signal detection in nearly gaussian skewed noise the problem of detecting a constant signal in nearly gaussian skewed noise is studied in this paper sensitivity of both the linear detector and the sign detector are examined it is shown that the performance of the linear detector deteriorates when the noise distribution becomes skewed on the other hand the sign detector appears to be less sensitive to the noise skewness furthermore a modification of the linear detector is proposed this modified scheme when the noise skewness measure is small will behave almost like the linear detector in gaussian noise

explicit combinatorial interpretation of kerov character polynomials as numbers of permutation factorizations abstract we find an explicit combinatorial interpretation of the coefficients of kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups s n in terms of free cumulants r 2 r 3 of the corresponding young diagram our interpretation is based on counting certain factorizations of a given permutation

computation in finitary stochastic and quantum processes abstract we introduce stochastic and quantum finite state transducers as computation theoretic models of classical stochastic and quantum finitary processes formal process languages representing the distribution over a process behaviors are recognized and generated by suitable specializations we characterize and compare deterministic and nondeterministic versions summarizing their relative computational power in a hierarchy of finitary process languages quantum finite state transducers and generators are a first step toward a computation theoretic analysis of individual repeatedly measured quantum dynamical systems they are explored via several physical systems including an iterated beam splitter an atom in a magnetic field and atoms in an ion trap a special case of which implements the deutsch quantum algorithm we show that these systems behaviors and so their information processing capacity depends sensitively on the measurement protocol

random sorting networks abstract a sorting network is a shortest path from 12 n to n 21 in the cayley graph of s n generated by nearest neighbour swaps we prove that for a uniform random sorting network as n the space time process of swaps converges to the product of semicircle law and lebesgue measure we conjecture that the trajectories of individual particles converge to random sine curves while the permutation matrix at half time converges to the projected surface measure of the 2 sphere we prove that in the limit the trajectories are holder 1 2 continuous while the support of the permutation matrix lies within a certain octagon a key tool is a connection with random young tableaux

jack polynomials and free cumulants abstract we study the coefficients in the expansion of jack polynomials in terms of power sums we express them as polynomials in the free cumulants of the transition measure of an anisotropic young diagram we conjecture that such polynomials have nonnegative integer coefficients this extends recent results about normalized characters of the symmetric group

jeu de taquin dynamics on infinite young tableaux and second class particles we study an infinite version of the jeu de taquin sliding game which can be thought of as a natural measure preserving transformation on the set of infinite young tableaux equipped with the plancherel probability measure we use methods from representation theory to show that the robinson schensted knuth rsk algorithm gives an isomorphism between this measure preserving dynamical system and the one sided shift dynamics on a sequence of independent and identically distributed random variables distributed uniformly on the unit interval we also show that the jeu de taquin paths induced by the transformation are asymptotically straight lines emanating from the origin in a random direction whose distribution is computed explicitly and show that this result can be interpreted as a statement on the limiting speed of a second class particle in the plancherel tasep particle system a variant of the totally asymmetric simple exclusion process associated with plancherel growth in analogy with earlier results for second class particles in the ordinary tasep

characterizations of quantum automata we define q quantum finite automata qqfas and q quantum regular grammars qqrgs and verify that they are exactly equivalent to those measure once quantum finite automata mo qfas in the literature in particular we define q quantum pushdown automata qqpdas and qpdas that are at least as powerful as those defined by moore and crutchfield and especially we focus on demonstrating the equivalence between qqpdas and qpdas also we discuss some of the properties of languages accepted by qqpdas for example every cut point language accepted by qqpda is independent of the cut point

quantum finite automata we first outline main steps and achievements along bertoni s research path in quantum finite automata theory from the very basic definitions of the models of quantum finite automata throughout the investigation of their computational and descriptional power next we choose to focus on bertoni s studies on quantum finite automata descriptional complexity in particular we expand on a statistical framework for the synthesis of succinct quantum finite automata discussing its adaptation to the case of multiperiodic events and languages we then improve such a framework to obtain even more succinct quantum finite automata for some multiperiodic languages finally we introduce some promise problems for multiperiodic inputs showing that even on this class of problems the descriptional power of quantum finite automata greatly outperforms that of equivalent classical finite automata we build small size monte carlo quantum finite automata for multiperiodic languages we analyze the modular architecture of such monte carlo quantum finite automata we solve promise problems on multiperiodic inputs by small quantum finite automata

sequential cfar detectors using a dead zone limiter the performances of some proposed sequential constant false alarm rate cfar detectors are evaluated the observations are passed through a dead zone limiter the output of which is 1 0 or 1 depending on whether the input is less than c between c and c or greater than c where c is a constant the test statistic is the sum of the outputs the test is performed on a reduced set of data those with absolute value larger than c with the test statistic being the sum of the signs of the reduced set of data both constant and linear boundaries are considered numerical results show a significant reduction of the average number of observations needed to achieve the same false alarm and detection probabilities as compared to a fixed sample size cfar detector using the same kind of test statistic

regular languages accepted by quantum automata in this paper we analyze some features of the behaviour of quantum automata in particular we prove that the class of languages recognized by quantum automata with isolated cut point is the class of reversible regular languages as a more general result we give a bound on the inverse error that implies the regularity of the language accepted by a quantum automaton

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

gaussian fluctuations of jack deformed random young diagrams we introduce a large class of random young diagrams which can be regarded as a natural one parameter deformation of some classical young diagram ensembles a deformation which is related to jack polynomials and jack characters we show that each such a random young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically gaussian

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

a random matrix model from two dimensional yang mills theory we use fat graphs to give a unified treatment of various asymptotic freeness results in particular a random matrix model from two dimensional yang mills theory on the plane is presented

determining the equivalence for one way quantum finite automata two quantum finite automata are equivalent if for any input string x the two automata accept x with equal probability in this paper we first focus on determining the equivalence for one way quantum finite automata with control language cl 1qfas defined by bertoni et al and then as an application we address the equivalence problem for measure many one way quantum finite automata mm 1qfas introduced by kondacs and watrous more specifically we obtain that i two cl 1qfas a 1 and a 2 with control languages regular languages l 1 and l 2 respectively are equivalent if and only if they are c 1n 1 2 c 2n 2 2 1 equivalent where n 1 and n 2 are the numbers of states in a 1 and a 2 respectively and c 1 and c 2 are the numbers of states in the minimal dfas that recognize l 1 and l 2 respectively furthermore if l 1 and l 2 are given in the form of dfas with m 1 and m 2 states respectively then there exists a polynomial time algorithm running in time o m 1n 1 2 m 2n 2 2 4 that takes as input a 1 and a 2 and determines whether they are equivalent ii as an application of item i two mm 1qfas a 1 and a 2 with n 1 and n 2 states respectively are equivalent if and only if they are 3n 1 2 3n 2 2 1 equivalent furthermore there is a polynomial time algorithm running in time o 3n 1 2 3n 2 2 4 that takes as input a 1 and a 2 and determines whether a 1 and a 2 are equivalent

asymptotic spectral analysis of growing regular graphs we propose the quantum probabilistic techniques to obtain the asymptotic spectral distribution of the adjacency matrix of a growing regular graph we prove the quantum central limit theorem for the adjacency matrix of a growing regular graph in the vacuum and deformed vacuum states the condition for the growth is described in terms of simple statistics arising from the stratification of the graph the asymptotic spectral distribution of the adjacency matrix is obtained from the classical reduction

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

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

characters of symmetric groups sharp bounds and applications we provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible it follows from our general bound that if a permutation s n has at most n o 1 cycles of length m then 1 1 m o 1 for all irreducible characters of s n this is a far reaching generalization of a result of fomin and lulov

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

positivity results for stanley s character polynomials in stanley r p stanley irreducible symmetric group characters of rectangular shape sem lothar combin 50 b50d 2003 11 pp the author introduces expressions for the normalized characters of the symmetric group and states some positivity conjectures for these expressions here we give an affirmative partial answer to stanley s positivity conjectures about the expressions using results on kerov polynomials in particular we use new positivity results in goulden and rattan i p goulden a rattan an explicit form for kerov s character polynomials trans amer math soc in press math co 0505317 november 2005 we shall see that the generating series c t introduced in i p goulden a rattan an explicit form for kerov s character polynomials trans amer math soc in press math co 0505317 november 2005 is critical to our discussion

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

shifted symmetric functions and multirectangular coordinates of young diagrams abstract in this paper we study shifted schur functions s as well as a new family of shifted symmetric functions k linked to kostka numbers we prove that both are polynomials in multi rectangular coordinates with nonnegative coefficients when written in terms of falling factorials we then propose a conjectural generalization to the jack setting this conjecture is a lifting of knop and sahi s positivity result for usual jack polynomials and resembles recent conjectures of lassalle we prove our conjecture for one part partitions

an explicit formula for the characters of the symmetric group we give an explicit expression of the normalized characters of the symmetric group in terms of the contents of the partition labelling the representation

two stage wilcoxon detectors using conditional tests the idea of the wilcoxon detector using a conditional test is extended to two stage detectors which constitute a class of sequential detectors three different two stage detectors are considered using the wilcoxon statistic with conditional tests the three detectors differ in the usage of the samples of the first stage for computing the test statistic in the second stage in computing the test statistic in the second stage the first detector disregards all the samples of the first stage the second detector uses the value of the test statistic of the first stage and the third detector uses all the samples of the first stage the asymptotic relative efficiency improves from the first to the second detector comparisons of these detectors with fixed sample size fss detectors show that they are better than both the fss linear detector and the fss conditional wilcoxon detector under the same false alarm and detection probabilities

