a thermodynamic like approach for the study of probabilistic systems abstract this paper starts by considering an arbitrary system for which the probability distribution p p 1 p 2 p n of a measurable property with n possible independent results is known by using the hartley information pseudo energy the escort probabilities of p and its shannon entropy a thermodynamic like formalism is developed including the deduction of the fundamental equation for the mean value of the pseudo energies thus the method allows a macrodescription of the system that is based on a purely mathematical procedure and involves thermodynamic like variables as a specific example we consider a bank that measures customer satisfaction by doing a statistical survey satisfied or unsatisfied customers

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

orthogonal polynomials and fluctuations of random matrices in this paper we establish a connection between the fluctuations of wishart random matrices shifted chebyshev poly nomials and planar diagrams whose linear span form a basis for the irreducible representations of the annular temperly lieb alge bra

quasi hereditary twisted category algebras we give a sufficient criterion for when a twisted finite category algebra over a field is quasi hereditary in terms of the partially ordered set of l classes in the morphism set of the category we show that this is a common generalisation of a long list of results in the context of ei categories regular monoids brauer algebras temperley lieb algebras partition algebras

asymptotics of unitary and orthogonal matrix integrals abstract in this paper we prove that in small parameter regions arbitrary unitary matrix integrals converge in the large n limit and match their formal expansion secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the hciz integral our convergence result also leads us to new results of smoothness of microstates we finally generalize our approach to integrals over the orthogonal group

on the global fluctuations of block gaussian matrices in this paper we study the global fluctuations of block gaussian matrices within the framework of second order free probability theory in order to compute the second order cauchy transform of these matrices we introduce a matricial second order conditional expectation and compute the matricial second order cauchy transform of a certain type of non commutative random variables as a by product using the linearization technique we obtain the second order cauchy transform of non commutative rational functions evaluated on selfadjoint gaussian matrices

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

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

asymptotic normality for traces of polynomials in independent complex wishart matrices we derive a non asymptotic expression for the moments of traces of monomials in several independent complex wishart matrices extending some explicit formulas available in the literature we then deduce the explicit expression for the cumulants from the latter we read out the multivariate normal approximation to the traces of finite families of polynomials in independent complex wishart matrices

matrix group integrals surfaces and mapping class groups i u n since the 1970 s physicists and mathematicians who study random matrices in the gue or goe models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces we establish a new aspect of this theory for random matrices sampled from the group u n of unitary matrices more concretely we study measures induced by free words on u n let fr be the free group on r generators to sample a random element from u n according to the measure induced by w fr one substitutes the r letters in w by r independent haar random elements from u n the main theme of this paper is that every moment of this measure is determined by families of pairs f where is an orientable surface with boundary and f is a map from to the bouquet ofr circles which sends the boundary components of to powers of w a crucial role is then played by euler characteristics of subgroups of the mapping class group of as corollaries we obtain asymptotic bounds on the moments we show that the measure on u n bears information about the number of solutions to the equation u1 v1 ug vg w in the free group and deduce that one can hear the stable commutator length of a word through its unitary word measures

second order freeness and fluctuations of random matrices i gaussian and wishart matrices and cyclic fock spaces abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we introduce the concept of second order freeness and interpret the global fluctuations of gaussian and wishart random matrices by a general limit theorem for second order freeness by introducing cyclic fock space we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation annihilation and preservation operators we show that orthogonal families of gaussian and wishart random matrices are asymptotically free of second order

central limit theorem for the heat kernel measure on the unitary group we prove that for a finite collection of real valued functions f1 fn on the group of complex numbers of modulus 1 which are derivable with lipschitz continuous derivative the distribution of trf1 trfn under the properly scaled heat kernel measure at a given time on the unitary group u n has gaussian fluctuations as n tends to infinity with a covariance for which we give a formula and which is of order n 1 in the limit where the time tends to infinity we prove that this covariance converges to that obtained by p diaconis and s n evans in a previous work on uniformly distributed unitary matrices finally we discuss some combinatorial aspects of our results

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

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

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

cumulants in noncommutative probability theory i noncommutative exchangeability systems cumulants linearize convolution of measures we use a formula of good to define noncommutative cumulants in a very general setting it turns out that the essential property needed is exchangeability of random variables roughly speaking the formula says that cumulants are moments of a certain discrete fourier transform of a random variable this provides a simple unified method to understand the known examples of cumulants like classical free and various q cumulants

real second order freeness and haar orthogonal matrices we demonstrate the asymptotic real second order freeness of haar distributed orthogonal matrices and an independent ensemble of random matrices our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix then the two ensembles are asymptotically real second order free this captures the known examples of asymptotic real second order freeness introduced by redelmeier

second order freeness and fluctuations of random matrices ii unitary random matrices abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we show how the concept of second order freeness which was introduced in part i allows one to understand global fluctuations of haar distributed unitary random matrices in particular independence between the unitary ensemble and another ensemble goes in the large n limit over into asymptotic second order freeness two important consequences of our general theory are i we obtain a natural generalization of a theorem of diaconis and shahshahani to the case of several independent unitary matrices ii we can show that global fluctuations in unitarily invariant multi matrix models are not universal

some orbits of free words that are determined by measures on finite groups abstract every word in a free group f induces a probability measure on every finite group in a natural manner it is an open problem whether two words that induce the same measure on every finite group necessarily belong to the same orbit of aut f a special case of this problem when one of the words is the primitive word x was settled positively by the third author and parzanchevski 15 here we extend this result to the case where one of the words is x d or x y d for an arbitrary d z

simple maps hurwitz numbers and topological recursion we introduce the notion of fully simple maps which are maps with non self intersecting disjoint boundaries in contrast maps where such a restriction is not imposed are called ordinary we study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder we show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks we also obtain an elegant formula for cylinders these relations reproduce the relation between moments and higher order free cumulants established by collins et al 22 and implement the symplectic transformation x leftrightarrow y on the spectral curve in the context of topological recursion we conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of x and y we propose an argument to prove this statement conditionally to a mild version of the symplectic invariance for the 1 hermitian matrix model which is believed to be true but has not been proved yet our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion our argument relies on an unconditional matrix model interpretation of fully simple maps via the formal hermitian matrix model with external field we also deduce a universal relation between generating series of fully simple maps and of ordinary maps which involves double monotone hurwitz numbers in particular ordinary maps without internal faces which are generated by the gaussian unitary ensemble and with boundary perimeters lambda 1 ldots lambda n are strictly monotone double hurwitz numbers with ramifications lambda above infty and 2 ldots 2 above 0 combining with a recent result of dubrovin et al 24 this implies an elsv like formula for these hurwitz numbers

fluctuations of particle systems determined by schur generating functions abstract we develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems we introduce and study the notion of the schur generating function of a random discrete configuration our main result provides a central limit theorem clt for such a configuration given certain conditions on the schur generating function as applications of this approach we prove clt s for several probabilistic models coming from asymptotic representation theory and statistical physics including random lozenge and domino tilings non intersecting random walks decompositions of tensor products of representations of unitary groups

second order cumulants of products we derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors this extends to the second order case the formula of krawczyk and speicher we apply our result to the problem of calculating the second order cumulants of a semi circular and haar unitary operator