random wavelet series based on a tree indexed markov chain we study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree indexed markov chain we determine the law of the spectrum of singularities of these series thereby performing their multifractal analysis we also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given holder exponent form a set with large intersection

conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

conformal invariance and finite one dimensional quantum chains based on previous work of cardy 1984 the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions the method is illustrated in the case of the three and four state potts models where the anomalous dimensions of the para fermionic operators are found

surface critical behaviour of the smoothly inhomogeneous planar ising model the pfaffian method a semi infinite nearest neighbour square ising lattice is investigated whose couplings j l at a distance l from the boundary differ from homogeneity by an amount j l j infinity approximately a l on the basis of the pfaffian method the critical behaviour at the surface of this system is investigated the exponents eta11 nu11 beta1 gamma11 and delta11 all display rich non universal behaviour as a function of the amplitude a for a below a critical value there is a spontaneous surface magnetisation when the bulk l infinity is critical and an asymmetry between the exponents on either side of the critical point

solvability of some operator equations and periodic solutions of nonlinear functional differential equations on donne des conditions suffisantes pour la resolubilite d equations operateurs lx nx dans des espaces normes avec l operateur lineaire non inversible mais a noyau de dimension finie et n un operateur non lineaire

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

multifractal analysis of infinite products we construct a family of measures called infinite products which generalize gibbs measures in the one dimensional lattice gas model the multifractal properties of these measures are studied under some regularity conditions in particular if the function is differentiable we prove a formula which gives the hausdorff dimension and packing dimension of the set of singularity points of a given order mathematical examples include riesz products g measures andg measures

on multifractality and time subordination for continuous functions let be a continuous function this paper relates to the existence of a decomposition of z as z g f where is a monofractal function with exponent 0 h 1 and f 0 1 0 1 is a time subordinator i e the integral of a positive borel measure supported by 0 1 an equivalent question consists of searching for a multifractal parametrization of z which transforms z into a monofractal function we establish that such a decomposition can be found for a large class of functions which includes the usual examples of multifractal functions we find an interesting relationship between self similar functions and self similar measures as an application of our results our theorems yield new insights in the understanding of the multifractal behaviour of functions giving a significant role to the regularity analysis of borel measures

exact results for vicious walker models of domain walls non intersecting or vicious random walker models in one dimension can be interpreted as models of domain walls in two dimensions three problems pertaining to vicious walker models are solved the first is the exact evaluation of the partition function for the random turns model of vicious walkers on a lattice in this model at each tick of the clock a randomly chosen walker must move one step to the left or one step to the right the second problem is the calculation of the mean spacing between walls in terms of the chemical potential for a brownian motion model of continuous domain walls in a strip while the final problem solved is the calculation of the correlation between defects which occur when two domain walls meet and end without crossing the whole system

the multifractal nature of levy processes we show that the sample paths of most levy processes are multifractal functions and we determine their spectrum of singularities

the singularity spectrum of levy processes in multifractal time abstract let x x t t 0 be a levy process and a positive borel measure on r suppose that the integral of defines a continuous increasing multifractal time f t 0 0 t under suitable assumptions on we compute the singularity spectrum of the sample paths of the process x in time defined as the process x f t t 0 a fundamental example consists in taking a measure equal to an independent random cascade and independently of a suitable stable levy process x then the associated process x in time is naturally related to the so called fixed points of the smoothing transformation in interacting particles systems our results rely on recent heterogeneous ubiquity theorems

exact expressions for row correlation functions in the isotropic d 2 ising model we present exact explicit expressions for the row spin spin correlation functions 00 n0 in the isotropicd 2 ising model in terms of elliptic integrals forn 5 we also give a general structural formula for 00 n0

hausdorff and packing spectra large deviations and free energy for branching random walks in mathbb r d consider an mathbb r d valued branching random walk brw on a supercritical galton watson tree without any assumption on the distribution of this brw we compute almost surely and simultaneously the hausdorff and packing dimensions of the level sets e k of infinite branches in the boundary of the tree endowed with its standard metric along which the averages of the brw have a given closed connected set of limit points k this goes beyond multifractal analysis which only considers those level sets when k ranges in the set of singletons alpha alpha in mathbb r d we also give a 0 law for the hausdorff and packing measures of the level sets e and compute the free energy of the associated logarithmically correlated random energy model in full generality moreover our results complete the previous works on multifractal analysis by including the levels which do not belong to the range of the gradient of the free energy this covers in particular a situation that was until now badly understood namely the case where a first order phase transition occurs as a consequence of our study we can also describe the whole singularity spectrum of mandelbrot measures as well as the associated free energy function or lq spectrum when a first order phase transition occurs

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

further exact solutions of the eight vertex sos model and generalizations of the rogers ramanujan identities the restricted eight vertex solid on solid sos model is an exactly solvable class of two dimensional lattice models to each sitei of the lattice there is associated an integer heightli restricted to the range 1 li r 1 the boltzmann weights of the model are expressed in terms of elliptic functions of period 2k and involve a parameter in an earlier paper we considered the case k r here we generalize those considerations to the case sk r s an integer relatively prime tor we are again led to generalizations of the rogers ramanujan identities

critical scaling for monodromy fields the large scale asymptotics of the correlations for a family of two dimensional lattice field theories is calculated at the critical temperature

random vicious walks and random matrices a lock step walk is a one dimensional integer lattice walk in discrete time suppose that initially there are infinitely many walkers on the nonnegative even integer sites at each moment of time every walker moves either to its left or to its right with equal probability the only constraint is that no two walkers can occupy the same site at the same time hence we describe the walk as vicious it is proved that as time tends to infinity a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the scaled largest eigenvalue of a random goe matrix goe tracy widom distribution the proof is based on the bijection between path configurations and semistandard young tableaux established recently by guttmann owczarek and viennot the distribution of semistandard young tableaux is analyzed using the hankel determinant expression for the probability obtained from the work of rains and the author the asymptotics of the hankel determinant are then obtained by applying the deift zhou steepest descent method to the riemann hilbert problem for the related orthogonal polynomials 2000 john wiley sons inc

a multifractal formalism let x be a metric space and a borel probability measure on x for q t r and e x write formula then hq t and pq t are borel measures hq t is a multifractal generalization of the centered hausdorff measure and pq t is a multifractal generalization of the packing measure the measures hq t and pq t define for a fixed q in the usual way a generalized hausdorff dimension dimq e and a generalized packing dimension dimq e of subsets e of x we study the functions b q dimq supp b q dimq supp and their relation to the so called multifractal spectra functions of formula we prove among other things that f f is bounded from above by the legendre transform of b b and that equality holds for graph directed self similar measures and cookie cutter measures finally we discuss the connection with generalized renyi dimensions

mean field approximation for spacing distribution functions in classical systems we propose a mean field method to calculate approximately the spacing distribution functions p n s in one dimensional classical many particle systems we compare our method with two other commonly used methods the independent interval approximation and the extended wigner surmise in our mean field approach p n s is calculated from a set of langevin equations which are decoupled by using a mean field approximation we find that in spite of its simplicity the mean field approximation provides good results in several systems we offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system the physical interpretation of each method is also discussed

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

macdonald denominators for affine root systems orthogonal theta functions and elliptic determinantal point processes rosengren and schlosser introduced notions of rn theta functions for the seven types of irreducible reduced affine root systems rn an 1 bn bn cn cn bcn dn n n and gave the macdonald denominator formulas we prove that if the variables of the rn theta functions are properly scaled with n they construct seven sets of biorthogonal functions each of which has a continuous parameter t 0 t with given 0 t following the standard method in random matrix theory we introduce seven types of one parameter t 0 t families of determinantal point processes in one dimension in which the correlation kernels are expressed by the biorthogonal theta functions we demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions in the scaling limits associated with n we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t 0 t we give new expressions for the macdonald denominators using the karlin mcgregor lindstrom gessel viennot determinants for noncolliding brownian paths and show the realization of the associated elliptic determinantal point processes as noncolliding brownian brides with a time duration t which are specified by the pinned configurations at time t 0 and t t rosengren and schlosser introduced notions of rn theta functions for the seven types of irreducible reduced affine root systems rn an 1 bn bn cn cn bcn dn n n and gave the macdonald denominator formulas we prove that if the variables of the rn theta functions are properly scaled with n they construct seven sets of biorthogonal functions each of which has a continuous parameter t 0 t with given 0 t following the standard method in random matrix theory we introduce seven types of one parameter t 0 t families of determinantal point processes in one dimension in which the correlation kernels are expressed by the biorthogonal theta functions we demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions in the scaling limits associated with n we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter t

scaling exponents and multifractal dimensions for independent random cascades this paper is concerned with mandelbrot s stochastic cascade measures the problems of i scaling exponents of structure functions of the measure q and ii multifractal dimensions are considered for cascades with a generator vector w1 wc of the general type these problems were previously studied for independent strongly bounded variableswi 0 a wi c consequently a broad class of models used in applications including kolmogorov s long normal model in turbulence long stable universal cascades in atmospheric dynamics has not been covered roughly speaking problems i ii are here solved under the condition that the scaling exists the function is calculated for all arguments previously this was done for positiveq and a new effect emerges the function can generally involve discontinuities in the first derivative as well as in the second

elliptic determinantal process of type a we introduce an elliptic extension of dyson s brownian motion model which is a temporally inhomogeneous diffusion process of noncolliding particles defined on a circle using elliptic determinant evaluations related to the reduced affine root system of type a we give determinantal martingale representation dmr for the process when it is started at the configuration with equidistant spacing on the circle dmr proves that the process is determinantal and the spatio temporal correlation kernel is determined by taking temporally homogeneous limits of the present elliptic determinantal process trigonometric and hyperbolic versions of noncolliding diffusion processes are studied

moments of vicious walkers and mobius graph expansions a system of brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval 0 t is studied the spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the gaussian unitary ensemble gue statistics to the gaussian orthogonal ensemble goe statistics as the time t goes on from 0 to t in the present paper we characterize this gue to goe transition by presenting the graphical expansion formula for the moments of positions of vicious walkers in the gue limit stackrel ensuremath rightarrow t 0 only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus following the time evolution of the vicious walkers however the graphs with twisted ribbons called m obius graphs increase their contribution to our expansion formula and we have to deal with the topology of nonorientable surfaces application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the m obius expansion using the stirling numbers of the first kind

a localized jarnik besicovitch theorem abstract fundamental questions in diophantine approximation are related to the hausdorff dimension of sets of the form x r x where 1 and x is the diophantine approximation exponent of an irrational number x we go beyond the classical results by computing the hausdorff dimension of the sets x r x f x where f is a continuous function our theorem applies to the study of the approximation exponents by various approximation families it also applies to functions f which are continuous outside a set of prescribed hausdorff dimension

random walks and random permutations a connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the gue random hermitian matrices the scaling of the distribution gives the maximum mean displacement after t time steps as 2t 1 2 with standard deviation proportional to 1 3 the exponent 1 3 is typical of a large class of two dimensional growth problems

vicious walkers and young tableaux i without walls we rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of young tableaux and integer partitions in this way we provide a proof of a result previously only conjectured for the total number of stars

ising field theory quadratic difference equations for the n point green s functions on the lattice for the two dimensional ising model at arbitrary temperature a system of quadratic difference equations is presented involving the correlations of n order variables and that of n ensuremath 2 order and two disorder variables with suitable boundary conditions these equations specify the correlations

correlation functions for the heisenberg xxz antiferromagnet the general method for calculation of correlation functions in integrable quantum models has been given in papers 1 2 the correlation function of the third components of local spins for the heisenberg one dimensionalxxz antiferromagnet is calculated in this paper the answer is a series which gives in particular an improved version of the usual perturbative expansion in the anisotropy parameter the remarkable property of the series obtained is that the long distance asymptotics of the correlator is given already by the first term the arguments are given in favour of the convergence of the series

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

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

a combinatorial interpretation of the free fermion condition of the six vertex model the free fermion condition of the six vertex model provides a five parameter sub manifold on which the bethe ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple hence allowing explicit solutions such conditions arose originally in early field theoretic s matrix approaches here we provide a combinatorial explanation for the condition in terms of a generalized gessel viennot involution by doing so we extend the use of the gessel viennot theorem originally devised for non intersecting walks only to a special weighted type of intersecting walk and hence express the partition function of n such walks starting and finishing at fixed endpoints in terms of the single walk partition functions

combining multifractal additive and multiplicative chaos the purpose of this article is the study of the new class of multifrac tal measures which combines additive and multiplicative chaos defined by x j 1 b j j 2 x

existence theorems for equations in normed spaces and boundary value problems for nonlinear vector ordinary differential equations on demontre l existence des solutions des equations dans des espaces normes quand la partie non lineaire de l equation satisfait des conditions de croissance et d asymptote

inside singularity sets of random gibbs measures we evaluate the scale at which the multifractal structure of some random gibbs measures becomes discernible the value of this scale is obtained through what we call the growth speed in holder singularity sets of a borel measure this growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self similar gibbs measures our results are useful to understand the multifractal nature of various heterogeneous jump processes

statistical behavior of domain systems we study the statistical behavior of two out of equilibrium systems the first one is a quasi one dimensional gas with two species of particles under the action of an external field which drives each species in opposite directions the second one is a one dimensional spin system with nearest neighbor interactions also under the influence of an external driving force both systems show a dynamical scaling with domain formation the statistical behavior of these domains is compared with models based on the coalescing random walk and the interacting random walk we find that the scaling domain size distribution of the gas and the spin systems is well fitted by the wigner surmise which lead us to explore a possible connection between these systems and the circular orthogonal ensemble of random matrices however the study of the correlation function of the domain edges shows that the statistical behavior of the domains in both gas and spin systems is not completely well described by a circular orthogonal ensemble nor it is by other models proposed such as the coalescing random walk and the interacting random walk nevertheless we find that a simple model of independent intervals describes more closely the statistical behavior of the domains formed in these systems

