dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition the one dimensional totally asymmetric simple exclusion process tasep is considered we study the time evolution property of a tagged particle in the tasep with the step initial condition calculated is the multi time joint distribution function of its position using the relation of the dynamics of the tasep to the schur process we show that the function is represented as the fredholm determinant we also study the scaling limit the universality of the largest eigenvalue in the random matrix theory is realized in the limit when the hopping rates of all particles are the same it is found that the joint distribution function converges to that of the airy process after the time at which the particle begins to move on the other hand when there are several particles with small hopping rate in front of a tagged particle the limiting process changes at a certain time from the airy process to the process of the largest eigenvalue in the hermitian multi matrix model with external sources

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

the arctic circle boundary and the airy process we prove that the appropriately rescaled boundary of the north polar region in the aztec diamond converges to the airy process the proof uses certain determinantal point processes given by the extended krawtchouk kernel we also prove a version of propp s conjecture concerning the structure of the tiling at the center of the aztec diamond 1 introduction and results domino tilings of the aztec diamond were introduced in 8 9 asymptotic properties of random domino tilings of the aztec diamond have been studied in 5 12 15 in particular in 12 the existence of the so called arctic circle was proved the arctic circle is the asymptotic boundary of the disordered so called temperate region of the tiling outside this boundary the tiling forms a completely regular brick wall pattern the methods in 12 combined with the results in 13 show that the fluctuations of the point of intersection of the boundary of the temperate region with a line converge to the tracy widom distribution of random matrix theory in this paper we extend this result to show that the fluctuations of the boundary around the arctic circle converges to the airy process introduced in 23 the paper is a continuation of the approach used in 14 and 15 where certain point processes with determinantal correlation functions 24 and the krawtchouk ensemble were used we will use the general techniques developed in 16 and investigate an extended point process which also has determinantal correlation functions given by a kernel which we call the extended krawtchouk kernel the aztec diamond an of order n is the union of all lattice squares m m 1 l l 1 m l 2 z that lie inside the region x1 y1 x1 y1

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

shuffling algorithm for boxed plane partitions we introduce discrete time markov chains that preserve uniform measures on boxed plane partitions elementary markov steps change the size of the box from a b c to a 1 b 1 c or a 1 b 1 c algorithmic realization of each step involves o a b c operations one application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions trajectories of our markov chains can be viewed as random point configurations in the three dimensional lattice we compute the bulk limits of the correlation functions of the resulting random point process on suitable two dimensional sections the limiting correlation functions define a two dimensional determinantal point processes with certain gibbs properties

the z measures on partitions pfaffian point processes and the matrix hypergeometric kernel abstract we consider a point process on one dimensional lattice originated from the harmonic analysis on the infinite symmetric group and defined by the z measures with the deformation jack parameter 2 we derive an exact pfaffian formula for the correlation function of this process namely we prove that the correlation function is given as a pfaffian with a 2 2 matrix kernel the kernel is given in terms of the gauss hypergeometric functions and can be considered as a matrix analogue of the hypergeometric kernel introduced by a borodin and g olshanski 2000 5 our result holds for all values of admissible complex parameters

on thermodynamics of rational mapsi negative spectrum we study the pressure spectrum p t of the maximal measure for arbitrary rational maps we also consider its modified version which is defined by means of the variational principle with respect to non atomic invariant measures it is shown that for negative values of t the modified spectrum has all major features of the hyperbolic case analyticity the existence of a spectral gap for the corresponding transfer operator rigidity properties etc the spectrum p t can be computed in terms of their legendre transforms are the hausdorff and the box counting dimension spectra of the maximal measure respectively this work is closely related to a paper 32 by d ruelle

phase transitions for countable markov shifts we study the analyticity of the topological pressure for some one parameter families of potentials on countable markov shifts we relate the non analyticity of the pressure to changes in the recurrence properties of the system we give sufficient conditions for such changes to exist and not to exist we apply these results to the manneville pomeau map and use them to construct examples with different critical behavior

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

commutation relations and markov chains it is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain markov chains examples studied include random walk on irreducible representations a local random walk on partitions whose stationary distribution is the ewens distribution and some birth death chains

nice inducing schemes and the thermodynamics of rational maps we study the thermodynamic formalism of a complex rational map f of degree at least two viewed as a dynamical system acting on the riemann sphere more precisely for a real parameter t we study the existence of equilibrium states of f for the potential t rm ln left f prime right and the analytic dependence on t of the corresponding pressure function we give a fairly complete description of the thermodynamic formalism for a large class of rational maps including well known classes of non uniformly hyperbolic rational maps such as topological collet eckmann maps and much beyond in fact our results apply to all non renormalizable polynomials without indifferent periodic points to infinitely renormalizable quadratic polynomials with a priori bounds and all quadratic polynomials with real coefficients as an application for these maps we describe the dimension spectrum for lyapunov exponents and for pointwise dimensions of the measure of maximal entropy and obtain some level 1 large deviations results for polynomials as above we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in mathbb r with at most two exceptions

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

markov processes on the path space of the gelfand tsetlin graph and on its boundary we construct a four parameter family of markov processes on infinite gelfand tsetlin schemes that preserve the class of central gibbs measures any process in the family induces a feller markov process on the infinite dimensional boundary of the gelfand tsetlin graph or equivalently the space of extreme characters of the infinite dimensional unitary group u the process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on u posed in olshanski 2003 44 as was shown in borodin and olshanski 2005 11 this measure can also be described as a determinantal point process with a correlation kernel expressed through the gauss hypergeometric function

measures with positive lyapunov exponent and conformal measures in rational dynamics ergodic properties of rational maps are studied generalising the work of f ledrappier a new construction allows for simpler proofs of stronger results very general conformal measures are considered equivalent conditions are given for an ergodic invariant probability measure with positive lyapunov exponent to be absolutely continuous with respect to a general conformal measure if they hold we can construct an induced expanding markov map with integrable return time which generates the invariant measure

markov processes on partitions we introduce and study a family of markov processes on partitions the processes preserve the so called z measures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group we show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels we also compute the scaling limits of the kernels in two different regimes the limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random young diagrams and the behavior of the young diagrams near the diagonal

conditioned random walks and the rsk correspondence we consider the stochastic evolution of three variants of the rsk algorithm giving both analytic descriptions and probabilistic interpretations symmetric functions play a key role and the probabilistic interpretations are obtained by elementary doob hunt theory in each case the evolution of the shape of the tableau obtained via the rsk algorithm can be interpreted as a conditioned random walk this is intuitively appealing and can be used for example to obtain certain relationships between orthogonal polynomial ensembles in a certain scaling limit there is a continous version of the rsk algorithm which inherits much of the structure exhibited in the discrete settings intertwining relationships between conditioned and unconditioned random walks are also given in the continuous limit these are related to the harish chandra itzyksen zuber integral

stein s method and plancherel measure of the symmetric group we initiate a stein s method approach to the study of the plancherel measure of the symmetric group a new proof of kerov s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained the proof gives an error term the construction of an exchangeable pair needed for applying stein s method arises from the theory of harmonic functions on bratelli diagrams we also find the spectrum of the markov chain on partitions underlying the construction of the exchangeable pair this yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group

homogenization of dirichlet parabolic problems for coefficients and open sets simultaneously variable and applications to optimal design in a previous paper we studied the homogenization of a sequence of parabolic linear dirichlet problems when the coefficients and the domains vary arbitrarily here we improve the convergence result given in this paper by showing the strong convergence in l2 every time this is applied to obtain an existence result for control problems in optimal design written in a relaxed form the control variables are the material and the shape

two dimensional yang mills theory painleve equations and the six vertex model we show that the chiral partition function of two dimensional yang mills theory on the sphere can be mapped to the partition function of the homogeneous six vertex model with domain wall boundary conditions in the ferroelectric phase a discrete matrix model description in both cases is given by the meixner ensemble leading to a representation in terms of a stochastic growth model we show that the partition function is a particular case of the z measure on the set of young diagrams yielding a unitary matrix model for chiral yang mills theory on s2 and the identification of the partition function as a tau function of the painlev v equation we describe the role played by generalized non chiral yang mills theory on s2 in relating the meixner matrix model to the toda chain hierarchy encompassing the integrability of the six vertex model we also argue that the thermodynamic behaviour of the six vertex model in the disordered and antiferroelectric phases are captured by particular q deformations of two dimensional yang mills theory on the sphere

asymptotic behaviour of linear dirichlet parabolic problems with variable operators depending on time in varying domains abstract we study the asymptotic behaviour of the solutions of linear parabolic dirichlet problems when the coefficients and the domains where the problems are posed vary simultaneously in the limit problem it appear the h limit of the operators and as it is usual in the homogenization of dirichlet problems a new term of order zero we also obtain a corrector result

