebras is constructed from hom commutative algebras together with a derivation generalizing a construction due to dorfman and gel fand two other classes of hom novikov algebras are constructed from hom lie algebras together with a suitable linear endomorphism generalizing a construction due to bai and meng

list decoding of generalized reed solomon codes over commutative rings we show that the list decoding procedure of guruswami and sudan may be used to decode generalized reed solomon rs codes defined over commutative rings with identity an algorithm for performing the first of the two phases of this decoding procedure is also given in particular we show that for generalized rs codes over chain rings the probability of picking an incorrect codeword from a given list can be reduced by increasing the size of the code alphabet while keeping the length and minimum hamming distance of the code constant

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

infinite dimensional stochastic differential equations for dyson s model in this paper we show the strong existence and the pathwise uniqueness of an infinite dimensional stochastic differential equation sde corresponding to the bulk limit of dyson s brownian motion for all beta ge 1 our construction applies to an explicit and general class of initial conditions including the lattice configuration x i mathbb z and the sine process we further show the convergence of the finite to infinite dimensional sde this convergence concludes the determinantal formula of katori and tanemura commun math phys 293 2 469 497 2010 for the solution of this sde at beta 2

gradient dynamics of infinite point systems nonequilibrium gradient dynamics of d dimensional particle systems is investigated the interaction is given by a superstable pair potential of finite range solutions are constructed in the well defined set of locally finite configurations with a logarithmic order of energy fluctuations if the system is deterministic and d 2 then singular potentials are also allowed for stochastic models with a smooth interaction we need d 4 in order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium we investigate smoothness of the markov semigroup and describe some properties of its generator 0 introduction the purpose of this paper is to study existence and regularity properties of solutions to the following infinite system of stochastic differential equations consider a countable set s of d dimensional particles suspended in a fluid where the interaction is given by a pair potential u rgd 00 oo in a quasi microscopic description of such systems the effect of collisions with the particles of the fluid can be represented by uncorrelated stochastic forces and the soft resistance of the liquid medium reduces the order of the equations of motion from two to one see 13 23 and 24 for some further references configurations of these systems are countable subsets of rd such that any bounded domain contains a finite number of points only particles of a configuration will be identified by labelling points by elements of s thus a