exact solution of the lock step model of vicious walkers the lock step model of vicious walkers on a one dimensional lattice allows each walker at the tick of a clock to move either one lattice site to the left or one lattice site to the right the only restriction is that no two walkers may arrive at the same lattice site or pass one another in periodic boundary conditions the partition function and correlation function for this model are calculated exactly taking the continuum limit gives an exactly solvable model of vicious walkers undergoing brownian motion

existence of solutions for some nonlinear problems with p laplacian like operators abstract the purpose of this paper is to obtain some existence results of solutions for the nonlinear boundary value problems with p laplacian like operators

ordering of random walks the leader and the laggard we investigate two complementary problems related to maintaining the relative positions of n random walks on the line i the leader problem that is the probability n t that the leftmost particle remains the leftmost as a function of time and ii the laggard problem the probability n t that the rightmost particle never becomes the leftmost we map these ordering problems onto an equivalent n 1 dimensional electrostatic problem from this construction we obtain a very accurate estimate for n t for n 4 the first case that is not exactly solvable 4 t t 4 with 4 0 91342 8 the probability of being the laggard also decays algebraically n t t n we derive 2 1 2 3 3 8 and argue that n n 1 ln n as n

on non scale invariant infinitely divisible cascades multiplicative processes multifractals and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models ranging from hydrodynamic turbulence to computer network traffic from image processing to economics the methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales while appealing due to their simplicity these constructions have limited applicability as they lead by default to power law progression of moments through scales to nonstationary increments and often to inherent log periodic scaling which favors an exponential set of scales this paper studies and develops a wide class of infinitely divisible cascades idc thereby establishing the first reported cases of controllable scaling of moments in non power law form embedded in the framework of idc these processes exhibit stationary increments and scaling over a continuous range of scales criteria for convergence further statistical properties as well as matlab routines are provided

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

the graph and range singularity spectra of b adic independent cascade functions abstract with the iso holder sets of a function we naturally associate subsets of the graph and the range of the function we compute the hausdorff dimension of these subsets for a class of statistically self similar multifractal functions namely the b adic independent cascade functions

nonintersecting brownian motions on the unit circle we consider an ensemble of n nonintersecting brownian particles on the unit circle with diffusion parameter n 1 2 which are conditioned to begin at the same point and to return to that point after time t but otherwise not to intersect there is a critical value of t which separates the subcritical case in which it is vanishingly unlikely that the particles wrap around the circle and the supercritical case in which particles may wrap around the circle in this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution we also give a streamlined approach to identifying the pearcey and tacnode processes in scaling limits the formula of the tacnode correlation kernel is new and involves a solution to a lax system for the painleve ii equation of size 2 2 the proofs are based on the determinantal structure of the ensemble asymptotic results for the related system of discrete gaussian orthogonal polynomials and a formulation of the correlation kernel in terms of a double contour integral

some exact correlations in the dyson brownian motion model for transitions to the cue abstract the dyson brownian motion model for transitions to the cue is considered for initial eigenvalue probability density functions corresponding to the coe and cse the density density correlation function between an eigenvalue at position initially 0 and an eigenvalue at position for general is calculated theoretical predictions for the asymptotic behaviour are verified and the distribution for the initial rate of change of the eigenvalues is calculated also for initial conditions corresponding to equispaced eigenvalues the n point equal parameter distribution is computed the extension of the calculation of these quantities to a multiparameter version of the theory is given

advanced determinant calculus a complement abstract this is a complement to my previous article advanced determinant calculus c krattenthaler advanced determinant calculus seminaire lotharingien combin 42 1999 the andrews festschrift article b42q 67 pp in the present article i share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory g almkvist c krattenthaler j petersson some new formulas for experiment math 12 2003 441 456 moreover i add a list of determinant evaluations which i consider as interesting which have been found since the appearance of the previous article or which i failed to mention there including several conjectures and open problems

the quantum inverse scattering method approach to correlation functions the inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with ther matrix of xxx type these models include the one dimensional bose gas and the heisenberg xxx model the algebraic questions of the problem are considered

vicious walkers friendly walkers and young tableaux iii between two walls we derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls as well as corresponding results for the analogous problem of friendly walkers our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series