uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

the optimal projection equations for reduced order state estimation the singular measurement noise case the optimal projection equations for reduced order state estimation are generalized to allow for singular i e colored measurement noise the noisy and noise free measurements serve as inputs to dynamic and static estimators respectively the optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity

robust observer based tracking control of hodgkin huxley neuron systems under environmental disturbances a nervous system consists of a large number of highly interconnected nerve cells nerve cells communicate by generation and transmission of short electrical pulses action potential in addition membrane voltage is the only measurable state in nervous systems a robust observer based model reference tracking control is proposed for hodgkin huxley hh neuron systems to generate a desired reference response in spite of environmental noises uncertain initial values and diffusion currents from other interconnected nerve cells in order to simplify the robust tracking control design of nonlinear stochastic hh neuron systems a fuzzy interpolation method is employed to interpolate several linear stochastic systems to approximate a nonlinear stochastic hh neuron system so that the nonlinear robust tracking control problem can be solved by the linear matrix inequality lmi technique with the help of robust control toolbox in matlab the proposed robust observer based tracking control scheme can provide new methods for desired action potential generation suppression of oscillations and blockage of action potential transmission under environmental noise and diffusion currents these new methods are useful for patients with different neuron system dysfunctions finally three simulation examples of tracking control of nervous systems are given to illustrate the design procedure and confirm the tracking performance of the proposed method

random matrix model with external source and a constrained vector equilibrium problem we consider the random matrix model with external source in the case where the potential v x is an even polynomial and the external source has two eigenvalues a of equal multiplicity we show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures 1 2 that solve a constrained vector equilibrium problem the proof is based on the steepest descent analysis of the associated riemann hilbert problem for multiple orthogonal polynomials we illustrate our results in detail for the case of a quartic double well potential v x 1 4x 4 t 2x 2 we are able to determine the precise location of the phase transitions in the ta plane where either the constraint becomes active or the two intervals in the support come together or both 2010 wiley periodicals inc

relative positions of limit cycles in a kolmogorov type system to estimate the relative position of limit cycles for a system is always useful in the qualitative analysis of the system if the system has only one limit cycle determining the position of the limit cycle is more important the author constructs an annular region containing all the limit cycles for a kolmogorov type system since the system contains the lotka volterra gausse kuang and freedman and huang models as special cases the results obtained are also valid for those models furthermore the approach to the location of limit cycles is very applicable because the region is easy to compute

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

h spl infin decentralized fuzzy model reference tracking control design for nonlinear interconnected systems in general due to the interactions among subsystems it is difficult to design an h sub spl infin decentralized controller for nonlinear interconnected systems the model reference tracking control problem of nonlinear interconnected systems is studied via h sub spl infin decentralized fuzzy control method first the nonlinear interconnected system is represented by an equivalent takagi sugeno type fuzzy model a state feedback decentralized fuzzy control scheme is developed to override the external disturbances such that the h spl infin model reference tracking performance is achieved furthermore the stability of the nonlinear interconnected systems is also guaranteed if states are not all available a decentralized fuzzy observer is proposed to estimate the states of each subsystem for decentralized control consequently a fuzzy observer based state feedback decentralized fuzzy controller is proposed to solve the h sub spl infin tracking control design problem for nonlinear interconnected systems the problem of h sub spl infin decentralized fuzzy tracking control design for nonlinear interconnected systems is characterized in terms of solving an eigenvalue problem evp the evp can be solved very efficiently using convex optimization techniques finally simulation examples are given to illustrate the tracking performance of the proposed methods

critical measures quadratic differentials and weak limits of zeros of stieltjes polynomials we investigate the asymptotic zero distribution of heine stieltjes polynomials polynomial solutions of second order differential equations with complex polynomial coefficients in the case when all zeros of the leading coefficients are all real zeros of the heine stieltjes polynomials were interpreted by stieltjes as discrete distributions minimizing an energy functional in a general complex situation one deals instead with a critical point of the energy we introduce the notion of discrete and continuous critical measures saddle points of the weighted logarithmic energy on the plane and prove that a weak limit of a sequence of discrete critical measures is a continuous critical measure thus the limit zero distributions of the heine stieltjes polynomials are given by continuous critical measures we give a detailed description of such measures showing their connections with quadratic differentials in doing that we obtain some results on the global structure of rational quadratic differentials on the riemann sphere that have an independent interest

the pearcey process the extended airy kernel describes the space time correlation functions for the airy process which is the limiting process for a polynuclear growth model the airy functions themselves are given by integrals in which the exponents have a cubic singularity arising from the coalescence of two saddle points in an asymptotic analysis pearcey functions are given by integrals in which the exponents have a quartic singularity arising from the coalescence of three saddle points a corresponding pearcey kernel appears in a random matrix model and a brownian motion model for a fixed time this paper derives an extended pearcey kernel by scaling the brownian motion model at several times and a system of partial differential equations whose solution determines associated distribution functions we expect there to be a limiting nonstationary process consisting of infinitely many paths which we call the pearcey process whose space time correlation functions are expressible in terms of this extended kernel

universality in the two matrix model with a monomial quartic and a general even polynomial potential in this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with the probability measure z 1 n rm exp left n left rm tr v m 1 w m 2 tau m 1m 2 right right rm d m 1 rm d m 2 in the case where w frac y 4 4 and v is a general even polynomial we studied the correlation kernel for the eigenvalues of the matrix m1 in the limit as n we extended the results of duits and kuijlaars in 14 to the case when the limiting eigenvalue density for m1 is supported on multiple intervals the results are achieved by constructing the parametrix to a riemann hilbert problem obtained in 14 with theta functions and then showing that this parametrix is well defined for all n by studying the theta divisor

random skew plane partitions and the pearcey process we study random skew 3d partitions weighted by qvol and specifically the q 1 asymptotics of local correlations near various points of the limit shape we obtain sine kernel asymptotics for correlations in the bulk of the disordered region airy kernel asymptotics near a general point of the frozen boundary and a pearcey kernel asymptotics near a cusp of the frozen boundary

orthogonal polynomials in the normal matrix model with a cubic potential abstract we consider the normal matrix model with a cubic potential the model is ill defined and in order to regularize it elbau and felder introduced a model with a cut off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut off region on the complex plane in the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane without any cut off which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of elbau and felder the main goal of this paper is to develop the riemann hilbert rh approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity as the first step in the rh approach we introduce an auxiliary vector equilibrium problem for a pair of measures 1 2 on the complex plane we then formulate a 3 3 matrix valued rh problem for the orthogonal polynomials in hand and we apply the nonlinear steepest descent method of deift zhou to the asymptotic analysis of the rh problem the central steps in our study are a sequence of transformations of the rh problem based on the equilibrium vector measure 1 2 and the construction of a global parametrix the main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane we prove that the distribution of zeros of the orthogonal polynomials converges to the measure 1 the first component of the equilibrium measure we also obtain analytical results for the measure 1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve

critical measures for vector energy global structure of trajectories of quadratic differentials abstract saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector valued critical measures a notion that finds a natural motivation in several branches of analysis we study in depth the case of measures 1 2 3 when the mutual interaction comprises both attracting and repelling forces for arbitrary vector polynomial external fields we establish general structural results about critical measures such as their characterization in terms of an algebraic equation solved by an appropriate combination of their cauchy transforms and the symmetry properties or the s properties exhibited by such measures in consequence we conclude that vector valued critical measures are supported on a finite number of analytic arcs that are trajectories of a quadratic differential globally defined on a three sheeted riemann surface the complete description of the so called critical graph for such a differential is the key to the construction of the critical measures we illustrate these connections studying in depth a one parameter family of critical measures under the action of a cubic external field this choice is motivated by the asymptotic analysis of a family of non hermitian multiple orthogonal polynomials that is subject of a forthcoming paper here we compute explicitly the riemann surface and the corresponding quadratic differential and analyze the dynamics of its critical graph as a function of the parameter giving a detailed description of the occurring phase transitions when projected back to the complex plane this construction gives us the complete family of vector valued critical measures that in this context turn out to be vector valued equilibrium measures

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

necessary and sufficient conditions for robust stabilization of perturbed observer based compensating systems abstract the design of a robust observer based compensator for imprecisely known linear multivariable systems is considered necessary and sufficient conditions for robust stability are derived with respect to additive plant perturbation multiplicative plant perturbation and fractional plant perturbation necessary and sufficient conditions are also given for the existence of such a robust observer based compensator

universality of the double scaling limit in random matrix models we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit the limiting kernels are constructed out of functions associated with the second painleve equation this extends a result of bleher and its for the special case of a critical quartic potential the two main tools we use are equilibrium measures and riemann hilbert problems in our treatment of equilibrium measures we allow a negative density near the critical point which enables us to treat all cases simultaneously the asymptotic analysis of the riemann hilbert problem is done with the deift zhou steepest descent analysis for the construction of a local parametrix at the critical point we introduce a modification of the approach of baik deift and johansson so that we are able to satisfy the required jump properties exactly c 2005 wiley periodicals inc

asymptotics for multiple meixner polynomials abstract we study the asymptotic behavior of multiple meixner polynomials of first and second kind respectively 6 we use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution moreover analyzing the limiting behavior of the coefficients of the recurrence relations for multiple meixner polynomials we obtain the main term of their asymptotics

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

robust stabilizer synthesis for feedback systems containing time varying nonlinear perturbations a new circle criterion is developed to solve the robust stabilization problem of a feedback system with time varying nonlinear perturbations which is different than previous results theorems presented in this note can be employed to treat systems with stable or unstable plants the generally stabilizing controller proposed by youla et al and the schur function class s will be employed to synthesize the robust stabilizer of a feedback system subjected to time varying nonlinear perturbations

the largest eigenvalue of real symmetric hermitian and hermitian self dual random matrix models with rank one external source part i we consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric 1 hermitian 2 and hermitian self dual 4 random matrix models with rank 1 external source they are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues assuming the one band condition and certain regularities of the potential function we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value when the nonzero eigenvalue of the external source matrix is less than or equal to the critical value the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper in this paper we also give a definition of the external source model for all 0

non intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights we study a model of n non intersecting squared bessel processes in the confluent case all paths start at time t 0 at the same positive value x a remain positive and are conditioned to end at time t t at x 0 in the limit n after appropriate rescaling the paths fill out a region in the tx plane that we describe explicitly in particular the paths initially stay away from the hard edge at x 0 but at a certain critical time t the smallest paths hit the hard edge and from then on are stuck to it for t t we obtain the usual scaling limits from random matrix theory namely the sine airy and bessel kernels a key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble corresponding to a system of two modified bessel type weights as a consequence there is a 3 3 matrix valued riemann hilbert problem characterizing this model that we analyze in the large n limit using the deift zhou steepest descent method there are some novel ingredients in the riemann hilbert analysis that are of independent interest

double scaling limit in the random matrix model the riemann hilbert approach we prove the existence of the double scaling limit in the unitary matrix model with quartic interaction and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the function for the hastings mcleod solution to the painleve ii equation the proof is based on the riemann hilbert approach and the central point of the proof is an analysis of analytic semiclassical asymptotics for the function at the critical point in the presence of four coalescing turning points 2003 wiley periodicals inc

the cauchy two matrix model we introduce a new class of two multi matrix models of positive hermitian matrices coupled in a chain the coupling is related to the cauchy kernel and differs from the exponential coupling more commonly used in similar models the correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate riemann hilbert problems thus paving the way to a steepest descent analysis and universality results the interpretation of the formal expansion of the partition function in terms of multicolored ribbon graphs is provided and a connection to the o 1 model a steepest descent analysis of the partition function reveals that the model is related to a trigonal curve three sheeted covering of the plane much in the same way as the hermitian matrix model is related to a hyperelliptic curve

large n limit of gaussian random matrices with external source part iii double scaling limit we consider the double scaling limit in the random matrix ensemble with an external source 1 over z n e n hbox tr 1 over 2 m 2 am dm defined on n n hermitian matrices where a is a diagonal matrix with two eigenvalues a of equal multiplicities the value a 1 is critical since the eigenvalues of m accumulate as n on two intervals for a 1 and on one interval for 0 a 1 these two cases were treated in parts i and ii where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles for the critical case a 1 new limiting behavior occurs which is described in terms of pearcey integrals as shown by brezin and hikami and tracy and widom we establish this result by applying the deift zhou steepest descent method to a 3 3 matrix valued riemann hilbert problem which involves the construction of a local parametrix out of pearcey integrals we resolve the main technical issue of matching the local pearcey parametrix with a global outside parametrix by modifying an underlying riemann surface

critical measures for vector energy asymptotics of non diagonal multiple orthogonal polynomials for a cubic weight abstract we consider the type i multiple orthogonal polynomials mops a n m b n m deg a n m n 1 deg b n m m 1 and type ii mops p n m deg p n m n m satisfying non hermitian orthogonality with respect to the weight e z 3 on two unbounded contours 1 and 2 on c with in the case of type ii mops n conditions on 1 and m on 2 under the assumption that n m n n m 0 1 we find the detailed rescaled asymptotics of a n m b n m and p n m on c and describe the phase transitions of this limit behavior as a function of this description is given in terms of the vector critical measure 1 2 3 the saddle point of an energy functional comprising both attracting and repelling forces this critical measure is characterized by a cubic equation spectral curve and its components j live on trajectories of a canonical quadratic differential on the riemann surface of this equation the structure of these trajectories and their deformations as functions of was object of study in our previous paper 42 and some of the present results strongly rely on the analysis carried out there we conclude that the asymptotic zero distribution of the polynomials a n m and p n m are given by appropriate combinations of the components j of the vector critical measure however in the case of the zeros of b n m the behavior is totally different and can be described in terms of the balayage of the signed measure 2 3 onto certain curves on the plane these curves are constructed with the aid of the canonical quadratic differential and their topology has three very distinct characters depending on the value of and are obtained from the critical graph of once the trajectories and vector critical measures are studied the main asymptotic technical tool is the deift zhou nonlinear steepest descent analysis of a 3 3 matrix valued riemann hilbert problem characterizing both a n m b n m and p n m which allows us to obtain the limit behavior of both types of mops simultaneously we illustrate our findings with results of several numerical experiments explain the computational methodology and formulate some conjectures and empirical observations based on these experiments

level spacing of random matrices in an external source in an earlier work we had considered a gaussian ensemble of random matrices in the presence of a given external matrix source the measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials or on the coulomb gas representation are not available nevertheless the n point correlation functions are still given in terms of the determinant of a kernel known through an explicit integral representation this kernel is no longer symmetric though and is not readily accessible to standard methods in particular finding the level spacing probability is always a delicate problem in fredholm theory and we have to reconsider the problem within our model we find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state the problem is solved through coupled non linear differential equations which turn out to form a hamiltonian system as a result we find that the level spacing probability p s behaves like exp c s 8 over 3 for large spacing s this is consistent with the asymptotic behavior exp c s 2 beta 2 whenever the density of state behaves near the edge as rho lambda sim lambda beta

necessary and sufficient conditions for robust stabilization of an observer based compensating system suffering nonlinear time varying perturbation it is demonstrated that the robust stabilization against additive nonlinear time varying uncertainty presented by b s chen and w s you ibid vol ac 32 p 1131 5 dec 1987 is not just a sufficient but also a necessary condition an additional necessary and sufficient condition that arises in the case of multiplicative uncertainty is also discussed

pdes for the gaussian ensemble with external source and the pearcey distribution the present paper studies a gaussian hermitian random matrix ensemble with external source given by a fixed diagonal matrix with two eigenvalues a as a first result the probability that the eigenvalues of the ensemble belong to an interval e satisfies a fourth order pde with quartic nonlinearity the variables are the eigenvalue a and the boundary of e this equation enables one to find a pde for the pearcey distribution the latter describes the statistics of the eigenvalues near the closure of a gap i e when the support of the equilibrium measure for large size random matrices has a gap that can be made to close the gaussian hermitian random matrix ensemble with external source described above has this feature the pearcey distribution is shown to satisfy a fourth order pde with cubic nonlinearity this also gives the pde for the transition probability of the pearcey process a limiting process associated with nonintersecting brownian motions on r c 2006 wiley periodicals inc

universality in the two matrix model a riemann hilbert steepest descent analysis the eigenvalue statistics of a pair m 1 m 2 of n x n hermitian matrices taken randomly with respect to the measure 1 z n exp n tr v m 1 w m 2 tau m 1 m 2 dm 1 dm 2 can be descri

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

random matrices with equispaced external source we study hermitian random matrix models with an external source matrix which has equispaced eigenvalues and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity we obtain strong asymptotics for the multiple orthogonal polynomials associated to these models and as a consequence for the average characteristic polynomials one feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree nevertheless we are able to characterize them in terms of a pair of 2 1 vector valued riemann hilbert problems and to perform an asymptotic analysis of the riemann hilbert problems

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

robust stabilization in an observer controller feedback system under nonlinear time varying perturbations or unmodeled dynamics in this note a new robust criterion is proposed to analyze multivariable observer controller compensator feedback systems under additive nonlinear perturbations in order to stabilize these systems a more general parameterized observer controller compensator is introduced and a design algorithm is proposed for robust stabilization against the nonlinear perturbations finally an example is given to illustrate our results

universality of a double scaling limit near singular edge points in random matrix models we consider unitary random matrix ensembles z n s t 1 e n tr v s t m dm on the space of hermitian n n matrices m where the confining potential vs t is such that the limiting mean density of eigenvalues as n and s t 0 vanishes like a power 5 2 at a singular endpoint of its support the main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit the limiting kernel is built out of functions associated with a special solution of the pi2 equation which is a fourth order analogue of the painleve i equation in order to prove our result we use the well known connection between the eigenvalue correlation kernel and the riemann hilbert rh problem for orthogonal polynomials together with the deift zhou steepest descent method to analyze the rh problem asymptotically the key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint for which we use the model rh problem for the special solution of the pi2 equation

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

universality of correlation functions of hermitian random matrices in an external field the behavior of correlation functions is studied in a class of matrix models characterized by a measure exp s containing a potential term and an external source term s n tr v m ma in the large n limit the short distance behavior is found to be identical to the one obtained in previously studied matrix models thus extending the universality of the level spacing distribution the calculation of correlation functions involves finite n determinant formulae reducing the problem to the large n asymptotic analysis of a single kernel k this is performed by an appropriate matrix integral formulation of k multi matrix generalizations of these results are discussed

the ellipse law kirchhoff meets dislocations in this paper we consider a nonlocal energy i whose kernel is obtained by adding to the coulomb potential an anisotropic term weighted by a parameter alpha in mathbb r the case 0 corresponds to purely logarithmic interactions minimised by the circle law 1 corresponds to the energy of interacting dislocations minimised by the semi circle law we show that for alpha in 0 1 the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi axes sqrt 1 alpha and sqrt 1 alpha this result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed for the proof we borrow techniques from fluid dynamics in particular those related to kirchhoff s celebrated result that domains enclosed by ellipses are rotating vortex patches called kirchhoff ellipses

universal singularity at the closure of a gap in a random matrix theory we consider a hamiltonian h h 0 v in which h 0 is a given non random hermitian matrix and v is an n times n hermitian random matrix with a gaussian probability distribution we had shown before that dyson s universality of the short range correlations between energy levels holds at generic points of the spectrum independently of h 0 we consider here the case in which the spectrum of h 0 is such that there is a gap in the average density of eigenvalues of h which is thus split into two pieces when the spectrum of h 0 is tuned so that the gap closes a new class of universality appears for the energy correlations in the vicinity of this singular point