raney distributions and random matrix theory recent works have shown that the family of probability distributions with moments given by the fuss catalan numbers permit a simple parameterized form for their density we extend this result to the raney distribution which by definition has its moments given by a generalization of the fuss catalan numbers such computations begin with an algebraic equation satisfied by the stieltjes transform which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the raney distribution for the fuss catalan distribution an equilibrium problem characterizing the density is identified the stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard gaussian matrices and s standard gaussian matrices is shown to satisfy a variant of the algebraic equation relating to the raney distribution supported on 0 infty we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization as an application the leading asymptotic form of the density as the endpoints of the support are approached is computed and is shown to have some universal features

error threshold in optimal coding numerical criteria and classes of universalities for complexity the free energy of the random energy model at the transition point between the ferromagnetic and spin glass phases is calculated at this point equivalent to the decoding error threshold in optimal codes the free energy has finite size corrections proportional to the square root of the number of degrees the response of the magnetization to an external ferromagnetic phase is maximal at values of magnetization equal to one half we give several criteria of complexity and define different universality classes according to our classification at the lowest class of complexity are random graphs markov models and hidden markov models at the next level is the sherrington kirkpatrick spin glass connected to neuron network models on a higher level are critical theories the spin glass phase of the random energy model percolation and self organized criticality the top level class involves highly optimized tolerance design error thresholds in optimal coding language and maybe financial markets living systems are also related to the last class the concept of antiresonance is suggested for complex systems

noncolliding brownian motion with drift and time dependent stieltjes wigert determinantal point process using the determinantal formula of biane bougerol and o connell we give multitime joint probability densities to the noncolliding brownian motion with drift where the number of particles is finite we study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well ordered with a scale we show that at each time t 0 the single time probability density of particle system is exactly transformed to the biorthogonal stieltjes wigert matrix model in the chern simons theory introduced by dolivet and tierz here one parameter extensions extensions of the stieltjes wigert polynomials which are themselves q extensions of the hermite polynomials play an essential role the two parameters a and of the process combined with time t are mapped to the parameters q and of the biorthogonal polynomials by the transformation of normalization factor of our probability density the partition function of the chern simons matrix model is readily obtained we study the determinantal structure of the matrix model and prove that at each time t 0 the present noncolliding brownian motion with drift is a determinantal point process in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel using the obtained correlation kernel we study time evolution of the noncolliding brownian motion with drift

critical behaviour of a surface reaction model with infinitely many absorbing states in a recent letter yaldram et al 1993 studied the critical behaviour of a simple lattice gas model of the co no catalytic reaction the model exhibits a second order non equilibrium phase transition from an active state into one out of infinitely many absorbing states estimates for the critical exponent beta suggested that the model belongs to a new universality class the results reported in this article contradict this notion as estimates for various critical exponents show that the model belongs to the universality class of directed percolation

approximation of analytic sets by nash tangents of higher order the aim of this paper is to show that for every locally analytic subset x of cm and every a in x there exist a neighborhood v of a in cm and a sequence x nu of nash subsets of v converging to x cap v such that x and x satisfy a certain condition for tangency of order next it is shown that this condition implies that for sufficiently large the multiplicities of x and x at a are equal

janossy densities ii pfaffian ensembles we extend the main result of the companion paper j stat phys 113 595 610 to the case of the pfaffian ensembles

asymptotic level spacing distribution for a q deformed random matrix ensemble we obtain the asymptotic behaviour of the nearest neighbour level spacing distribution for a q deformed unitary random matrix ensemble

hierarchic trees with branching number close to one noiseless kardar parisi zhang equation with additional linear term for imitating two dimensional and three dimensional phase transitions an imitation of two dimensional 2d field theory is formulated by means of a model on the hierarchic tree with branching number close to one with the same potential and the free correlators identical to 2d correlator ones such a model carries on some features of the original model for certain scale invariant theories the renormalization group equation for the free energy is the noiseless kardar parisi zhang equation with an additional linear term

janossy densities i determinantal ensembles we derive an elementary formula for janossy densities for determinantal point processes with a finite rank projection type kernel in particular for 2 polynomial ensembles of random matrices we show that the janossy densities on an interval i can be expressed in terms of the christoffel darboux kernel for the orthogonal polynomials on the complement of i