representations of symmetric groups and free probability abstract we consider representations of symmetric groupssqfor largeq we give the asymptotic behaviour of the characters when the corresponding young diagrams rescaled by a factorq 1 2 converge to some prescribed shape this behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram we also show that the basic operations of representation theory like taking tensor products restriction or induction have a limiting behavior which can be described using the free probability theory of d voiculescu

quantum mechanical hamiltonian models of turing machines quantum mechanical hamiltonian models which represent an aribtrary but finite number of steps of any turing machine computation are constructed here on a finite lattice of spin 1 2 systems different regions of the lattice correspond to different components of the turing machine plus recording system successive states of any machine computation are represented in the model by spin configuration states both time independent and time dependent hamiltonian models are constructed here the time independent models do not dissipate energy or degrade the system state as they evolve they operate close to the quantum limit in that the total system energy uncertainty computation speed is close to the limit given by the time energy uncertainty relation however the model evolution is time global and the hamiltonian is more complex the time dependent models do not degrade the system state also they are time local and the hamiltonian is less complex

asymptotic efficiencies of truncated sequential tests truncation of a sequential test with constant boundaries is considered for the problem of testing a location hypothesis f x theta 0 versus f x theta 1 a test design procedure is developed by using bounds for the error probabilities under the hypothesis and alternative by viewing the truncated sequential test as a mixture of a sequential probability ratio test and a fixed sample size test its boundaries and truncation point can be obtained once the degree of mixture is specified asymptotically correct approximations for the operating characteristic function and the average sample number function of the resulting test are derived numerical results show that an appropriately designed truncated sequential test performs favorably as compared to both the fixed sample size test and the sequential probability ratio test with the same error probabilities the average sample number function of the truncated test is uniformly smaller than that of the fixed sample size test and the truncated test maintains average sample sizes under the hypothesis and the alternative that are close to those optimum values achieved by wald s sequential probability ratio test moreover the truncated test is more favorable than the sequential probability ratio test in the sense that is has smaller average sample size when the actual location parameter is between theta 0 and theta 1 this behavior becomes more pronounced as the error probabilities become smaller implying that the truncated sequential test becomes more favorable as the error probabilities become smaller

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

locally optimum rank detection of correlated random signals in additive noise nonparametric detection of a zero mean random signal in additive noise is considered the locally optimum detector based on signs and ranks of observations is derived for good weak signal detection performance under any specified noise probability density function this detector is shown to have interesting similarities to the locally optimum detector for random signals it may also be viewed as a generalization of the locally optimum rank detector for known signals examples of the test statistic of the detector are given for some specific noise probability density functions asymptotic and finite sample size performance of the locally optimum rank detector is also considered

zonal polynomials via stanleyʼs coordinates and free cumulants abstract we study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power sum symmetric functions we show that the zonal characters just like the characters of the symmetric groups admit a nice combinatorial description in terms of stanley s multirectangular coordinates of young diagrams we also study the analogue of kerov polynomials namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients in this way we prove two recent conjectures of lassalle for jack polynomials in the special case of zonal polynomials

asymptotics of characters of symmetric groups related to stanley character formula we prove an upper bound for characters of the symmetric groups in particular we show that there exists a constant a 0 with a property that for every young diagram with n boxes r rows and c columns tr tr e i amax a r n c n j j n aoj j where j j is the minimal number of factors needed to write 2 sn as a product of transpositions we also give uniform estimates for the error term in the vershik kerov s and biane s character formulas and give a new formula for free cumulants of the transition measure

analogies and differences between quantum and stochastic automata abstract we analyze some features of the behaviour of quantum automata providing analogies and differences with the corresponding stochastic models in particular we prove there is a quantum automaton where the change of state depends on unitary transformations defined by matrices with nonnull amplitudes that accepts a non regular language with cut point zero and inverse error polynomially bounded stochastic automata with matrices having nonnull elements and with polynomial bounds on the inverse error recognize only regular languages the class of stochastic languages contains the class of quantum languages quantum languages are empty or contain an infinite number of words the class of quantum languages is not closed under complementation

an explicit form for kerov s character polynomials kerov considered the normalized characters of irreducible representations of the symmetric group evaluated on a cycle as a polynomial in free cumulants biane has proved that this polynomial has integer coefficients and made various conjectures recently sniady has proved biane s conjectured explicit form for the first family of nontrivial terms in this polynomial in this paper we give an explicit expression for all terms in kerov s character polynomials our method is through lagrange inversion