eynard mehta theorem schur process and their pfaffian analogs we give simple linear algebraic proofs of the eynard mehta theorem the okounkov reshetikhin formula for the correlation kernel of the schur process and pfaffian analogs of these results we also discuss certain general properties of the spaces of all determinantal and pfaffian processes on a given finite set

quantum random walk on the dual of su n we study a quantum random walk ona su n the von neumann algebra of su n obtained by tensoring the basic representation of su n two classical markov chains are derived from this quantum random walk by restriction to commutative subalgebras ofa su n and the main result of the paper states that these two markov chains are related by means of doob sh processes

harmonic analysis on the infinite symmetric group the infinite symmetric group s whose elements are finite permutations of 1 2 3 is a model example of a big group by virtue of an old result of murray von neumann the one sided regular representation of s in the hilbert space l2 s generates a type ii1 von neumann factor while the two sided regular representation is irreducible this shows that the conventional scheme of harmonic analysis is not applicable to s for the former representation decomposition into irreducibles is highly non unique and for the latter representation there is no need of any decomposition at all we start with constructing a compactification mathfrak s supset s infty which we call the space of virtual permutations although mathfrak s is no longer a group it still admits a natural two sided action of s thus mathfrak s is a g space where g stands for the product of two copies of s on mathfrak s there exists a unique g invariant probability measure 1 which has to be viewed as a true haar measure for s more generally we include 1 into a family t t 0 of distinguished g quasiinvariant probability measures on virtual permutations by making use of these measures we construct a family tz z of unitary representations of g called generalized regular representations each representation tz with z 0 can be realized in the hilbert space l 2 mathfrak s mu t where t z 2 as z the generalized regular representations tz approach in a suitable sense the naive two sided regular representation of the group g in the space l2 s in contrast with the latter representation the generalized regular representations tz are highly reducible and have a rich structure we prove that any tz admits a unique decomposition into a multiplicity free continuous integral of irreducible representations of g for any two distinct and not conjugate complex numbers z1 z2 the spectral types of the representations t z 1 and t z 2 are shown to be disjoint in the case z a complete description of the spectral type is obtained further work on the case z reveals a remarkable link with stochastic point processes and random matrix theory

infinite dimensional diffusions as limits of random walks on partitions starting with finite markov chains on partitions of a natural number n we construct via a scaling limit transition as n a family of infinite dimensional diffusion processes the limit processes are ergodic their stationary distributions the so called z measures appeared earlier in the problem of harmonic analysis for the infinite symmetric group the generators of the processes are explicitly described

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

positive lyapunov exponents in families of one dimensional dynamical systems we give a generalization of jakobson s theorem ja and a precise estimate for the density of parameters corresponding to systems with absolutely continuous invariant measures

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

continuous phase transitions for dynamical systems we study the asymptotic expansion of the topological pressure of one parameter families of potentials at a point of non analyticity the singularity is related qualitatively and quantitatively to non gaussian limit laws and to slow decay of correlations with respect to the equilibrium measure

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

hyperbolicity sinks and measure in one dimensional dynamics letf be ac2 map of the circle or the interval and let f denote the complement of the basins of attraction of the attracting periodic orbits we prove that f is a hyperbolic expanding set if and obviously only if every periodic point is hyperbolic and f doesn t contain the critical point this is the real one dimensional version of fatou s hyperbolicity criteria for holomorphic endomorphisms of the riemann sphere we also explore other applications of the techniques used for the result above proving for instance that for everyc2 immersionf of the circle i e a map of the circle onto itself without critical points either its julia set has measure zero or it is the whole circle and thenf is ergodic i e positively invariant borel sets have zero or full measure

low temperature phase transitions in the quadratic family abstract we give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function this implies that several dimension spectra and large deviation rate functions associated to this map are not expected to be real analytic in contrast to the uniformly hyperbolic case the quadratic map we study has a non recurrent critical point so it is non uniformly hyperbolic in a strong sense

renormalisation induced phase transitions for unimodal maps the thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential t log df multiple and indeed infinitely many phase transitions at positive t can occur for some quadratic maps all unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

distributions on partitions point processes and the hypergeometric kernel we study a 3 parametric family of stochastic point processes on the one dimensional lattice originated from a remarkable family of representations of the infinite symmetric group we prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel the kernel can be expressed through the gauss hypergeometric function we call it the hypergeometric kernel in a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles as we showed in previous works the correlation functions of these limit processes also have determinantal form with so called whittaker kernel we show that the scaling limit of the hypergeometric kernel is the whittaker kernel integrable operator as defined by its izergin korepin and slavnov we argue that the hypergeometric kernel can be considered as a kernel defining a discrete integrable operator we also show that the hypergeometric kernel degenerates for certain values of parameters to the christoffel darboux kernel for meixner orthogonal polynomials this fact is parallel to the degeneration of the whittaker kernel to the christoffel darboux kernel for laguerre polynomials

the problem of harmonic analysis on the infinite dimensional unitary group abstract the goal of harmonic analysis on a noncommutative group is to decompose the most natural unitary representations of this group like the regular representation on irreducible ones the infinite dimensional unitary group u is one of the basic examples of big groups whose irreducible representations depend on infinitely many parameters our aim is to explain what the harmonic analysis on u consists of we deal with unitary representations of a reasonable class which are in 1 1 correspondence with characters central positive definite normalized functions on u the decomposition of any representation of this class is described by a probability measure called spectral measure on the space of indecomposable characters the indecomposable characters were found by dan voiculescu in 1976 the main result of the present paper consists in explicitly constructing a 4 parameter family of natural representations and computing their characters we view these representations as a substitute of the nonexisting regular representation of u we state the problem of harmonic analysis on u as the problem of computing the spectral measures for these natural representations a solution to this problem is given in the next paper harmonic analysis on the infinite dimensional unitary group and determinantal point processes math 0109194 to appear in ann math joint with alexei borodin we also prove a few auxiliary general results in particular it is proved that the spectral measure of any character of u can be approximated by a sequence of discrete spectral measures for the restrictions of the character to the compact unitary groups u n this fact is a starting point for computing spectral measures

renormalization and conjugacy of piecewise linear lorenz maps abstract we investigate the uniform piecewise linearizing question for a family of lorenz maps let f be a piecewise linear lorenz map with different slopes and positive topological entropy we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy it is either bi lipschitz or singular depending on whether f is renormalizable or not f is renormalizable if and only if its rotation interval degenerates to be a rational point furthermore if the endpoints are periodic points with the same rotation number then the conjugacy is quasisymmetric

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

homogenization of dirichlet parabolic systems with variable monotone operators in general perforated domains we consider the homogenization of parabolic systems with dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously we assume the operators do not depend on t then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one from this result we easily obtain the limit problem in the parabolic case

the quasi invariance property for the gamma kernel determinantal measure abstract the gamma kernel is a projection kernel of the form a x b y b x a y x y where a and b are certain functions on the one dimensional lattice expressed through euler s function the gamma kernel depends on two continuous parameters its principal minors serve as the correlation functions of a determinantal probability measure p defined on the space of infinite point configurations on the lattice as was shown earlier a borodin g olshanski adv math 194 2005 141 202 arxiv math ph 0305043 p describes the asymptotics of certain ensembles of random partitions in a limit regime theorem the determinantal measure p is quasi invariant with respect to finitary permutations of the nodes of the lattice this result is motivated by an application to a model of infinite particle stochastic dynamics

random partitions and the gamma kernel abstract we study the asymptotics of certain measures on partitions the so called z measures and their relatives in two different regimes near the diagonal of the corresponding young diagram and in the intermediate zone between the diagonal and the edge of the young diagram we prove that in both cases the limit correlation functions have determinantal form with a correlation kernel which depends on two real parameters in the first case the correlation kernel is discrete and it has a simple expression in terms of the gamma functions in the second case the correlation kernel is continuous and translationally invariant and it can be written as a ratio of two suitably scaled hyperbolic sines

natural equilibrium states for multimodal maps this paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps this class contains but it is larger than collet eckmann for a map in this class we prove existence and uniqueness of equilibrium states for the geometric potentials t log df for the largest possible interval of parameters t we also study the regularity and convexity properties of the pressure function completely characterising the first order phase transitions results concerning the existence of absolutely continuous invariant measures with respect to the lebesgue measure are also obtained

chaos butterflies also generate phase transitions we exhibit examples of mixing subshifts of finite type and of continuous potentials such that there are phase transitions but the pressure is always strictly convex more surprisingly we show that the pressure can be analytic on some interval although there exist several equilibrium states

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