replica symmetry breaking discrete and continuous schemes in the sherrington kirkpatrick model we study hierarchies of replica symmetry breaking solutions of the sherrington kirkpatrick model stationarity equations for order parameters of solutions with an arbitrary number of hierarchies are set and the limit to infinite number of hierarchical levels is discussed in particular we demonstrate how the continuous replica symmetry breaking scheme of parisi emerges and how the limit to infinite many hierarchies leads to equations for the order parameter function of the continuous solution the general analysis is accompanied by an explicit asymptotic solution near the de almeida thouless instability line in the nonzero magnetic field

rigorous solution of the gardner problem we prove rigorously the well known result of gardner about the typical fractional volume of interactions between n spins which solve the problem of storing a given set of p random patterns the gardner formula for this volume in the limit n p p n is proven for all values of besides we prove a useful criterion for the factorisation of all correlation functions for a class of models of classical statistical mechanics

gibbs states of the hopfield model with extensively many patterns we consider the hopfield model withm n n patterns wheren is the number of neurons we show that if is sufficiently small and the temperature sufficiently low then there exist disjoint gibbs states for each of the stored patterns almost surely with respect to the distribution of the random patterns this solves a provlem left open in previous work the key new ingredient is a self averaging result on the free energy functional this result has considerable additional interest and some consequences are discussed a similar result for the free energy of the sherrington kirkpatrick model is also given

the replica symmetric solution for potts models on d regular graphs we establish an explicit formula for the limiting free energy density log partition function divided by the number of vertices for ferromagnetic potts models on uniformly sparse graph sequences converging locally to the d regular tree for d even covering all temperature regimes this formula coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d regular tree the so called replica symmetric solution for uniformly random d regular graphs we further show that the replica symmetric bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions

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

the free energy of a class of hopfield models we show that in the limitp n 0 p n 0 the limit free energy of the hopfield model equals in probability the curie weiss free energy we prove also that the free energy of the hopfield model is self averaging for any finite

griffith kelly sherman correlation inequalities a useful tool in the theory of error correcting codes it is shown that a correlation inequality of statistical mechanics can be applied to linear low density parity check codes thanks to this tool we prove that under a natural assumption the exponential growth rate of regular low density parity check ldpc codes can be computed exactly by iterative methods at least on the interval where it is a concave function of the relative weight of code words then considering communication over a binary input additive white gaussian noise channel with a poisson ldpc code we prove that under a natural assumption part of the gexit curve associated to map decoding can also be computed exactly by the belief propagation algorithm the correlation inequality yields a sharp lower bound on the gexit curve we also make an extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the sat problem

the parisi formula has a unique minimizer in 1979 parisi phys rev lett 43 1754 1756 1979 predicted a variational formula for the thermodynamic limit of the free energy in the sherrington kirkpatrick model and described the role played by its minimizer this formula was verified in the seminal work of talagrand ann math 163 1 221 263 2006 and later generalized to the mixed p spin models by panchenko ann probab 42 3 946 958 2014 in this paper we prove that the minimizer in parisi s formula is unique at any temperature and external field by establishing the strict convexity of the parisi functional

completeness of the dirac oscillator eigenfunctions completeness of the dirac oscillator eigenfunctions is proved in one and three spatial dimensions proofs are based on standard properties of the hermite and the generalized laguerre polynomials

large system analysis of joint channel and data estimation for mimo ds cdma systems this paper presents a large system analysis of the performance of joint channel estimation multiuser detection and per user decoding ce mudd for randomly spread multiple input multiple output mimo direct sequence code division multiple access ds cdma systems a suboptimal receiver based on successive decoding in conjunction with linear minimum mean squared error lmmse channel estimation is investigated the replica method developed in statistical mechanics is used to evaluate the performance in the large system limit where the number of users and the spreading factor tend to infinity while their ratio and the number of transmit and receive antennas are kept constant joint ce mudd based on lmmse channel estimation is compared to several methods of one shot channel estimation i e methods that do not utilize decoded data symbols to refine the initial channel estimates in terms of spectral efficiency joint ce mudd is found to significantly reduce the rate loss caused by transmission of pilot signals when compared to the receivers based on one shot channel estimation this holds particularly for multiple antenna systems as a result joint ce mudd can provide significant performance gains compared to the receivers based on one shot channel estimation

replica bounds for optimization problems and diluted spin systems in this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by guerra cond mat 0205123 to prove that the replica method generates variational bounds for disordered systems we analyze a family of models that includes the viana bray model the diluted p spin model or random xor sat problem and the random k sat problem showing that the replica cavity method at the various levels of approximation provides systematic schemes to obtain lower bounds of the free energy at all temperatures and of the ground state energy in the case of k sat and xor sat it thus gives upper bounds of the satisfiability threshold our analysis underlines deep connections with the cavity method which are not evident in the long range case

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

the legendre transform of two replicas of the sherrington kirkpatrick spin glass model we prove a variational inequality linking the values of the free energy per site at different temperatures this inequality is based on the legendre transform of the free energy of two replicas of the system we prove that equality holds when 1 sqrt 2 and fails when 1 sqrt 2 1 sqrt 2 we also use our inequality numerically within the low temperature region to improve slightly the best previously known lower bounds for the free energy and the ground state energy per site

bounds for diluted mean fields spin glass models in an important recent paper 2 s franz and m leone prove rigorous lower bounds for the free energy of the diluted p spin model and the k sat model at any temperature we show that the results for these two models are consequences of a single general principle our calculations are significantly simpler than those of 2 even in the replica symmetric case

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

behaviour of wavepackets of the dirac oscillator dirac representation versus foldy wouthuysen representation the behaviour of wavepackets of the dirac oscillator in 1 1 dimensions is examined the dirac oscillator is a relativistic extension of the harmonic oscillator we construct a wavepacket in the usual dirac representation and another one in the foldy wouthuysen fw representation in the nonrelativistic limit these wavepackets are both reduced to schrodinger s coherent wavepacket for the harmonic oscillator the wavepacket in the dirac representation exhibits zitterbewegung whereas the one in the fw representation does not we find that the wavepacket in the fw representation behaves much more like a classical particle than that of the dirac representation this gives insights into ehrenfest s theorem for the dirac equation

the sherrington kirkpatrick model of spin glasses and stochastic calculus the high temperature case we study the fluctuations of free energy energy and entropy in the high temperature regime for the sherrington kirkpatrick model of spin glasses we introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter thus formulating the thermodynamic formalism in terms of random processes the well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments a generalization of a result proved by aizenman lebowitz and ruelle 1

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

on the free energy of the hopfield model the general theory of inhomogeneous mean field systems of raggio and werner provides a variational expression for the almost sure limiting free energy density of the hopfield model h n p xi s frac 1 2n sum limits i j 1 n sum limits mu 1 n xi i mu xi j mu s i s j for ising spinssi andp random patterns 1 2 n under the assumption that mathop lim limits n to gamma n 1 sum limits i 1 n delta xi i lambda xi i xi i 1 xi i 2 xi i p exists almost surely in the space of probability measures overp copies of 1 1 including an external field term p h i 1 n i si we give a number of general properties of the free energy density and compute it for a p 2 in general and b p arbitrary when is uniform and at most the two componentsh 1 andh 2 are nonzero obtaining the almost sure formula f beta h tfrac 1 2 f ew beta h mu 1 h mu 2 tfrac 1 2 f ew beta h mu 1 h mu 2 for the free energy wherefcw denotes the limiting free energy density of the curie weiss model with unit interaction constant in both cases we obtain explicit formulas for the limiting almost sure values of the so called overlap parameters m n mu beta h n 1 sum limits i 1 n xi i mu left langle s i right rangle in terms of the curie weiss magnetizations for the general i i d case with prob i 1 1 2 we obtain the lower bound 1 4 2 p 1 for the temperaturetc separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero this lower bound is exact forp 2 or e 0 or e 1 2 forp 2 we identify an intermediate temperature region between t 1 4 2 and tc 1 4 2 where the overlap vector is homogeneous i e all its components are equal and nonzero t marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct we conjecture that the homogeneous nonzero regime exists forp 3 and that t max 1 4 2 p 1 0

algebraic properties of the dirac oscillator an algebraic representation independent analysis is presented for the dirac oscillator in an angular momentum basis the analysis is based on shift operators for energy and angular momentum and it is similar to that for a non relativistic isotropic harmonic oscillator the shift operators generate all the eigenkets of the dirac oscillator from a vacuum ket the shift operations yield energy eigenvalues and certain matrix elements the relationship to the factorization method is discussed

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

exact solution of the gauge symmetric p spin glass model on a complete graph we consider a gauge symmetric version of the p spin glass model on a complete graph the gauge symmetry guarantees the absence of replica symmetry breaking and allows to fully use the interpolation scheme of guerra fields inst commun 30 161 2001 to rigorously compute the free energy in the case of pairwise interactions p 2 where we have a gauge symmetric version of the sherrington kirkpatrick model we get the free energy and magnetization for all values of external parameters our analysis also works for even p 4 except in a range of parameters surrounding the phase transition line and for odd p 3 in a more restricted region we also obtain concentration estimates for the magnetization and overlap parameter that play a crucial role in the proofs for odd p and justify the absence of replica symmetry breaking our initial motivation for considering this model came from problems related to communication over a noisy channel and is briefly explained

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

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

spin glass computations and ruelle s probability cascades we study the parisi functional appearing in the parisi formula for the pressure of the sk model as a functional on ruelle s probability cascades rpc computation techniques for the rpc formulation of the functional are developed they are used to derive continuity and monotonicity properties of the functional retrieving a theorem of guerra we also detail the connection between the aizenman sims starr variational principle and the parisi formula as a final application of the techniques we rederive the almeida thouless line in the spirit of toninelli but relying on the rpc structure

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

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

rigorous results for the free energy in the hopfield model we prove that the free energy of the hopfield model with a finite number of patterns can be represented in terms of an asymptotic series expansion in inverse powers of the neurons number the series is borel summable for large temperatures we also establish mathematically some other interesting properties partly used before in a seminal paper by amit gutfreund and sompolinsky

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

replica symmetry breaking complexity and spin representation in the generalized random energy model we study the random energy model with a hierarchical structure known as the generalized random energy model grem in contrast to the original analysis by the microcanonical ensemble formalism we investigate the grem by the canonical ensemble formalism in conjunction with the replica method in this analysis spin glass order parameters are defined for the respective hierarchy level and all possible patterns of replica symmetry breaking rsb are taken into account as a result we find that the higher step rsb ansatz is useful for describing spin glass phases in this system for investigating the nature of the higher step rsb we generalize the notion of complexity developed for the one step rsb to the higher step and demonstrate how the grem is characterized by the generalized complexity in addition we propose a novel mean field spin glass model with a hierarchical structure which is equivalent to the grem at a certain limit we also show that the same hierarchical structure can be implemented to other mean field spin models than the grem such models with hierarchy exhibit phase transitions of multiple steps in common

fluctuations of the free energy in the mixed p spin models with external field we show that the free energy in the mixed p spin models of spin glasses does not superconcentrate in the presence of external field which means that its variance is of the order suggested by the poincare inequality this complements the result of chatterjee who showed that the free energy superconcentrates when there is no external field for models without odd p spin interactions for p geqslant 3 we prove the central limit theorem for the free energy at any temperature and give an explicit formula for the limiting variance although we only deal with the case of ising spins all our results can be extended to the spherical models as well

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

on the distribution of overlaps in the sherrington kirkpatrick spin glass model this paper describes some of the analytic tools developed recently by ghirlanda and guerra in the investigation of the distribution of overlaps in the sherrington kirkpatrick spin glass model and of parisi s ultrametricity in particular we introduce to this task a simplified but also generalized model on which the gaussian analysis is made easier moments of the hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps under the essential tool of self averaging we describe with full rigour various overlap identities and replica independence that actually hold in a rather large generality the results are presented in a language accessible to probabilists and analysts

quenched large deviation principle for the overlap of a p spins system in this note we prove a quenched large deviation result for the overlap of a p spins interaction system at high temperature the rate function of the large deviation principle is proved to be deterministic and some of its basic properties are studied our result is based on a pure state result for a multidimensional p spins system combined with a careful application of the gartner ellis theorem

