global properties of invariant measures abstract we study global regularity properties of invariant measures associated with second order differential operators in r n under suitable conditions we prove global boundedness of the density sobolev regularity a harnack inequality and pointwise upper and lower bounds

absence of self averaging of the order parameter in the sherrington kirkpatrick model we prove that if n is the sherrington kirkpatrick sk hamiltonian and the quantity bar q n n 1 sum left langle s l right rangle h 2 converges in the variance to a nonrandom limit asn then the mean free energy of the model converges to the expression obtained by sk since this expression is known not to be correct in the low temperature region our result implies the non self averaging of the order parameter of the sk model this fact is an important ingredient of the parisi theory which is widely believed to be exact we also prove that the variance of the free energy of the sk model converges to zero asn i e the free energy has the self averaging property

right convergence of sparse random graphs the paper is devoted to the problem of establishing right convergence of sparse random graphs this concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper graphs mathbb g n n ge 1 to some target graph w the theory of dense graph convergence including random dense graphs is now well understood borgs et al in ann math 176 151 219 2012 borgs et al in adv math 219 1801 1851 2008 chatterjee and varadhan in eur j comb 32 1000 1017 2011 lovasz and szegedy in j comb theory ser b 96 933 957 2006 but its counterpart for sparse random graphs presents some fundamental difficulties phrased in the statistical physics terminology the issue is the existence of the limits of appropriately normalized log partition functions also known as free energy limits for the gibbs distribution associated with w in this paper we prove that the sequence of sparse erdos renyi graphs is right converging when the tensor product associated with the target graph w satisfies a certain convexity property we treat the case of discrete and continuous target graphs w the latter case allows us to prove a special case of talagrand s recent conjecture more accurately stated as level iii research problem 6 7 2 in his recent book talagrand in mean field models for spin glasses volume i basic examples springer berlin 2010 concerning the existence of the limit of the measure of a set obtained from mathbb r n by intersecting it with linearly in n many subsets generated according to some common probability law our proof is based on the interpolation technique introduced first by guerra and toninelli commun math phys 230 71 79 2002 and developed further in abbe and montanari in on the concentration of the number of solutions of random satisfiability formulas 2013 bayati et al in ann probab conference version in proceedings of 42nd ann symposium on the theory of computing stoc 2010 contucci et al in antiferromagnetic potts model on the erdos renyi random graph 2011 franz and leone in j stat phys 111 3 4 535 564 2003 franz et al in j phys a math gen 36 10967 10985 2003 montanari in ieee trans inf theory 51 9 3221 3246 2005 panchenko and talagrand in probab theory relat fields 130 312 336 2004 specifically bayati et al ann probab conference version in proceedings of 42nd ann symposium on the theory of computing stoc 2010 establishes the right convergence property for erdos renyi graphs for some special cases of w in this paper most of the results in bayati et al ann probab conference version in proceedings of 42nd ann symposium on the theory of computing stoc 2010 follow as a special case of our main theorem

stability of continuous time positive switched linear systems a weak common copositive lyapunov functions approach in this paper we study the problem of asymptotic stability of continuous time positive switched linear systems under both arbitrary and restricted switchings it is well known that asymptotic stability under arbitrary switching can be implied by several classes of strong common copositive lyapunov functions clfs i e functions whose derivative along the nontrivial system trajectories is negative however asymptotically stable positive switched systems may not admit strong common clfs the main contribution of this paper is to study the stability problem by requiring only weak common clfs firstly necessary and sufficient conditions are established for asymptotic stability under arbitrary switching among them an easily verifiable graphical stability criterion based on the connectivity of the digraphs associated with the subsystem matrices is proposed secondly we further relax the obtained graphical condition to derive a relaxed weak excitation condition for asymptotic stability under dwell time switching finally two examples are provided to illustrate the effectiveness of our theoretical results

stochastic symmetry breaking in a gaussian hopfield model we study a two pattern hopfield model with gaussian disorder we find that there are infinitely many pure states at low temperatures in this model and that the metastate is supported on an infinity of symmetric pairs of pure states the origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder variables

tight bounds on the capacity of binary input random cdma systems in this paper we consider code division multiple access cdma communication over a binary input additive white gaussian noise awgn channel using random spreading for a general class of symmetric distributions for spreading sequences in the limit of a large number of users we prove an upper bound to the capacity the bound matches the formula obtained by tanaka using the replica method we also show concentration of various relevant quantities including mutual information and free energy the mathematical methods are quite general and allow us to discuss extensions to other multiuser scenarios

about the ergodic regime in the analogical hopfield neural networks moments of the partition function in this paper we introduce and exploit the real replica approach for a minimal generalization of the hopfield model by assuming the learned patterns to be distributed according to a standard unit gaussian we consider the high storage case when the number of patterns linearly diverges with the number of neurons we study the infinite volume behavior of the normalized momenta of the partition function we find a region in the parameter space where the free energy density in the infinite volume limit self averages around its annealed approximation as well as the entropy and the internal energy density moreover we evaluate the corrections to their extensive counterparts with respect to their annealed expressions the fluctuations of properly introduced overlaps which act as order parameters are also discussed

rigorous results for hierarchical models of structural glasses we consider two non mean field models of structural glasses built on a hierarchical lattice first we consider a hierarchical version of the random energy model and we prove the existence of the thermodynamic limit and self averaging of the free energy furthermore we prove that the infinite volume entropy is positive in a high temperature region bounded from below thus providing an upper bound on the kauzmann critical temperature in addition we show how to improve this bound by leveraging the hierarchical structure of the model finally we introduce a hierarchical version of the p spin model of a structural glass and we prove the existence of the thermodynamic limit and self averaging of the free energy

on the overlap in the multiple spherical sk models in order to study certain questions concerning the distribution of the overlap in sherrington kirkpatrick type models such as the chaos and ultrametricity problems it seems natural to study the free energy of multiple systems with constrained overlaps one can write analogues of guerra s replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds we were able to make some progress for spherical pure p spin sk models where many computations can be made explicitly for pure 2 spin model we prove ultrametricity and chaos in an external field for the pure p spin model for even p 4 without an external field we describe two possible values of the overlap of two systems at different temperatures we also prove a somewhat unexpected result which shows that in the 2 spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and for example ultrametricity holds only in a weak sense

random overlap structures properties and applications to spin glasses random overlap structures rost s are random elements on the space of probability measures on the unit ball of a hilbert space where two measures are identified if they differ by an isometry in spin glasses they arise as natural limits of gibbs measures under the appropriate algebra of functions we prove that the so called cavity mapping on the space of rost s is continuous leading to a proof of the stochastic stability conjecture for the limiting gibbs measures of a large class of spin glass models similar arguments yield the proofs of a number of other properties of rost s that may be useful in future attempts at proving the ultrametricity conjecture lastly assuming that the ultrametricity conjecture holds the setup yields a constructive proof of the parisi formula for the free energy of the sherrington kirkpatrick model by making rigorous a heuristic of aizenman sims and starr

surface terms on the nishimori line of the gaussian edwards anderson model for the edwards anderson model we find an integral representation for some surface terms on the nishimori line among the results are expressions for the surface pressure for free and periodic boundary conditions and the adjacency pressure i e the difference between the pressure of a box and the sum of the pressures of adjacent sub boxes in which the box can been decomposed we show that all those terms indeed behave proportionally to the surface size and prove the existence in the thermodynamic limit of the adjacency pressure

the thermodynamic limit in mean field spin glass models we present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities for a large class of mean field disordered models as for example the sherrington kirkpatrick model and the derrida p spin model the main argument is based on a smooth interpolation between a large system made of n spin sites and two similar but independent subsystems made of n 1 and n 2 sites respectively with n 1 n 2 n the quenched average of the free energy turns out to be subadditive with respect to the size of the system this gives immediately convergence of the free energy per site in the infinite volume limit moreover a simple argument based on concentration of measure gives the almost sure convergence with respect to the external noise similar results hold also for the ground state energy per site

notes on the polynomial identities in random overlap structures in these notes we review first in some detail the concept of random overlap structure rost applied to fully connected and diluted spin glasses we then sketch how to write down the general term of the expansion of the energy part from the boltzmann rost for the sherrington kirkpatrick model and the corresponding term from the ramost which is the diluted extension suitable for the viana bray model

approaching the rate distortion limit with spatial coupling belief propagation and decimation we investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled low density generator matrix codes the degree of the check nodes is regular and the one of code bits is poisson distributed with an average depending on the compression rate the performance of a low complexity belief propagation guided decimation algorithm is excellent the algorithmic rate distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows moreover as the check degree grows both curves approach the ultimate shannon rate distortion limit the belief propagation guided decimation encoder is based on the posterior measure of a binary symmetric test channel this measure can be interpreted as a random gibbs measure at a temperature directly related to the noise level of the test channel we investigate the links between the algorithmic performance of the belief propagation guided decimation encoder and the phase diagram of this gibbs measure the phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds in particular the dynamical and condensation phase transition temperatures equivalently test channel noise thresholds are computed we observe that 1 the dynamical temperature of the spatially coupled construction saturates toward the condensation temperature and 2 for large degrees the condensation temperature approaches the temperature i e noise level related to the information theoretic shannon test channel noise parameter of rate distortion theory this provides heuristic insight into the excellent performance of the belief propagation guided decimation algorithm this paper contains an introduction to the cavity method

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

griffiths inequalities for the gaussian spin glass the griffiths inequalities for ising spin glass models with gaussian randomness of non vanishing mean are proved using properties of the gaussian distribution and gauge symmetry of the system these inequalities imply that correlation functions are non negative and monotonic along the nishimori line in the phase diagram from this result the existence of thermodynamic limit for correlation functions and pressure is proved under free and fixed boundary conditions relations between the location of multicritical points are also derived for different lattices

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

estimates on path delocalization for copolymers at selective interfaces starting from the simple symmetric random walk s n n we introduce a new process whose path measure is weighted by a factor exp open image in new window with h 0 w n n a typical realization of an iid process and n a positive integer we are looking for results in the large n limit this factor favors s n 0 if w n h 0 and s n 0 if w n h 0 the process can be interpreted as a model for a random heterogeneous polymer in the proximity of an interface separating two selective solvents it has been shown 6 that this model undergoes a de localization transition more precisely there exists a continuous increasing function h c such that if h h c then the model is localized while it is delocalized if h h c however localization and delocalization were not given in terms of path properties but in a free energy sense later on it has been shown that free energy localization does indeed correspond to a strong form of path localization 3 on the other hand only weak results on the delocalized regime have been known so far

vector precoding for gaussian mimo broadcast channels impact of replica symmetry breaking the replica method of statistical physics is employed for the large system analysis of vector precoding for the gaussian multiple input multiple output broadcast channel the transmitter comprises a linear front end combined with nonlinear precoding minimizing transmit energy by means of input alphabet relaxation for the common discrete lattice based relaxation the problem violates replica symmetry and a replica symmetry breaking rsb ansatz is taken the limiting empirical distribution of the precoder s output and the limiting transmit energy are derived for one step rsb particularizing to a zero forcing zf linear front end a decoupling result is derived for discrete lattice based relaxations the impact of rsb is demonstrated for the transmit energy the spectral efficiencies of the aforementioned precoding methods are compared to linear zf and tomlinson harashima precoding thp focusing on quaternary phase shift keying qpsk significant performance gains of both lattice and convex relaxations are revealed for medium to high signal to noise ratios snrs when compared to linear zf precoding thp is shown to be outperformed as well comparing certain lattice based relaxations for qpsk against a convex counterpart the latter is found to be superior for low and high snrs but slightly inferior for medium snrs in terms of spectral efficiency

environmental variability in a stochastic epidemic model abstract in this paper we investigate the stochastic dynamics of a simple epidemic model incorporating the mean reverting ornstein uhlenbeck process analytically and numerically we define two threshold parameters the stochastic demographic reproduction number r d s and the stochastic basic reproduction number r 0 s to utilize in identifying the stochastic extinction and persistence of the disease we find that the stochastic disease dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion the larger intensity of volatility or the smaller speed of reversion can suppress the outbreak of the disease the smaller intensity of volatility or the the higher speed of reversion can enhance the outbreak of the disease furthermore via numerical simulations we find that the stochastic model has an endemic stationary distribution which leads to the stochastic persistence of the disease our results show that mean reverting process is a well established way of introducing stochastic environmental noise into biologically realistic population dynamic models

on nonuniqueness of solutions to elliptic equations for probability measures we consider the set of all probability measures on rd satisfying an elliptic equation l 0 in the weak worm we give sufficient conditions in order that this set contains at least two different elements we also construct new examples of nonuniqueness for such equations

antiferromagnetic potts model on the erdős renyi random graph we study the antiferromagnetic potts model on the poissonian erd s renyi random graph by identifying a suitable interpolation structure and an extended variational principle together with a positive temperature second moment analysis we prove the existence of a phase transition at a positive critical temperature upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution recovered in the framework of derrida ruelle probability cascades and from an entropy positivity argument

neural networks retrieving boolean patterns in a sea of gaussian ones restricted boltzmann machines are key tools in machine learning and are described by the energy function of bipartite spin glasses from a statistical mechanical perspective they share the same gibbs measure of hopfield networks for associative memory in this equivalence weights in the former play as patterns in the latter as boltzmann machines usually require real weights to be trained with gradient descent like methods while hopfield networks typically store binary patterns to be able to retrieve the investigation of a mixed hebbian network equipped with both real e g gaussian and discrete e g boolean patterns naturally arises we prove that in the challenging regime of a high storage of real patterns where retrieval is forbidden an additional load of boolean patterns can still be retrieved as long as the ratio between the overall load and the network size does not exceed a critical threshold that turns out to be the same of the standard amit gutfreund sompolinsky theory assuming replica symmetry we study the case of a low load of boolean patterns combining the stochastic stability and hamilton jacobi interpolating techniques the result can be extended to the high load by a non rigorous but standard replica computation argument

