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

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

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

a rotation invariant spherical harmonic decomposition method for mapping intravoxel multiple fiber structures a new rotation invariant spherical harmonic decomposition shd method is proposed in this paper for analyzing high angular resolution diffusion hard imaging regular shd methods have been used to characterize the features of the apparent diffusion coefficient adc profile measured by the hard technique however these regular shd methods are rotation variant i e the magnitude and or the phase of the harmonic components changes with the rotation of the adc profile we propose a new rotation invariant shd ri shd method based on the rotation invariant property of a diffusion tensor model the basic idea of the proposed method is to reorient the measured adc profile into a local coordinate system determined by the three eigenvectors of the diffusion tensor in each imaging voxel and then apply a shd to the adc profile both simulations and in vivo experiments were carried out to validate the method comparisons were made between the component maps from a regular shd method diffusion circular spectrum mapping dcsm method and the proposed ri shd method the results indicate that the regular shd maps vary significantly with the rotation of the diffusion encoding scheme whereas the maps of the dcsm and the proposed method remain unchanged in particular the 0 0 th 2 2 th and 4 4 th component maps from the ri shd method exhibited good consistency with the 0th 2nd and 4th order maps of the dcsm method respectively compared with the regular shd methods used in hard imaging the proposed ri shd method is superior in characterizing the diffusion patterns of multiple fiber structures between different brain regions or across subjects

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

a tight degree 4 sum of squares lower bound for the sherrington kirkpatrick hamiltonian we show that if varvec w is an n times n matrix drawn from the gaussian orthogonal ensemble then with high probability the degree 4 sum of squares relaxation cannot certify an upper bound on the objective n 1 cdot varvec x top varvec w varvec x under the constraints x i 2 1 0 i e varvec x in pm 1 n that is asymptotically smaller than lambda max varvec w approx 2 we also conjecture a proof technique for lower bounds against sum of squares relaxations of any degree held constant as n rightarrow infty by proposing an approximate pseudomoment construction

replica symmetry of the minimum matching we establish the soundness of the replica symmetric ansatz introduced by m m ezard and g parisi for the minimum matching problem in the pseudo dimensiond mean eld model for d 1 the cased 1 corresponds to the 2 6 limit for the assignment problem proved by d aldous in 2001 we introduce a game theoretical framework by which we establish the analogous limit also for d 1

on the energy landscape of the mixed even p spin model we investigate the energy landscape of the mixed even p spin model with ising spin configurations we show that for any given energy level between zero and the maximal energy with overwhelming probability there exist exponentially many distinct spin configurations such that their energies stay near this energy level furthermore their magnetizations and overlaps are concentrated around some fixed constants in particular at the level of maximal energy we prove that the hamiltonian exhibits exponentially many orthogonal peaks this improves the results of chatterjee disorder chaos and multiple valleys in spin glasses 2009 and ding et al ann probab 43 6 3468 3493 2015 where the former established a logarithmic size of the number of the orthogonal peaks while the latter proved a polynomial size our second main result obtains disorder chaos at zero temperature and at any external field as a byproduct this implies that the fluctuation of the maximal energy is superconcentrated when the external field vanishes and obeys a gaussian limit law when the external field is present

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

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

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

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

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

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

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

buchsbaumness in rees modules associated to ideals of minimal multiplicity in the equi i invariant case abstract the present article gives certain conditions for rees modules to obtain buchsbaumness suppose a given m primary ideal is of minimal multiplicity in the equi i invariant case then it is shown that the positively graded submodule of the rees module must be buchsbaum and moreover that the rees module itself is also buchsbaum if the dimension of a given module is greater than one

universality of the mean field for the potts model we consider the potts model with q colors on a sequence of weighted graphs with adjacency matrices a n allowing for both positive and negative weights under a mild regularity condition on a n we show that the mean field prediction for the log partition function is asymptotically correct whenever mathrm tr a n 2 o n in particular our results are applicable for the ising and the potts models on any sequence of graphs with average degree going to infty using this we establish the universality of the limiting log partition function of the ferromagnetic potts model for a sequence of asymptotically regular graphs and that of the ising model for bi regular bipartite graphs in both ferromagnetic and anti ferromagnetic domain we also derive a large deviation principle for the empirical measure of the colors for the potts model on asymptotically regular graphs

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

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

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

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

the sherrington kirkpatrick model a challenge for mathematicians the sherrington kirkpatrick sk model for spin glasses is deceptively simple to state yet its rigorous study represents a considerable challenge we report here some modest progresses obtained through elementary methods even in the supposedly simple high temperature region a number of basic questions remain unsolved

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

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

on the hardness of sampling independent sets beyond the tree threshold we consider local markov chain monte carlo algorithms for sampling from the weighted distribution of independent sets with activity where the weight of an independent set i is i a recent result has established that gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and c it is np hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph