moment matrices and multi component kp with applications to random matrix theory questions on random matrices and non intersecting brownian motions have led to the study of moment matrices with regard to several weights the main result of this paper is to show that the determinants of such moment matrices satisfy upon adding one set of time deformations for each weight the multi component kp hierarchy these determinants are thus tau functions for these integrable hierarchies the tau functions so obtained with appropriate shifts of the time parameters forward and backwards will be expressed in terms of multiple orthogonal polynomials for these weights and their cauchy transforms the main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials it concerns an identity between the orthogonality of polynomials on the real line the bilinear identity in kp theory and a generating functional for the full kp theory an additional fact not discussed in this paper is that these functions satisfy virasoro constraints with respect to these time parameters

rich dynamics in a spatial predator prey model with delay in this paper we study the spatiotemporal dynamics of a diffusive holling tanner predator prey model with discrete time delay via analytically and numerically analysis we unveil six types of patterns with and without time delay among them of particular novel is the observation of linear pattern consisting of a series of parallel lines whose formation is closely related with the temporal hopf bifurcation threshold moreover we also find that larger time delay or diffusion of predator may induce the extinction of both prey and predator theoretical analysis and numerical simulations validate the well known conclusion diffusion is usually beneficial for stabilizing pattern formation yet discrete time delay plays a destabilizing role in the generation of pattern

on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

dyson s nonintersecting brownian motions with a few outliers consider n nonintersecting brownian particles on r dyson brownian motions all starting from the origin at time t 0 and forced to return to x 0 at time t 1 for large n the average mean density of particles has its support for each 0 0 does it affect the brownian fluctuations along the curve c for large n in this paper we show that no new process appears as long as one considers points y t is an element of c such that 0 infinity this kernel is closely related to one found by baik ben arous and peche in the context of multivariate statistics c 2008 wiley periodicals inc

airy processes with wanderers and new universality classes consider n m nonintersecting brownian bridges with n of them leaving from 0 at time t 1 and returning to 0 at time t 1 while the m remaining ones wanderers go from m points a i to m points b i first we keep m fixed and we scale a i b i appropriately with n in the large n limit we obtain a new airy process with wanderers in the neighborhood of root 2n the approximate location of the rightmost particle in the absence of wanderers this new process is governed by an airy type kernel with a rational perturbation letting the number m of wanderers tend to infinity as well leads to two pearcey processes about two cusps a closing and an opening cusp the location of the tips being related by an elliptic curve upon tuning the starting and target points one can let the two tips of the cusps grow very close this leads to a new process which might be governed by a kernel represented as a double integral involving the exponential of a quintic polynomial in the integration variables

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

nonlinear response with dichotomous noise dichotomous noise appears in a wide variety of physical and mathematical models it has escaped attention that the standard results for the long time properties cannot be applied when unstable fixed points are crossed in the asymptotic regime we show how calculations have to be modified to deal with these cases and present as a first application full analytic results for hypersensitive transport

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

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

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

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

motif distributions in phase space networks for characterizing experimental two phase flow patterns with chaotic features the dynamics of two phase flows have been a challenging problem in nonlinear dynamics and fluid mechanics we propose a method to characterize and distinguish patterns from inclined water oil flow experiments based on the concept of network motifs that have found great usage in network science and systems biology in particular we construct from measured time series phase space complex networks and then calculate the distribution of a set of distinct network motifs to gain insight we first test the approach using time series from classical chaotic systems and find a universal feature motif distributions from different chaotic systems are generally highly heterogeneous our main finding is that the distributions from experimental two phase flows tend to be heterogeneous as well suggesting the underlying chaotic nature of the flow patterns calculation of the maximal lyapunov exponent provides further support for this motif distributions can thus be a feasible tool to understand the dynamics of realistic two phase flow patterns

communication scheme based on evolutionary spatial 2 2 games abstract a visual communication scheme based on evolutionary spatial 2 2 games is proposed in this paper self organizing patterns induced by complex interactions between competing individuals are exploited for hiding and transmitting secret visual information properties of the proposed communication scheme are discussed in details it is shown that the hiding capacity of the system the minimum size of the detectable primitives and the minimum distance between two primitives is sufficient for the effective transmission of digital dichotomous images also it is demonstrated that the proposed communication scheme is resilient to time backwards plain image attacks and is highly sensitive to perturbations of private and public keys several computational experiments are used to demonstrate the effectiveness of the proposed communication scheme

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

the dilute potts model on random surfaces we present a new solution of the asymmetric two matrix model in the large n limit which only involves a saddle point analysis the model can be interpreted as ising in the presence of a magnetic field on random dynamical lattices with the topology of the sphere resp the disk for closed resp open surfaces we elaborate on the resulting phase diagram the method can be equally well applied to a more general q 1 matrix model which represents the dilute potts model on random dynamical lattices we discuss in particular duality of boundary conditions for open random surfaces

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

multiple orthogonal polynomials for classical weights a new set of special functions which has a wide range of applications from number theory to integrability of nonlinear dynamical systems is described we study multiple orthogonal polynomials with respect to p 1 weights satisfying pearson s equation in particular we give a classification of multiple orthogonal polynomials with respect to classical weights which is based on properties of the corresponding rodrigues operators we show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p 1 we also obtain explicit formulas and recurrence relations for these polynomials

correlation functions of eigenvalues of multi matrix models and the limit of a time dependent matrix the universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems and proved in some particular cases as the hermitian one matrix model with polynomial potential here we consider the more difficult case of a unidimensional chain of hermitian matrices with first neighbour couplings and polynomial potentials an asymptotic expression of the orthogonal polynomials and a generalization of the darboux christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain eventually we consider the limit of the infinite chain of matrices which can be interpreted as a time dependent hermitian one matrix model and give the correlation functions of eigenvalues at different times

sums of random matrices and the potts model on random planar maps we compute the partition function of the q states potts model on a random planar lattice with p leq q allowed equally weighted colours on a connected boundary to this end we employ its matrix model representation in the planar limit generalising a result by voiculescu for the addition of random matrices to a situation beyond free probability theory we show that the partition functions with p and q p colours on the boundary are related algebraically finally we investigate the phase diagram of the model when 0 leq q leq 4 and comment on the conformal field theory description of the critical points

random matrices with external source and kp τ functions in this paper we prove that the partition function in the random matrix model with external source is a kp function

principal bifurcations and symmetries in the emergence of reaction diffusion advection patterns on finite domains pattern formation mechanisms of a reaction diffusion advection system with one diffusivity differential advection and robin boundary conditions of danckwerts type are being studied pattern selection requires mapping the domains of coexistence and stability of propagating or stationary nonuniform solutions which for the general case of far from instability onsets is conducted using spatial dynamics and numerical continuations the selection is determined by the boundary conditions which either preserve or destroy the translational symmetry of the model accordingly we explain the criterion and the properties of stationary periodic states if the system is bounded and show that propagation of nonlinear waves including solitary against the advective flow corresponds to coexisting family that emerges nonlinearly from a distinct oscillatory hopf instability consequently the resulting pattern selection is qualitatively different from the symmetric finite wavenumber turing or hopf instabilities

large deviations asymptotics for spherical integrals consider the spherical integral i n dn en exp n tr udnu en dm n u where m n denote the haar measure on the orthogonal group on when 1 and on the unitary group un when 2 and dn en are diagonal real matrices whose spectral measures converge to d e in this paper we prove the existence and represent as solution to a variational problem the limit i d e lim n 2 log i n dn en this limit appears in so called matrix models but also in the evaluation of large deviations of the spectral measure of generalized wishart matrices our technique is based on stochastic calculus large deviations and elements from free probability

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

first order asymptotics of matrix integrals a rigorous approach towards the understanding of matrix models we investigate the large n limit of spectral measures of matrices which relate to the gibbs measures of a number of statistical mechanical systems on random graphs these include the ising and potts models on random graphs for most of these models we prove that the spectral measures converge almost surely and describe their limit via solutions to an euler equation for isentropic flow with negative pressure p 3 1 2 3

secure steganographic communication algorithm based on self organizing patterns a secure steganographic communication algorithm based on patterns evolving in a beddington de angelis type predator prey model with self and cross diffusion is proposed in this paper small perturbations of initial states of the system around the state of equilibrium result in the evolution of self organizing patterns small differences between initial perturbations result in slight differences also in the evolving patterns it is shown that the generation of interpretable target patterns cannot be considered as a secure mean of communication because contours of the secret image can be retrieved from the cover image using statistical techniques if only it represents small perturbations of the initial states of the system an alternative approach when the cover image represents the self organizing pattern that has evolved from initial states perturbed using the dot skeleton representation of the secret image can be considered as a safe visual communication technique protecting both the secret image and communicating parties

forced patterns near a turing hopf bifurcation we study time periodic forcing of spatially extended patterns near a turing hopf bifurcation point a symmetry based normal form analysis yields several predictions including that i weak forcing near the intrinsic hopf frequency enhances or suppresses the turing amplitude by an amount that scales quadratically with the forcing strength and ii the strongest effect is seen for forcing that is detuned from the hopf frequency to apply our results to specific models we perform a perturbation analysis on general two component reaction diffusion systems which reveals whether the forcing suppresses or enhances the spatial pattern for the suppressing case our results are consistent with features of previous experiments on the chlorine dioxide iodine malonic acid chemical reaction however we also find examples of the enhancing case which has not yet been observed in experiment numerical simulations verify the predicted dependence on the forcing parameters

riemann hilbert approach to multi time processes the airy and the pearcey cases we prove that matrix fredholm determinants related to multi time processes can be expressed in terms of determinants of integrable kernels a la its izergin korepin slavnov iiks and hence related to suitable riemann hilbert problems thus extending the known results for the single time case we focus on the airy and pearcey processes as an example of applications we re deduce a third order pde found by adler and van moerbeke for the two time airy process

universality for the pearcey process consider non intersecting brownian motions on the line leaving from the origin and forced to two arbitrary points letting the number of brownian particles tend to infinity and upon rescaling there is a point of bifurcation where the support of the density of particles goes from one interval to two intervals in this paper we show that at that very point of bifurcation a cusp appears near which the brownian paths fluctuate like the pearcey process this is a universality result within this class of problems tracy and widom obtained such a result in the symmetric case when the two target points are symmetric with regard to the origin this asymmetry enabled us to improve considerably a result concerning the non linear partial differential equations governing the transition probabilities for the pearcey process obtained by adler and van moerbeke c 2010 elsevier b v all rights reserved

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

the internal description of a causal set what the universe looks like from the inside we describe an algebraic way to code the causal information of a discrete spacetime the causal set is transformed to a description in terms of the causal pasts of the events in this is done by an evolving set a functor which to each event of assigns its causal past evolving sets obey a heyting algebra which is characterised by a non standard notion of complement conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set a causal quantum theory can be based on the quantum version of evolving sets which we briefly discuss

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

enriched categories and models for spaces of evolving states partially ordered sets causets partially ordered spaces and their local counterparts are now often used to model systems in computer science and theoretical physics the order models time which is often not globally given in this setting directed paths are important objects of study as they correspond to an evolving state or particle traversing the system we model both the space and the directed paths by a simplicially enriched category and show how to adapt some classical constructions to produce a differential graded enrichment

deadlocks and dihomotopy in mutual exclusion models higher dimensional automata hda represent a promising tool for modelling true concurrency in a both combinatorial and topological framework within these models fast algorithms investigating deadlocks and unreachable regions have been devised previously on a background of easily understandable directed geometric ideas in this article we modify notions and methods from homotopy theory to define and investigate essentially different schedules in a hda and to detect whether two given runs are essentially different using an algorithm again based on directed geometry

cross diffusion induced turing instability for a competition model with saturation effect abstract competitive behavior like predation widely exists in the world which has influences on the dynamics of ecosystems though a good knowledge for the patten formation and selection of predator prey models with diffusion is known in the literature little is known for that of a diffused competition model as a result a competition model with saturation effect and self and cross diffusion is proposed and analyzed in this paper firstly by making use of nullcline analysis we derive the condition of existence and stability of positive equilibrium and prove global stability then using the linear stability analysis the turing bifurcation critical value and the condition of the occurrence of turing pattern are obtained when control parameter is selected finally we deduce the amplitude equations around the turing bifurcation point by using the standard multiple scale analysis meanwhile a series of numerical simulations are given to expand our theoretical analysis this work shows that cross diffusion plays a key role in the formation of spatial patterns for competitive model with saturation effect

competitively coupled maps for hiding secret visual information abstract a novel digital image hiding scheme based on competitively coupled maps is presented in this paper self organizing patterns produced by an array of non diffusively coupled nonlinear maps are exploited to conceal the secret the secret image is represented in the form of a dot skeleton representation and is embedded into a spatially homogeneous initial state far below the noise level self organizing patterns leak the secret image at a predefined set of system parameters computational experiments are used to demonstrate the effectiveness and the security of the proposed image hiding scheme

parametric spectral statistics in unitary random matrix ensembles from distribution functions to intra level correlations we establish a general framework to explore parametric statistics of individual energy levels in unitary random matrix ensembles for a generic confinement potential tr w h we i find the joint distribution functions of the eigenvalues of h and h h v for an arbitrary fixed v both for finite matrix size n and in the thermodynamic n limit ii derive many point parametric correlation functions of the two sets of eigenvalues and show that they are naturally parametrized by the eigenvalues of the reactance matrix for scattering off the potential v iii prove the universality of the correlation functions in unitary ensembles with non gaussian non invariant confinement potential tr w h v iv establish a general scheme for exact calculation of level number dependent parametric correlation functions and apply the scheme to the calculation of intra level velocity autocorrelation function and the distribution of parametric level shifts

cyclic competition of mobile species on continuous space pattern formation and coexistence we propose a model for cyclically competing species on continuous space and investigate the effect of the interplay between the interaction range and mobility on coexistence a transition from coexistence to extinction is uncovered with a strikingly nonmonotonic behavior in the coexistence probability about the minimum in the probability switches between spiral and plane wave patterns arise a strong mobility can either promote or hamper coexistence depending on the radius of the interaction range these phenomena are absent in any lattice based model and we demonstrate that they can be explained using nonlinear partial differential equations our continuous space model is more physical and we expect the findings to generate experimental interest

a pde for non intersecting brownian motions and applications abstract consider n n 1 n 2 n p non intersecting brownian motions on the real line starting from the origin at t 0 with n i particles forced to reach p distinct target points i at time t 1 with 1 2 p this can be viewed as a diffusion process in a sector of r n this work shows that the transition probability that is the probability for the particles to pass through windows e k at times t k satisfies in a new set of variables a non linear pde which can be expressed as a near wronskian that is a determinant of a matrix of size p 1 with each row being a derivative of the previous except for the last column it is an interesting open question to understand those equations from a more probabilistic point of view as an application of these equations let the number of particles forced to the extreme points 1 and p tend to infinity keep the number of particles forced to intermediate points fixed inliers but let the target points themselves go to infinity according to a proper scale a new critical process appears at the point of bifurcation where the bulk of the particles forced to n depart from those going to n these statistical fluctuations near that point of bifurcation are specified by a kernel which is a rational perturbation of the pearcey kernel this work also shows that such equations are an essential tool in obtaining certain asymptotic results finally the paper contains a conjecture

polynuclear growth model with external source and random matrix model with deterministic source we present a random matrix interpretation of the distribution functions which have appeared in the study of the one dimensional polynuclear growth png model with external sources it is shown that the distribution goe2 which is defined as the square of the gaussian orthogonal ensemble goe tracy widom distribution can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source which have been studied in a different context previously compared to the original interpretation of the goe2 as the square of goe ours has an advantage in that it can also describe the transition from the gaussian unitary ensemble gue tracy widom distribution to the goe2 we further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain noncolliding brownian motion model and the multilayer png model with an external source this provides us with a multimatrix model interpretation of the multipoint height distributions of the png model with an external source

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

zooming in on local level statistics by supersymmetric extension of free probability we consider unitary ensembles of hermitian n n matrices governed by a confining potential nv with analytic and uniformly convex v from earlier work it is known that the large n limit of the characteristic function for a finite rank fourier variable k is determined by the voiculescu r transform a key object in free probability theory going beyond these results we argue that the same holds true when the finite rank operator k has the form that is required by the wegner efetov supersymmetry method this insight leads to a potent new technique for the study of local statistics e g level correlations we illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering

time delay induced instabilities in reaction diffusion systems time delay in the kinetic terms of reaction diffusion systems has been investigated it has been shown that short delay beyond a critical threshold may induce spatiotemporal instabilities for unequal diffusivities and appropriate parameter space delay may induce turing instability resulting in stationary patterns and also interesting turing hopf transition with the formation of spirals the theoretical scheme has been numerically explored in two different prototypical reaction diffusion systems