robust estimation for neural networks with randomly occurring distributed delays and markovian jump coupling this paper studies the issue of robust state estimation for coupled neural networks with parameter uncertainty and randomly occurring distributed delays where the polytopic model is employed to describe the parameter uncertainty a set of bernoulli processes with different stochastic properties are introduced to model the randomly occurrences of the distributed delays novel state estimators based on the local coupling structure are proposed to make full use of the coupling information the augmented estimation error system is obtained based on the kronecker product a new lyapunov function which depends both on the polytopic uncertainty and the coupling information is introduced to reduce the conservatism sufficient conditions which guarantee the stochastic stability and the l 2 l infty performance of the augmented estimation error system are established then the estimator gains are further obtained on the basis of these conditions finally a numerical example is used to prove the effectiveness of the results

about a solvable mean field model of a gaussian spin glass in a series of papers we have studied a modified hopfield model of a neural network with learned words characterized by a gaussian distribution the model can be represented as a bipartite spin glass with one party described by dichotomic ising spins and the other party by continuous spin variables with an a priori gaussian distribution by application of standard interpolation methods we have found it useful to compare the neural network model bipartite from one side with two spin glass models each monopartite from the other side of these the first is the usual sherrington kirkpatrick model the second is a spin glass model with continuous spins and inbuilt highly nonlinear smooth cut off interactions this model is an invaluable laboratory for testing all techniques which have been useful in the study of spin glasses the purpose of this paper is to give a synthetic description of the most peculiar aspects by stressing the necessary novelties in the treatment in particular it will be shown that the control of the infinite volume limit according to the well known guerra toninelli strategy requires in addition one to consider the involvement of the cut off interaction in the interpolation procedure moreover the control of the ergodic region the annealed case cannot be directly achieved through the standard application of the borel cantelli lemma but requires previous modification of the interaction this remark could find useful application in other cases the replica symmetric expression for the free energy can be easily reached through a suitable version of the doubly stochastic interpolation technique however this model shares the unique property that the fully broken replica symmetry ansatz can be explicitly calculated a very simple sum rule connects the general expression of the fully broken free energy trial function with the replica symmetric one the definite sign of the error term shows that the replica solution is optimal then we follow a deep strategy developed by talagrand to conclude that the replica symmetric ansatz not only gives a bound for the free energy which cannot be improved by symmetry breaking but gives also the true value for the free energy for the sake of completeness we study also the fluctuations of the centered and rescaled overlaps by following a method similar to that exploited for the sherrington kirkpatrick model in the replica symmetric region

exponential synchronization of complex valued complex networks with time varying delays and stochastic perturbations via time delayed impulsive control considering the fact that time delays are unavoidable in the control of practical systems this paper considers globally exponential synchronization of complex valued complex dynamical networks with multiple time varying delays and stochastic perturbations by designing a time delayed impulsive control scheme by taking the advantage of lyapunov method in complex field and utilizing an impulsive inequality with delays several synchronization criteria are obtained through strict mathematical proofs our results are general which extend some existing ones concerning impulsive synchronization a numerical example is given to illustrate the effectiveness of theoretical results

on an achievable rate of large rayleigh block fading mimo channels with no csi training based transmission over rayleigh block fading multiple input multiple output mimo channels is investigated as a training method a combination of a pilot assisted scheme and a biased signaling scheme is considered the achievable rates of successive decoding sd receivers based on the linear minimum mean squared error lmmse channel estimation are analyzed in the large system limit by using the replica method under the assumption of replica symmetry it is shown that negligible pilot information is best in terms of the achievable rates of the sd receivers in the large system limit the obtained analytical formulas of the achievable rates can improve the existing lower bound on the capacity of the mimo channel with no channel state information csi derived by hassibi and hochwald for all snrs the comparison between the obtained bound and a high snr approximation of the channel capacity derived by zheng and tse implies that the high snr approximation is unreliable unless quite high snr is considered energy efficiency in the low snr regime is also investigated in terms of the power per information bit required for reliable communication the required minimum power is shown to be achieved at a positive rate for the sd receiver with no csi whereas it is achieved in the zero rate limit for the case of perfect csi available at the receiver moreover numerical simulations imply that the presented large system analysis can provide a good approximation for not so large systems the results in this paper imply that sd schemes can provide a significant performance gain in the low to moderate snr regimes compared to conventional receivers based on one shot channel estimation

on asymptotic stability of linear time varying systems this paper is concerned with asymptotic stability analysis of linear time varying ltv systems with the help of the notion of stable functions some differential lyapunov inequalities dlis based necessary and sufficient conditions are derived for testing asymptotic stability exponential stability and uniformly exponential stability of general ltv systems with the help of the concept of non uniformly exponential stability a class of upper triangular ltv systems is carefully investigated based on the developed stability analysis approaches a couple of numerical examples with some of them borrowed from the literature is provided to illustrate the effectiveness of the proposed theoretical results

phase transitions in semidefinite relaxations statistical inference problems arising within signal processing data mining and machine learning naturally give rise to hard combinatorial optimization problems these problems become intractable when the dimensionality of the data is large as is often the case for modern datasets a popular idea is to construct convex relaxations of these combinatorial problems which can be solved efficiently for large scale datasets semidefinite programming sdp relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs it has been observed several times that when the statistical noise is small enough sdp relaxations correctly detect the underlying combinatorial structures in this paper we develop asymptotic predictions for several detection thresholds as well as for the estimation error above these thresholds we study some classical sdp relaxations for statistical problems motivated by graph synchronization and community detection in networks we map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions our results clarify the effectiveness of sdp relaxations in solving high dimensional statistical problems

a replica coupling approach to disordered pinning models we consider a renewal process 0 1 on the integers where the law of i i 1 has a power like tail p i i 1 n n 1 l n with 0 and l slowly varying we then assign a random n dependent reward penalty to the occurrence of the event that the site n belongs to in such generality this class of problems includes among others 1 d dimensional models of pinning of directed polymers on a one dimensional random defect 1 1 dimensional models of wetting of disordered substrates and the poland scheraga model of dna denaturation by varying the average of the reward the system undergoes a transition from a localized phase where occupies a finite fraction of mathbb n to a delocalized phase where the density of vanishes in absence of disorder i e if the reward is independent of n the transition is of first order for 1 and of higher order for 1 2 11 in physical terms disorder is relevant in this situation in agreement with the heuristic harris criterion on the other hand for 0 1 2 it has been proven recently by k alexander 2 that if disorder is sufficiently weak critical exponents are not modified by randomness disorder is irrelevant in this work generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation we give a new simpler proof of the main results of 2 moreover we partially justify a small disorder expansion worked out in 9 for 1 2 showing that it provides a free energy upper bound which improves the annealed one

statistical mechanics of optimization problems abstract here i will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques after presenting the general results in order to simplify the presentation i will describe in details the problems related to the coloring of a random graph

on input to state stability of stochastic retarded systems with markovian switching this note develops a razumikhin type theorem on pth moment input to state stability of hybrid stochastic retarded systems also known as stochastic retarded systems with markovian switching which is an improvement of an existing result an application to hybrid stochastic delay systems verifies the effectiveness of the improved result

a note on the guerra and talagrand theorems for mean field spin glasses the simple case of spherical models the aim of this paper is to discuss the main ideas of the talagrand proof of the parisi ansatz for the free energy of mean field spin glasses with a physicist s approach we consider the case of the spherical p spin model which has the following advantages 1 the parisi ansatz takes the simple one step replica symmetry breaking form 2 the replica free energy as a function of the order parameters is simple enough to allow for numerical maximization with arbitrary precision we present the essential ideas of the proof we stress its connections with the theory of effective potentials for glassy systems and we reduce the technically more difficult part of the talagrand s analysis to an explicit evaluation of the solution of a variational problem

bicolouring random hypergraphs we study the problem of bicolouring random hypergraphs both numerically and analytically we apply the zero temperature cavity method to find analytical results for the phase transitions dynamic and static in the one step replica symmetry breaking 1rsb approximation these points appear to be in agreement with the results of the numerical algorithm in the second part we implement and test the survey propagation algorithm for specific bicolouring instances in the so called hard sat phase

exponential control of overlap in the replica method for p spin sherrington kirkpatrick model in talagrand j stat phys 126 4 5 837 894 2007 the large deviations limit lim n to infty na 1 log mathbb e z n a for the moments of the partition function zn in the sherrington kirkpatrick model sherrington and kirkpatrick in phys rev lett 35 1792 1796 1972 was computed for all real a 0 for a 1 this result extends the classical physicist s replica method that corresponds to integer values of a we give a new proof for a 1 in the case of the pure p spin sk model that provides a strong exponential control of the overlap

tight bounds for ldpc and ldgm codes under map decoding a new method for analyzing low density parity check ldpc codes and low density generator matrix ldgm codes under bit maximum a posteriori probability map decoding is introduced the method is based on a rigorous approach to spin glasses developed by francesco guerra it allows one to construct lower bounds on the entropy of the transmitted message conditional to the received one based on heuristic statistical mechanics calculations we conjecture such bounds to be tight the result holds for standard irregular ensembles when used over binary input output symmetric bios channels the method is first developed for tanner graph ensembles with poisson left degree distribution it is then generalized to multi poisson graphs and by a completion procedure to arbitrary degree distribution

global stability problem for coupled systems of differential equations on networks the global stability problem of equilibria is investigated for coupled systems of differential equations on networks using results from graph theory we develop a systematic approach that allows one to construct global lyapunov functions for large scale coupled systems from building blocks of individual vertex systems the approach is applied to several classes of coupled systems in engineering ecology and epidemiology and is shown to improve existing results

the stability of stochastic coupled systems with time varying coupling and general topology structure we introduce a class of novel stochastic coupled systems in which the coupling structure is time varying and the topology structure is not strongly connected and first establish the system on a digraph with a time varying weight matrix motivated by du and li 2014 we give a hierarchical method to deal with digraphs without strong connectivity and establish the corresponding hierarchical algorithm to realize this approach also an example is given to illustrate our hierarchical algorithm and its feasibility in the sequel based on the theory of asymptotically autonomous systems kirchhoff s matrix tree theorem and lyapunov method several moment exponential stability criteria are presented including a lyapunov type theorem and a coefficient type criterion furthermore theoretical results are applied to stochastic coupled oscillators with time varying coupling structure sctcs and the stability criterion of sctcs is obtained finally the effectiveness of theoretical results is illustrated by two numerical examples

stability analysis of complex valued impulsive systems with time delay in this paper the global exponential stability of complex valued impulsive systems is addressed some new sufficient conditions are obtained to guarantee the global exponential stability by the lyapunov razumikhin theory which extend and improve most of recent results moreover the obtained razumikhin conditions are very simple and efficient to verify in real problems and helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation finally a numerical example is given to show the effectiveness of the obtained results

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

brief paper a unified synchronization criterion for impulsive dynamical networks this paper focuses on the problem of globally exponential synchronization of impulsive dynamical networks two types of impulses are considered synchronizing impulses and desynchronizing impulses in previous literature all of the results are devoted to investigating these two kinds of impulses separately thus a natural question arises is there any unified synchronization criterion which is simultaneously effective for synchronizing impulses and desynchronizing impulses in this paper a unified synchronization criterion is derived for directed impulsive dynamical networks by proposing a concept named average impulsive interval the derived criterion is theoretically and numerically proved to be less conservative than existing results numerical examples including scale free and small world structures are given to show that our results are applicable to large scale networks

remarks on the interpolation method we discuss a generalization of the classic condition of validity of the interpolation method for the density of quenched free energy of mean field spin glasses the condition is written just in terms of the l 2 metric structure of the gaussian random variables as an example of application we deduce the existence of the thermodynamic limit for a grem model with infinite branches for which the classic conditions of validity fail we underline the dependence of the density of quenched free energy just on the metric structure and discuss the models from a metric viewpoint

global stability criterion for delayed complex valued recurrent neural networks the stability problem for delayed complex valued recurrent neural networks is considered in this paper by separating complex valued neural networks into real and imaginary parts forming an equivalent real valued system and constructing appropriate lyapunov functional a sufficient condition to ascertain the existence uniqueness and globally asymptotical stability of the equilibrium point of complex valued systems is provided in terms of linear matrix inequality meanwhile the errors in the recent work are pointed out and even if the result therein is correct it is shown that our result not only improves but also generalizes in that work numerical examples are given to show the effectiveness and merits of the present result

exponential stability of complex valued memristor based neural networks with time varying delays abstract in this paper we propose a new type of complex valued memristor based neural networks with time varying delays and discuss their exponential stability firstly by using a matrix measure method the halanay inequality and some analytic techniques we derive a sufficient condition for the global exponential stability of this type of neural networks then we build a lyapunov functional and utilize the halanay inequality to establish several criteria for the exponential stability of such networks with time varying delays finally we show two numerical simulations to demonstrate the theoretical results

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

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

replica bounds for diluted non poissonian spin systems in this paper we extend replica bounds and free energy subadditivity arguments to diluted spin glass models on graphs with arbitrary non poissonian degree distribution the new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method as a byproduct we obtain self averaging identities that generalize the ghirlanda guerra ones to the multi overlap case

exponential inequalities and convergence of moments in the replica symmetric regime of the hopfield model in a previous work we proved the validity of the replica symmetric solution for the hopfield model in a nontrivial domain of parameters this was done at the accuracy of the lln in a somewhat larger domain we obtain a description at the level of the clt or in the terminology of physics we calculate the fluctuations around the mean field this is obtained by refining the tools we developed for a rigorous use of the cavity method and proving new a priori estimates about the localization of the gibbs measure

phase diagram of restricted boltzmann machines and generalized hopfield networks with arbitrary priors restricted boltzmann machines are described by the gibbs measure of a bipartite spin glass which in turn can be seen as a generalized hopfield network this equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities both at low and high load of pure states we study the paramagnetic spin glass and the spin glass retrieval phase transitions as the pattern i e weight distribution and spin i e unit priors vary smoothly from gaussian real variables to boolean discrete variables our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard hopfield model with boolean patterns the retrieval region becomes larger when the pattern entries and retrieval units get more peaked and conversely when the hidden units acquire a broader prior and therefore have a stronger response to high fields moreover at low load retrieval always exists below some critical temperature for every pattern distribution ranging from the boolean to the gaussian case

rigorous low temperature results for the mean field p spins interaction model we prove that just below the critical temperature the mean field p spins interaction model for p suitably large spontaneously decomposes into different states the asymptotic overlaps between any two different states are zero under a mild unproven hypothesis on the weight distribution of these states we prove that they are pure states this situation is called in physics one level of symmetry breaking

generalized random energy model the formulae for the free energy when the driving distributions in generalized random energy model grem are of the form ce x gamma for 1 are derived the large deviation technique allows the use of different distributions at different levels of the grem as an illustration we consider in detail a two level grem with exponential and gaussian distributions this simple case itself leads to interesting phenomena

localization transition for polymers in poissonian medium we study a model of directed polymers in random environment in dimension 1 d given by a brownian motion in a poissonian potential we study the effect of the density and the strength of inhomogeneities respectively the intensity parameter of the poisson field and the temperature inverse our results are i fine information on the phase diagram with quantitative estimates on the critical curve ii pathwise localization at low temperature and or large density iii complete localization in a favourite corridor for large 2 and bounded

the adaptive interpolation method a simple scheme to prove replica formulas in bayesian inference in recent years important progress has been achieved towards proving the validity of the replica predictions for the asymptotic mutual information or free energy in bayesian inference problems the proof techniques that have emerged appear to be quite general despite they have been worked out on a case by case basis unfortunately a common point between all these schemes is their relatively high level of technicality we present a new proof scheme that is quite straightforward with respect to the previous ones we call it the adaptive interpolation method because it can be seen as an extension of the interpolation method developped by guerra and toninelli in the context of spin glasses with an interpolation path that is adaptive in order to illustrate our method we show how to prove the replica formula for three non trivial inference problems the first one is symmetric rank one matrix estimation or factorisation which is the simplest problem considered here and the one for which the method is presented in full details then we generalize to symmetric tensor estimation and random linear estimation we believe that the present method has a much wider range of applicability and also sheds new insights on the reasons for the validity of replica formulas in bayesian inference

entropy landscape of solutions in the binary perceptron problem the statistical picture of the solution space for a binary perceptron is studied the binary perceptron learns a random classification of input random patterns by a set of binary synaptic weights the learning of this network is difficult especially when the pattern constraint density is close to the capacity which is supposed to be intimately related to the structure of the solution space the geometrical organization is elucidated by the entropy landscape from a reference configuration and of solution pairs separated by a given hamming distance in the solution space we evaluate the entropy at the annealed level as well as replica symmetric level and the mean field result is confirmed by the numerical simulations on single instances using the proposed message passing algorithms from the first landscape a random configuration as a reference we see clearly how the solution space shrinks as more constraints are added from the second landscape of solution pairs we deduce the coexistence of clustering and freezing in the solution space

finite range spin glasses in the kac limit free energy and local observables we study a finite range spin glass model in arbitrary dimension where the intensity of the coupling between spins decays to zero over some distance 1 we prove that under a positivity condition for the interaction potential the infinite volume free energy of the system converges to that of the sherrington kirkpatrick model in the kac limit 0 we study the implication of this convergence for the local order parameter i e the local overlap distribution function and a family of susceptibilities associated with it and we show that locally the system behaves like its mean field analogue similar results are obtained for models with p spin interactions finally we discuss a possible approach to the problem of the existence of long range order for finite based on a large deviation functional for overlap profiles this will be developed in future work

replica bounds by combinatorial interpolation for diluted spin systems in two papers franz et al proved bounds for the free energy of diluted random constraints satisfaction problems for a poisson degree distribution franz and leone in j stat phys 111 3 4 535 564 2003 and a general distribution franz et al in j phys a 36 43 10967 2003 panchenko and talagrand probab theo relat fields 130 3 319 336 2004 simplified the proof and generalized the result of franz and leone j stat phys 111 3 4 535 564 2003 for the poisson case we provide a new proof for the general degree distribution case and as a corollary we obtain new bounds for the size of the largest independent set also known as hard core model in a large random regular graph our proof uses a combinatorial interpolation based on biased random walks salez in combin probab comput 25 03 436 447 2016 and allows to bypass the arguments in franz et al j phys a 36 43 10967 2003 based on the study of the sherrington kirkpatrick sk model

a note on mixed matrix moments for the complex ginibre ensemble we consider the mixed matrix moments for the complex ginibre ensemble these are well known we consider the relation to the expected overlap functions of chalker and mehlig this leads to new asymptotic problems for the overlap we obtain some results but we also state some remaining open problems

exponential stability of impulsive stochastic functional differential equations in this paper we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using lyapunov method several stability theorems of impulsive stochastic functional differential equations with finite delay are derived these new results are employed to impulsive stochastic equations with bounded time varying delays and stochastically perturbed equations meanwhile an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay

stability of complex valued impulsive stochastic functional differential equations on networks with markovian switching abstract this paper investigates the stability problem of complex valued impulsive stochastic functional differential equations on networks with markovian switching cisfnm markovian switching and impulsive effects are considered into complex valued systems for the first time by virtue of complex generalized ito s formula the stability of cisfnm can be analyzed effectively without splitting the real and imaginary parts then based on razumikhin technique graph theoretical technique as well as average dwell time approach several novel stability criteria are derived which mainly depend on integral average value of time varying coefficients compared with some existing results our results gain not only less conservativeness but also lower calculation complexity subsequently the stability of complex valued impulsive stochastic networks with time varying delays and markovian switching is studied as an application complex valued coupled oscillators are considered and numerical simulations are also presented to show the effectiveness of our main results

global exponential convergence of fuzzy complex valued neural networks with time varying delays and impulsive effects abstract in this paper the global exponential convergence of t s fuzzy complex valued neural networks with time varying delays and impulsive effects is discussed by employing lyapunov functional method and matrix inequality technique we analyze a type of activation functions with lipschitz function and sufficient conditions in terms of complex valued linear matrix inequality are obtained to ensure the global exponential convergence moreover the framework of the exponential convergence ball in the state space of the considered neural networks and the exponential convergence rate index are also given out here the existence and uniqueness of the equilibrium points need not be considered and the results improve existing results on the lyapunov exponential stability as special cases finally one numerical example with simulations is given to illustrate the effectiveness of our theoretical results

variational bounds for the generalized random energy model we compute the pressure of the random energy model rem and generalized random energy model grem by establishing variational upper and lower bounds for the upper bound we generalize guerra s broken replica symmetry bounds and identify the random probability cascade as the appropriate random overlap structure for the model for the rem the lower bound is obtained in the high temperature regime using talagrand s concentration of measure inequality and in the low temperature regime using convexity and the high temperature formula the lower bound for the grem follows from the lower bound for the rem by induction while the argument for the lower bound is fairly standard our proof of the upper bound is new

factor models on locally tree like graphs we consider homogeneous factor models on uniformly sparse graph sequences converging locally to a unimodular random tree t and study the existence of the free energy density the limit of the log partition function divided by the number of vertices n as n tends to infinity we provide a new interpolation scheme and use it to prove existence of and to explicitly compute the quantity subject to uniqueness of a relevant gibbs measure for the factor model on t by way of example we compute for the independent set or hard core model at low fugacity for the ferromagnetic ising model at all parameter values and for the ferromagnetic potts model with both weak enough and strong enough interactions even beyond uniqueness regimes our interpolation provides useful explicit bounds on in the regimes in which we establish existence of the limit we show that it coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation bethe recursions on t in the special case that t has a galton watson law this formula coincides with the nonrigorous bethe prediction obtained by statistical physicists using the replica or cavity methods thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees we also provide a variational characterization for the bethe prediction in this general setting which is of independent interest

analogue neural networks on correlated random graphs we consider a generalization of the hopfield model where the entries of patterns are gaussian and diluted we focus on the high storage regime and we investigate analytically the topological properties of the emergent network as well as the thermodynamic properties of the model we find that by properly tuning the dilution in the pattern entries the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size the structure is also shown to exhibit a large degree of cliquishness even when very sparse moreover we obtain explicitly the replica symmetric free energy and the self consistency equations for the overlaps order parameters of the theory which turn out to be classical weighted sums of sub overlaps defined on all possible sub graphs finally a study of criticality is performed through a small overlap expansion of the self consistencies and through a whole fluctuation theory developed for their rescaled correlations both approaches show that the net effect of dilution in pattern entries is to rescale the critical noise level at which ergodicity breaks down

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

on the surface pressure for the edwards anderson model for the edwards anderson model we introduce an integral representation for the surface pressure per unit surface in terms of a quenched moment of the bond overlap on the surface we consider free periodic and antiperiodic boundary conditions by symmetry and prove the bounds open image in new windowopen image in new window we show moreover that at high temperatures is close to 2 4 and is close to 2 4 uniformly in the volume