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

new family of unitary random matrices we consider an exactly solvable random matrix model related to the random transfer matrix model for disordered conductors in the conventional random matrix models the spacing distribution of nearest neighbor eigenvalues when expressed in units of average spacing has a universal behavior known generally as the wigner distribution in contrast our model has a single parameter as a function of which the spacing distribution crosses over from a wigner to a distribution which is increasingly more poisson like a feature common to a wide variety of physical systems including disorder and chaos

multifractality and freezing phenomena in random energy landscapes an introduction abstract we start our lectures with introducing and discussing the general notion of multifractality spectrum for random measures on lattices and how it can be probed using moments of that measure then we show that the boltzmann gibbs probability distributions generated by logarithmically correlated random potentials provide a simple yet non trivial example of disorder induced multifractal measures the typical values of the multifractality exponents can be extracted from calculating the free energy of the associated statistical mechanics problem to succeed in such a calculation we introduce and discuss in some detail two analytically tractable models for logarithmically correlated potentials the first model uses a special definition of distances between points in space and is based on the idea of multiplicative cascades which originated in theory of turbulent motion it is essentially equivalent to statistical mechanics of directed polymers on disordered trees studied long ago by derrida and spohn 1988 in ref 12 in this way we introduce the notion of the freezing transition which is identified with an abrupt change in the multifractality spectrum second model which allows for explicit analytical evaluation of the free energy is the infinite dimensional version of the problem which can be solved by employing the replica trick in particular the latter version allows one to identify the freezing phenomenon with a mechanism of the replica symmetry breaking rsb and to elucidate its physical meaning the corresponding one step rsb solution turns out to be marginally stable everywhere in the low temperature phase we finish with a short discussion of recent developments and extensions of models with logarithmic correlations in particular in the context of extreme value statistics the first appendix summarizes the standard elementary information about gaussian integrals and related subjects and introduces the notion of the gaussian free field characterized by logarithmic correlations three other appendices provide the detailed exposition of a few technical details underlying the replica analysis of the model discussed in the lectures

universal scaling of lyapunov exponents we prove numerically for the first time that the lower part of the spectrum of the transfer matrix of the quasi one dimensional disordered system is strongly correlated in the neighbourhood of the critical point of the metal insulator transition in particular the disorder and the system size dependence of the spectrum is governed only by one parameter which we identify with the scaling parameter of mackinnon and kramer the strong correlation of the spectrum supports the idea of the universality of the random matrix like theory of the metal insulator transition and provides us with a new way of calculation of the critical parameters of this transition

edge effects in some perturbations of the gaussian unitary ensemble a bordering of gaussian unitary ensemble matrices is considered in which the bordered row consists of zero mean complex gaussians n 0 2 in 0 2 off the diagonal and the real gaussian n 2 on the diagonal we compute the explicit form of the eigenvalue probability function for such matrices as well as that for matrices obtained by repeating the bordering the correlations are in general determinantal and in the single bordering case the explicit form of the correlation kernel is computed in the large n limit it is shown that and or can be tuned to induce a separation of the largest eigenvalue this effect is shown to be controlled by a single parameter universal correlation kernel

energy correlations for a random matrix model of disordered bosons linearizing the heisenberg equations of motion around the ground state of an interacting quantum many body system one gets a time evolution generator in the positive cone of a real symplectic lie algebra the presence of disorder in the physical system determines a probability measure with support on this cone the present paper analyzes a discrete family of such measures of exponential type and does so in an attempt to capture by a simple random matrix model some generic statistical features of the characteristic frequencies of disordered bosonic quasiparticle systems the level correlation functions of the said measures are shown to be those of a determinantal process and the kernel of the process is expressed as a sum of biorthogonal polynomials while the correlations in the bulk scaling limit are in accord with sine kernel or gaussian unitary ensemble universality at the low frequency end of the spectrum an unusual type of scaling behavior is found

limit theorems for biorthogonal ensembles and related combinatorial identities abstract we study the fluctuations of certain biorthogonal ensembles for which the underlying family p q satisfies a finite term recurrence relation of the form x p x j p x for polynomial linear statistics of such ensembles we reformulate the cumulant method introduced in 53 in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix j in the spirit of 12 we show that the asymptotic fluctuations are described by the right limits of the matrix j moreover whenever the right limit is a laurent matrix we show that the clt is equivalent to soshnikov s main combinatorial lemma we discuss several applications to unitary invariant hermitian random matrices in particular we provide a general central limit theorem clt and a law of large numbers in the one cut regime we also prove a clt for the square singular values of the product of independent complex rectangular ginibre matrices as well as for the laguerre and jacobi biorthogonal ensembles introduced in 7 and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients finally we discuss the connection with the strong szeg limit theorem where this combinatorial method originates