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

dirichlet to neumann and neumann to dirichlet embedding methods for bound states of the dirac equation the embedding method inglesfield j e 1981 j phys c solid state phys 14 3795 is nowadays widely used for determining properties of bound and continuum eigenstates of the schrodinger equation recently crampin 2004 j phys condens matter 16 8875 extended inglesfield s formalism to systems described by the dirac equation in this paper we reformulate the embedding method for bound states of dirac particles in the language of the theory of surface integral operators which are natural analogues of the dirichlet to neumann dtn and neumann to dirichlet ntd operators used in the non relativistic theory a method of constructing kernels of these dirac counterparts of the dtn and ntd maps from solutions to an analogue of the steklov stekloff eigenproblem for an exterior dirac problem is presented a numerical example illustrating the utility of the dtn ntd embedding method is provided

self averaging in a class of generalized hopfield models we prove the almost sure convergence to zero of the fluctuations of the free energy in a class of disordered mean field spin systems that generalize the hopfield model in two ways i multi spin interactions are permitted and ii the random variables xi imu describing the patterns can have arbitrary distributions with mean zero and finite 4 epsilon th moments the number of patterns m is allowed to be an arbitrary multiple of the system size this generalizes a previous result of bovier gayrard and picco for the standard hopfield model and improves a result of feng and tirozzi that required m to be a finite constant note that the convergence of the mean of the free energy is not proven

free energy and some sample path properties of a random walk with random potential we study the asymptotic behavior of the free energy for a model defined by sinai of one dimensional random walk with random potential in particular we obtain a central limit theorem and a strong law of large numbers for this free energy we use some results on the free energy to study some sample path properties of this random walk which are related respectively to its recurrence and localization some exponents describing the recurrence and localization are found

cavity approach to sphere packing in hamming space in this paper we study the hard sphere packing problem in the hamming space by the cavity method we show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of gilbert and varshamov consistently with known numerical results the replica symmetric equations also suggest a crystalline solution where for even diameters the spheres are more likely to be found in one of the subspaces even or odd of the hamming space these crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra metric space finally we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres known maximum packings are reproduced efficiently i n non trivial ranges of dimensions and number of spheres

hierarchical exchangeability of pure states in mean field spin glass models the main result in this paper is motivated by the mezard parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models such as the random k sat model at positive temperature using the fact that one can safely assume the validity of the ghirlanda guerra identities in these models we prove hierarchical exchangeability of pure states for the asymptotic gibbs measures which allows us to apply a representation result for hierarchically exchangeable arrays recently proved in austin and panchenko in probab theory relat fields 2013 comparing this representation with the predictions of the mezard parisi ansatz one can see that the key property still missing is that the multi overlaps between pure states depend only on their overlaps

statistical mechanics of low density parity check codes we review recent theoretical progress on the statistical mechanics of error correcting codes focusing on low density parity check ldpc codes in general and on gallager and mackay neal codes in particular by exploiting the relation between ldpc codes and ising spin systems with multispin interactions one can carry out a statistical mechanics based analysis that determines the practical and theoretical limitations of various code constructions corresponding to dynamical and thermodynamical transitions respectively as well as the behaviour of error exponents averaged over the corresponding code ensemble as a function of channel noise we also contrast the results obtained using methods of statistical mechanics with those derived in the information theory literature and show how these methods can be generalized to include other channel types and related communication problems

the distribution of the partition function of the hopfield model with finite number of patterns we derive the leading term in the large n asymptotic expansion of the partition function of the hopfield model with finite number of patterns we show that this leading order term is deterministic in the high temperature region in the low temperature region and at the critical point it is random with the distribution governed by normal or iterated exponential distributions

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

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

rigorous results for mean field models for spin glasses in this paper i explain why i believe that it is important to prove rigorous results about mean field models for spin glasses and why i think it is difficult i also describe at a high level what mathematicians have been able to prove

large deviation principles for the hopfield model and the kac hopfield model we study the kac version of the hopfield model and prove a lebowitz penrose theorem for the distribution of the overlap parameters at the same time we prove a large deviation principle for the standard hopfield model with infinitely many patterns

fluctuation of the free energy in the hopfield model we prove that in the ergodic region t j2 1 r the deviation of the total free energy of the hopfield neural network converges in distribution asn to a shifted gaussian variable moreover the free energy per site converges in probability to lim 1 n ln n

structure of finite rsb asymptotic gibbs measures in the diluted spin glass models we suggest a possible approach to proving the mezard parisi formula for the free energy in the diluted spin glass models such as diluted k spin or random k sat model at any positive temperature in the main contribution of the paper we show that a certain small modification of the hamiltonian in any of these models forces all finite rsb asymptotic gibbs measures in the sense of the overlaps to satisfy the mezard parisi ansatz for the distribution of spins unfortunately what is still missing is a description of the general full rsb asymptotic gibbs measures if one could show that the general case can be approximated by finite rsb case in the right sense then one could a posteriori remove the small modification of the hamiltonian to recover the mezard parisi formula for the original model

