chaos in the mixed even spin models we consider a disordered system obtained by coupling two mixed even spin models together the chaos problem is concerned with the behavior of the coupled system when the external parameters in the two models such as temperature disorder or external field are slightly different it is conjectured that the overlap between two independently sampled spin configurations from respectively the gibbs measures of the two models is essentially concentrated around a constant under the coupled gibbs measure using the extended guerra replica symmetry breaking bound together with a recent development of controlling the overlap using the ghirlanda guerra identities as well as a new family of identities we present rigorous results on chaos in temperature in addition chaos in disorder and in external field are addressed

robust tracking relation between the numerators mode readability and inverse internal model in the robust multivariable asymptotic tracking problem with two output plants it is shown that if the numerator related to the controlled output is fixed while the other numerator and the denominator are perturbed then there is no solution to the problem if however the whole part of the plant related to the controlled output is stable and fixed while the other part is arbitrarily perturbed the problem has a solution and the compensator that solves the problem incorporates an inverse internal model of the signal to be tracked

the complexity of spherical p spin models a second moment approach recently auffinger ben arous and cern initiated the study of critical points of the hamiltonian in the spherical pure pp spin spin glass model and established connections between those and several notions from the physics literature denoting the number of critical values less than nunu by crtn u crtn u they computed the asymptotics of 1nlog ecrtn u 1nlog ecrtn u as nn the dimension of the sphere goes to we compute the asymptotics of the corresponding second moment and show that for p 3p 3 and sufficiently negative uu it matches the first moment e crtn u 2 e crtn u 2 1 as an immediate consequence we obtain that crtn u e crtn u 1crtn u e crtn u 1 in l2l2 and thus in probability for any uu for which ecrtn u ecrtn u does not tend to 00 we prove that the moments match on an exponential scale

general solution of robust tracking problem in two degree of freedom control systems in this note we study linear time invariant multivariable systems with two degree of freedom compensators first we give a new parametrization of all stabilizing compensators from the viewpoint of independent design of transfer function and robustness property second utilizing this parametrization we solve a general robust tracking problem including the case of nonsquare plants and show that tracking and robustness can be independently specified in two degree of freedom control systems

new representations of pade pade type and partial pade approximants pade approximants are rational functions with a denominator which does not vanish at zero and whose series expansion match a given series as far as possible these approximants are usually written under a rational form in this paper we will show how to write them also under two different barycentric forms and under a partial fraction form depending on free parameters according to the choice of these parameters pade type approximants can be obtained under a barycentric or a partial fraction form

spin glasses and stein s method we introduce some applications of stein s method in the high temperature analysis of spin glasses stein s method allows the direct analysis of the gibbs measure without having to eate a cavity another advantage is that it gives limit theorems with total variation error bounds although the bounds can be suboptimal a surprising byproduct of our analysis is a relatively transparent explanation of the thouless anderson palmer system of equations along the way we develop stein s method for mixtures of two gaussian densities

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

random matrices and complexity of spin glasses cern y abstract we give an asymptotic evaluation of the complexity of spherical p spin spin glass models via random matrix theory this study enables us to obtain detailed infor mation about the bottom of the energy landscape including the absolute minimum the ground state the other local minima and describe an interesting layered structure of the low critical values for the hamiltonians of these models we also show that our ap proach allows us to compute the related tap complexity and extend the results known in the physics literature as an independent tool we prove a ldp for the k th largest eigenvalue of the goe extending the results of bdg01 how many critical values of given index and below a given level does a typical random morse function have on a high dimensional manifold our work addresses this question in a very special case we look at certain natural random gaussian functions on the n dimensional sphere known as p spin spherical spin glass models we cannot yet answer the question above about the typical number but we can study thoroughly the mean number which we show is exponentially large in n we introduce a new identity based on the classical kac rice formula relating random matrix theory and the problem of counting these critical values using this identity and tools from random matrix theory we give an asymptotic evaluation of the complexity of these spherical spin glass models the complexity mentioned here is defined as the mean number of critical points of given index whose value is below or above a given level this includes the important question of counting the mean number of local minima below a given level and in particular the question of finding the ground state energy the minimal value of the hamiltonian we show that this question is directly related to the study of the edge of the spectrum of the gaussian orthogonal ensemble goe the question of computing the complexity of mean field spin glass models has recently been thoroughly studied in the physics literature see for example clr03 and the refer ences therein mainly for a different measure of the complexity i e the mean number of solutions to the thouless anderson palmer equations or tap complexity our approach to the complexity enables us to recover known results in the physics literature about tap complexity to compute the ground state energy when p is even and to describe an interesting layered structure of the low energy levels of the hamiltonians of these models which might prove useful for the study of the metastability of langevin dynamics for these models in longer time scales than those studied in bdg01 the paper is organised as follows in section 2 we give our main results in section 3 we prove two main formulas theorem 2 1 and 2 2 relating random matrix theory specif ically the goe and spherical spin glasses these formulas are direct consequences of the kac rice formula we learned the version needed here in the book at07 for another modern account see aw09 the main new ingredient is the fact that for spherical spin glass models the hessian of the hamiltonian at a critical point conditioned on the value of the hamiltonian is a symmetric gaussian random matrix with independent entries up to symmetry plus a diagonal matrix this implies in particular that it is possible to

an iterative construction of solutions of the tap equations for the sherrington kirkpatrick model we propose an iterative scheme for the solutions of the tap equations in the sherrington kirkpatrick model which is shown to converge up to and including the de almeida thouless line the main tool is a representation of the iterations which reveals an interesting structure of them this representation does not depend on the temperature parameter but for temperatures below the de almeida thouless line it contains a part which does not converge to zero in the limit

robust tracking error feedback and two degree of freedom controllers the purpose of this paper is to establish first that in the absence of error feedback asymptotic tracking is a property that is fragile with respect to the controller parameters this implies that arbitrary small perturbation of the parameters of such a controller will cause asymptotic tracking to fail second we propose a general 2 dof controller architecture we prove that in this structure asymptotic tracking is in general fragile with respect to the controller parameters f sub r f sub y and c while it is robust with respect to the plant parameters as well as the components h and l of the 2 dof controller moreover the only way to realize a 2 dof controller in a nonfragile way is to generate the error signal exactly which amounts to setting f sub r f sub y 1 the results reconcile the requirements of the internal model principle namely the necessity of error feedback with the additional design freedom potentially offered by 2 dof controllers it specifies the controller architecture necessary for robustness and identifies the blocks whose parameters are available for tuning in the design process

algebraic theory of linear time invariant feedback systems with two input two output plant and compensator this paper present an algebraic theory for linear time invariant multi input multi output systems with two input two output plant and compensator using the factorization approach this system configuration considered by doyle and nett is most general in that any interconnection of two systems can be represented in terms of this scheme other system configurations encountered in compensator design problems are special cases of this system configuration the analysis and synthesis applies to continuous time as well as discrete time systems and to lumped as well as distributed systems since the algebraic setting is completely general the compensator parametrization has four degrees of freedom the paper is self contained and is tutorial in the sense that it develops the main results of nett without introducing the concept of containment

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

the extremal process of critical points of the pure p spin spherical spin glass model recently sharp results concerning the critical points of the hamiltonian of the p spin spherical spin glass model have been obtained by means of moments computations in particular these moments computations allow for the evaluation of the leading term of the ground state i e of the global minimum in this paper we study the extremal point process of critical points that is the point process associated to all critical values in the vicinity of the ground state we show that the latter converges in distribution to a poisson point process of exponential intensity in particular we identify the correct centering of the ground state and prove the convergence in distribution of the centered minimum to a minus gumbel variable these results are identical to what one obtains for a sequence of i i d variables correctly normalized namely we show that the model is in the universality class of rem

complexity of random smooth functions on the high dimensional sphere we analyze the landscape of general smooth gaussian functions on the sphere in dimension n when n is large we give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean euler characteristic of level sets we then find two possible scenarios for the bottom landscape one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large we end the paper by discussing how these results can be interpreted in the language of spin glasses models

free energy of the spherical mean field model we compute at any temperature the free energy of the multi p spin spherical model when only terms for p even are considered

the parisi ultrametricity conjecture in this paper we prove that the support of a random measure on the unit ball of a separable hilbert space that satises the ghirlanda guerra identities must be ultrametric with probability one this implies the parisi ultrametricity conjecture in mean eld spin glass models such as the sherrington kirkpatrick and mixed p spin models for which gibbs measures are known to satisfy the ghirlanda guerra identities in the thermodynamic limit

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

an approach to chaos in some mixed p spin models we consider the problems of chaos in disorder and temperature for coupled copies of the mixed p spin models under certain assumptions on the parameters of the models first we prove a weak form of chaos by showing that the overlap is concentrated around its gibbs average depending on the disorder and then obtain several results toward strong chaos by providing control of the overlap between two systems in terms of their parisi measures

further results on the robust tracking problem in two degree of freedom control systems the robust tracking problem in two degree of freedom control systems is formulated in a general setting which includes the case where the controlled output is not available directly a complete solution of the problem which is an extension of previous results is derived in a considerably simplified and transparent manner furthermore we show what kind of information about the output is needed to achieve robust tracking

random matrices overcrowding estimates for the spectrum abstract we address overcrowding estimates for the singular values of random iid matrices as well as for the eigenvalues of random wigner matrices we show evidence of long range separation under arbitrary perturbation even in matrices of discrete entry distributions in many cases our method yields nearly optimal bounds