random matrix models with additional interactions it has been argued that despite remarkable success existing random matrix theories are not adequate at describing disordered conductors in the metallic regime due to the presence of certain two body interactions in the effective hamiltonian for the eigenvalues in addition to the standard logarithmic interaction that arises entirely from symmetry considerations we present a new method that allows exact solution of random matrix models with such additional two body interactions this should broaden the scope of random matrix models in general

integrable structure of products of finite complex ginibre random matrices abstract we consider the squared singular values of the product of m standard complex gaussian matrices since the squared singular values form a determinantal point process with a particular meijer g function kernel the gap probabilities are given by a fredholm determinant based on this kernel it was shown by strahov 2014 that a hard edge scaling limit of the gap probabilities is described by hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the kyoto school we generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form as a result we obtain the hamiltonian structure for finite size matrices and formulate it in terms of a m 1 m 1 matrix schlesinger system the case m 1 reproduces the tracy and widom theory which results in the painleve v equation for the 0 s gap probability some integrals of motion for m 2 are identified and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding 0 s gap probability

approximation in compact nash manifolds let rn be a compact nash manifold a b the rings of nash analytic global functions on the main result of this paper is the following theorem 1 let be a pair of nash submanifolds of some rn rq and let us suppose is compact let f1 fq r be nash functions then every analytic solution y f x of the system f1 x y fq x y 0 can be approximated in the whitney topology by the global nash solutions y g x the main tool used to prove the above results is this version of neron s desingularisation theorem any homomorphism of a algebras c b with c finitely generated over a factorizes through a finitely generated a algebra d such that a d is regular using theorem 1 the authors are able to solve several interesting problems that have been open for many years for example they prove i every analytic factorization of a global nash function defined over is equivalent to a nash factorization ii every semialgebraic subset of which is a global analytic subset is also a global nash subset iii every prime ideal of a generates a prime ideal in b iv every coherent ideal subsheaf of the sheaf n of nash functions on is generated by its global sections the case where is noncompact is only partially studied in this paper in the reviewer s opinion this article makes crucial progress in the theory of global nash functions

transitions in spectral statistics we present long range statistical properties of a recently introduced unitary random matrix ensemble whose short range correlations were found to describe a transition from wigner to poisson type as a function of a single parameter we argue by evaluating the two level correlation function of a different solvable model that the transition is perhaps a quite general feature for a class of models our analytic results compare well with numerical studies on a variety of physical systems we discuss the possibility of observing experimentally the signature of such transitions

large deviations for disordered bosons and multiple orthogonal polynomial ensembles we prove a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles that includes biorthogonal laguerre jacobi and hermite ensembles the matrix model of lueck sommers and zirnbauer for disordered bosons the stieltjes wigert matrix model of chern simons theory and angelesco ensembles

on wright s generalized bessel kernel abstract in this paper we consider the wright s generalized bessel kernel k x y defined by x 0 1 j 1 1 u x j 1 u y u d u 1 0 where j a b x j 0 x j j a b j a c b 1 is wright s generalization of the bessel function this non symmetric kernel which generalizes the classical bessel kernel corresponding to 1 in random matrix theory is the hard edge scaling limit of the correlation kernel for certain muttalib borodin ensembles we show that if is rational i e m n with m n n g c d m n 1 and m 1 m n the wright s generalized bessel kernel is integrable in the sense of its izergin korepin slavnov we then come to the fredholm determinant of this kernel over the union of several scaled intervals which can also be interpreted as the gap probability the probability of finding no particles on these intervals the integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding fredholm determinant as well as a hamiltonian interpretation as a consequence we are able to represent the gap probability over a single interval 0 s in terms of a solution of a system of nonlinear ordinary differential equations

brownian motion ensembles correlation functions of determinantal processes we calculate the n point correlation function for a large class of brownian motion ensembles of random matrix theory the corresponding fokker planck equation describes a determinantal process in the theory of matrix valued stochastic differential equations and can be mapped onto a schodinger equation of non interacting electrons in one dimension the correlation functions are obtained explicitly via a suitable generalization of the method of biorthogonal functions