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

some rigorous results on the hopfield neural network model we analyze the thermal equilibrium distribution of 2p mean field variables for the hopfield model withp stored patterns in the case where 2p is small compared to the number of spins in particular we give a full description of the free energy density in the thermodynamic limit and of the so called symmetric solutions for the mean field equations

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

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

fundamental limits of symmetric low rank matrix estimation we consider the high dimensional inference problem where the signal is a low rank symmetric matrix which is corrupted by an additive gaussian noise given a probabilistic model for the low rank matrix we compute the limit in the large dimension setting for the mutual information between the signal and the observations as well as the matrix minimum mean squared error while the rank of the signal remains constant we also show that our model extends beyond the particular case of additive gaussian noise and we prove an universality result connecting the community detection problem to our gaussian framework we unify and generalize a number of recent works on pca sparse pca submatrix localization or community detection by computing the information theoretic limits for these problems in the high noise regime in addition we show that the posterior distribution of the signal given the observations is characterized by a parameter of the same dimension as the square of the rank of the signal i e scalar in the case of rank one this allows to locate precisely the information theoretic thresholds for the above mentioned problems finally we connect our work with the hard but detectable conjecture in statistical physics

on the freezing of variables in random constraint satisfaction problems the set of solutions of random constraint satisfaction problems zero energy groundstates of mean field diluted spin glasses undergoes several structural phase transitions as the amount of constraints is increased this set first breaks down into a large number of well separated clusters at the freezing transition which is in general distinct from the clustering one some variables spins take the same value in all solutions of a given cluster in this paper we introduce and study a message passing procedure that allows to compute for generic constraint satisfaction problems the sizes of the rearrangements induced in response to the modification of a variable these sizes diverge at the freezing transition with a critical behavior which is also investigated in details we apply the generic formalism in particular to the random satisfiability of boolean formulas and to the coloring of random graphs the computation is first performed in random tree ensembles for which we underline a connection with percolation models and with the reconstruction problem of information theory the validity of these results for the original random ensembles is then discussed in the framework of the cavity method

the replica symmetric solution without replica trick for the hopfield model we derive the saddle point equations for the order parameters of the hopfield model in the case of replica symmetry without using the replica trick but assuming that the edwards anderson parameter is a self averaging quantity

integral forms in vertex operator algebras which are invariant under finite groups abstract for certain pairs consisting of a vertex operator algebra e g lattice type and a finite subgroup of its automorphism group we prove existence of a positive definite integral form invariant under the group applications include an integral form in the moonshine voa which is invariant under the monster and examples in other lattice type voas

lattice integrality of certain group invariant integral forms in vertex operator algebras abstract certain vertex operator algebras have integral forms integral spans of bases which are closed under the countable set of products it is unclear when they or integral multiples of them are integral as lattices under the natural bilinear form on the voa we show that lattice integrality may be arranged under some hypotheses including cases of integral forms invariant by finite groups in particular there exists a lattice integral monster invariant integral form in the moonshine voa

determinants for integral forms in lattice type vertex operator algebras abstract we prove a determinant formula for the standard integral form of a lattice vertex operator algebra

the legendre structure of the parisi formula we show that the parisi formula of the mixed p spin model is a concave function of the squared inverse temperature this allows us to derive a new expression for the parisi formula that involves the inverse temperature and the parisi measure as legendre conjugate variables

information theoretic thresholds from the cavity method abstract vindicating a sophisticated but non rigorous physics approach called the cavity method we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models we work out several concrete applications of these general results for instance we pinpoint the exact condensation phase transition in the potts antiferromagnet on the random graph thereby improving prior approximate results contucci et al 2013 34 further we prove the conjecture from krzakala et al 2007 55 about the condensation phase transition in the random graph coloring problem for any number q 3 of colors moreover we prove the conjecture on the information theoretic threshold in the disassortative stochastic block model decelle et al 2011 35 additionally our general result implies the conjectured formula for the mutual information in low density generator matrix codes montanari 2005 73

large time asymptotics for the fluctuation spde in the kuramoto synchronization model abstract we investigate the long time asymptotics of the fluctuation spde in the kuramoto synchronization model we establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its mckean vlasov limit this is carried out through a spectral analysis of the underlying unbounded evolution operator using arguments of perturbation of self adjoint operators and analytic semigroups we state in particular a jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the disordered kuramoto model are not self averaging