determination of equivalence between quantum sequential machines quantum sequential machines qsms that may be viewed as a quantum variant of stochastic sequential machines ssms are one of the important quantum computing models a very crucial result on ssms is that two ssms with n and n states respectively and the same input and output alphabets are equivalent if and only if they are n n 1 equivalent therefore gudder asked whether or not it holds for qsms as an open problem in this direction since the further study is closely related to this result qiu demonstrated that in qsms this result does not hold and therefore answered in part the problem however as qiu indicated the sufficient and necessary conditions for the equivalence between two qsms are still not discovered in this paper we show that if the condition of n n 1 equivalence is appropriately relaxed then we can give a sufficient and necessary condition for justifying the equivalence between two qsms more precisely we show that any two qsms m and m are equivalent if and only if they are n n 2 equivalent where n and n are respectively the numbers of states in m and m we therefore solve the open problem suggested by gudder and provide a basic result for further developing qsms as well we discuss strongly factorizable qsms and present the conditions for the equivalence between two strongly factorizable qsms

rapid solution of problems by quantum computation a class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method the quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation

the selberg jack symmetric functions abstract k aomoto has recently given a simple proof of an extension of a selberg s integral we prove the following generalization of aomoto s theorem for eachk 0 there exists a family sk t of homogeneous symmetric functions such that if the integrand of selberg s integral is multiplied bysk t1 tn then the integral has a certain closed form sincesk t is a renormalized jack symmetric function with 1 k we call sk t the selberg jack symmetric functions r p stanley and i g macdonald have extended many of the basic properties of the i j schur functions to the jack symmetric functions stanley s extensions of the pieri formula in a form obtained by macdonald s extension of the duality property and the combinatorial representation arise naturally in our proof we give proofs of these and other results of stanley and macdonald we give two constant term identities which are equivalent to the integration formula these results extend w g morris theorem foranand the macdonald morris constant term identity forbcn by exploring selberg s proof of his integral we give a recurrence relation outline a method for calculating the selberg jack symmetric functions and show that a constant term orthogonality relation recently proved by macdonald arises naturally we give an expansion which allows us to evaluate the normalization of this orthogonality thus giving an alternative proof of the integration formula

automata theory based on quantum logic reversibilities and pushdown automata automata theory based on quantum logic called l valued finite automata l vfas may be viewed as a logical approach to quantum computing this work is mainly divided into two parts one part deals with reversibility of l vfas and the other establishes a basic framework of l valued pushdown automata l vpdas first we provide some preliminaries concerning quantum logic and l vfas and we prove a useful property of l valued successor and source operators then we clarify the relationships between various reversibilities closely related to quantum finite automata in the literature in particular we define a reversibility of l vfas which is termed as retrievability and we clarify the relationships between a number of different fashions regarding retrievability of l vfas we prove that some of them are equivalent but for the others to be equivalent the truth value set is required to satisfy a certain condition this is an essential difference from the classical situation afterwards we introduce l vpdas and show that the class of the languages accepted by l vpdas by empty stack coincides with that accepted by l vpdas by final state finally we provide some examples of l vfas and conclude with some remarks

upper bound on the characters of the symmetric groups for balanced young diagrams and a generalized frobenius formula we study asymptotics of an irreducible representation of the symmetric group sn corresponding to a balanced young diagram a young diagram with at most rows and columns for some fixed constant c in the limit as n tends to infinity we show that there exists a constant d which depends only on c with a property that where p denotes the length of a permutation the minimal number of factors necessary to write p as a product of transpositions our main tool is an analogue of the frobenius character formula which holds true not only for cycles but for arbitrary permutations

quantum theory the church turing principle and the universal quantum computer it is argued that underlying the church turing hypothesis there is an implicit physical assertion here this assertion is presented explicitly as a physical principle every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means classical physics and the universal turing machine because the former is continuous and the latter discrete do not obey the principle at least in the strong form above a class of model computing machines that is the quantum generalization of the class of turing machines is described and it is shown that quantum theory and the universal quantum computer are compatible with the principle computing machines resembling the universal quantum computer could in principle be built and would have many remarkable properties not reproducible by any turing machine these do not include the computation of non recursive functions but they do include quantum parallelism a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it the intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than everett s some of the numerous connections between the quantum theory of computation and the rest of physics are explored quantum complexity theory allows a physically more reasonable definition of the complexity or knowledge in a physical system than does classical complexity theory

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

gaussian fluctuations of young diagrams and structure constants of jack characters in this paper we consider a deformation of plancherel measure linked to jack polynomials our main result is the description of the first and second order asymptotics of the bulk of a random young diagram under this distribution which extends celebrated results of vershik kerov logan and shepp for the first order asymptotics and kerov for the second order asymptotics this gives more evidence for the connection with the gaussian beta ensemble already suggested by a work of matsumoto our main tool is a polynomiality result for the structure constants of some quantities that we call jack characters recently introduced by lassalle we believe that this result is also interesting in itself and we give several other applications of it

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools