critical behaviour of anisotropic superelastic central force percolation the authors study central force percolation on a triangular lattice where the bonds are springs that can freely rotate around the nodes all bonds in one preferred direction are infinitely rigid in the other two directions a proportion p of the springs chosen at random are infinitely rigid while the rest have a finite strength for this specific problem the threshold at which the network becomes infinitely rigid corresponds to the usual bond percolation threshold of a square lattice and is therefore exactly equal to 0 5 they use a transfer matrix algorithm to study the elastic modulus of strips of width ranging from 2 to 32 and length 105 they obtain an estimate of the critical exponent relative to the scaling of the elastic moduli with the strip width 0 99 or 0 03 close to s v for the corresponding isotropic electric problem

rigid animals in three dimensions the number of rigid clusters c n is enumerated for the central force model on a site diluted fcc lattice the exponent vr governing the growth of the radius of gyration of the rigid animal is also calculated it is found that c n approximately n0 333n and vr 0 32 or 0 02

an efficient algorithm for testing the generic rigidity of graphs in the plane given a structure made up of n sites connected by b bars the problem of recognizing which subsets of sites form rigid units is not a trivial one because of the non local character of rigidity in central force systems even though this is a very old problem of statics no simple algorithms are available for it so the most usual approach has been to solve the elastic equations which is very time consuming for large systems recently an integer algorithm was proposed for this problem in two dimensions using matching methods from graph theory and laman s theorem for two dimensional graphs the method is relatively simple but its time complexity grows as in the worst case and almost as fast on practical cases so that an improvement is highly desirable in this paper we describe a further elaboration of that procedure which relies upon the description of the system as a collection of rigid bodies connected by bars instead of sites connected by bars sets of rigidly connected objects are replaced by a unique body and this is done recursively as more rigid connections between bodies are discovered at larger scales as a consequence of this rescaling transformation our algorithm has much improved average behaviour even when its worst case complexity remains the time complexity of the body bar algorithm is found to scale as for the randomly diluted triangular lattice while the original site bar version scales as for the same problem

lattice based random jammed configurations for hard particles a nontrivial subset of the jammed packings for rigid disks and spheres are those that can be obtained by sequential removal of particles from periodic crystalline arrays this paper considers the enumeration problems presented by such packings that are based on the close packed triangular disk lattice and the face centered and body centered cubic sphere lattices three distinct categories of packings have been distinguished depending on their behavior with respect to nonoverlap geometric constraints and or externally imposed virtual displacements locally jammed collectively jammed and strictly jammed each of these possesses an upper limiting vacancy concentration beyond which no packings of the types considered can exist for each of the three lattices specific vacancy clusters have been identified whose presence would destroy local jamming and some of the corresponding patterns that would destroy collective jamming in the triangular disk lattice have also been found within the allowable range of vacancy concentration for each case the number of distinct jammed packings is expected to rise exponentially with system size by using the concept of local attrition factors approximate enumerations have been constructed for the three lattice classes of locally jammed packings in the interests of later extension of this work we stress that at least some aspects of these enumeration problems might benefit from the formal transcription to a lattice gas ising model representation with vacancy interactions chosen to enforce the packing category of interest

towards a dynamical temperature of finite hamiltonian systems abstract with the aid of numerical simulations of the fermi pasta ulam fpu system we compare the different definitions of dynamical temperature for hamiltonian systems we have shown that each definition gives different values of temperature for a system with a small number of degrees of freedom dof only for systems with a sufficiently large number of dof do all the definitions of dynamical temperature approach the same value

possible physical meaning of the tsallis entropy parameter since the proposal of the tsallis generalized entropy the general explanation of the role played by the parameter q that defines which specific entropy to pick among a whole family remained somewhat partial and tentative although some particular examples were taken into account specifically the purpose of the present paper is to present a rigorous formal derivation of a mathematical expression giving the parameter q in terms of the momentum fluctuations of a stochastic process thus furnishing at least one of its possible physical origins

an exactly solvable random satisfiability problem we introduce a new model for the generation of random satisfiability problems it is an extension of the hyper sat model of ricci tersenghi weigt and zecchina which is a variant of the famous k sat model it is extended to q state variables and relates to a different choice of the statistical ensemble the model has an exactly solvable statistic the critical exponents and scaling functions of the sat unsat transition are calculable at zero temperature with no need of replicas also with exact finite size corrections we also introduce an exact duality of the model and show an analogy of thermodynamic properties with the random energy model of disordered spin system theory relations with error correcting codes are also discussed

rigidity transition in two dimensional random fiber networks rigidity percolation is analyzed in two dimensional random fibrous networks the model consists of central forces between the adjacent crossing points of the fibers two strategies are used to incorporate rigidity adding extra constraints between second nearest crossing points with a probability p sn and welding individual crossing points by adding there four additional constraints with a probability p weld and thus fixing the angles between the fibers these additional constraints will make the model rigid at a critical probability p sn p sn c and p weld p weld c respectively accurate estimates are given for the transition thresholds and for some of the associated critical exponents the transition is found in both cases to be in the same universality class as that of the two dimensional central force rigidity percolation in diluted lattices

comparison of rigidity and connectivity percolation in two dimensions using a recently developed algorithm for generic rigidity of two dimensional graphs we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to l 4096 we compare three different universality classes the generic rigidity class the connectivity class and the generic open quotes braced square net close quotes gbsn we analyze the spanning cluster density p sub infinity the backbone density p sub b and the density of dangling ends p sub d in the generic rigidity gr and connectivity cases the load carrying component of the spanning cluster the backbone is ital fractal at p sub c so that the backbone density behaves as b approximately p minus p sub c sup beta sup prime for p gt p sub c we estimate beta sub gr sup prime 0 25 plus minus 0 02 for generic rigidity and beta sub c sup prime 0 467 plus minus 0 007 for the connectivity case we find the correlation length exponents nu sub gr 1 16 plus minus 0 03 for generic rigidity compared to the exact value for connectivity nu sub c 4 3 in contrast the gbsn undergoes a first order rigidity transition with the backbone density being extensive at p sub c and undergoing a jump discontinuity on reducing ital p across the transition we define a model which tunes continuously between the gbsn and gr classes and show that the gr class is typical copyright ital 1999 italmore the american physical society less

a geometrical interpretation of the chaotic state of inhomogeneous deterministic cellular automata abstract we propose a geometrical interpretation of the chaotic state of inhomogeneous cellular automata from the rules of the cellular automaton we construct a network the percolating phase of this network coincides with the chaotic phase of the cellular automaton we also report numerical tests of these ideas on several cellular automata

source coding by efficient selection of ground state clusters we analyze the geometrical structure of clusters of ground states which appear in many frustrated systems over random graphs focusing on the regime of connectivities where the number of clusters is exponential in the size of the problems we identify an appropriate generalization of the survey propagation equations efficiently exploring the geometry the possibility of selecting different clusters has also computational consequences as a proof of concept here we show how a well known physical system can be used to perform nontrivial data compression for which we introduce a unique compression scheme performances are optimized when the number of well separated clusters is maximal in the underlying physical model

near optimal configurations in mean field disordered systems we present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems our approach is based on a generalization of the cavity method to a system interacting with its ground state with this technique we study the random matching problem as well as the mean field diluted spin glass as a by product of this approach we calculate the de almeida thouless transition line of the spin glass on a fixed connectivity random graph

on the universality of geometrical and transport exponents of rigidity percolation we develop a three parameter position space renormalization group method and investigate the universality of geometrical and transport exponents of rigidity vector percolation in two dimensions to do this we study site bond percolation in which sites and bonds are randomly and independently occupied with probabilitiess andb respectively the global flow diagram of the renormalization transformation is obtained which shows that thegeometrical exponents of the rigid clusters in both site and bond percolation belong to the same universality class and possibly that of random scalar percolation however if we use the same renormalization transformation to calculate the critical exponents of the elastic moduli of the system in bond and site percolation we find them to be very different although the corresponding values of the correlation length exponent are the same this indicates that the critical exponent of the elastic moduli of rigidity percolation may not be universal which is consistent with some of the recent numerical simulations

rigidity of random networks of stiff fibers in the low density limit rigidity percolation is analyzed in two dimensional random networks of stiff fibers as fibers are randomly added to the system there exists a density threshold q q min above which a rigid stress bearing percolation cluster appears this threshold is found to be above the connectivity percolation threshold q q c such that q min 1 1698 0 0004 q c the transition is found to be continuous and in the universality class of the two dimensional central force rigidity percolation on lattices at percolation threshold the rigid backbone of the percolating cluster was found to break into rigid clusters whose number diverges in the limit of infinite system size when a critical bond is removed the scaling with system size of the average size of these clusters was found to give a new scaling exponent delta 1 61 0 04

fluctuation of energy in the generalized thermostatistics abstract we calculate the fluctuation of a system s energy in tsallis statistics following the finite heat bath canonical ensemble approach we obtain this fluctuation as the second derivative of the logarithm of the partition function plus an additional term we also find an explicit expression for the relative fluctuation as related to the number of degrees of freedom of the bath and the composite system

statistical mechanics of maximal independent sets the graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory a maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints it is known that finding minimum and maximum density maximal independent sets are hard optimization problems in this paper we use cavity method of statistical physics and monte carlo simulations to study the corresponding constraint satisfaction problem on random graphs we obtain the entropy of maximal independent sets within the replica symmetric and one step replica symmetry breaking frameworks shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms this is of particular relevance for the application to the study of strategic interactions in social and economic networks where maximal independent sets correspond to pure nash equilibria of a graphical game of public goods allocation

spanning trees in random satisfiability problems working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree like structures to a given graph we find a correspondence between the solutions of random k satisfiability problem and those of spanning trees in the associated factor graph we introduce a modified survey propagation algorithm which returns null edges of the factor graph and helps us to find satisfiable spanning trees this allows us to study organization of satisfiable spanning trees in the space spanned by spanning trees

maximum independent set on diluted triangular lattices core percolation and maximum independent set on random graphs have recently been characterized using the methods of statistical physics here we present a statistical physics study of these problems on bond diluted triangular lattices core percolation critical behavior is found to be consistent with the standard percolation values though there are strong finite size effects a transfer matrix method is developed and applied to find accurate values of the density and degeneracy of the maximum independent set on lattices of limited width but large length an extrapolation of these results to the infinite lattice limit yields high precision results which are tabulated these results are compared to results found using both vertex based and edge based local probability recursion algorithms which have proven useful in the analysis of hard computational problems such as the satisfiability problem

rigidity percolation by next nearest neighbor bonds on generic and regular isostatic lattices we study rigidity percolation transitions in two dimensional central force isostatic lattices including the square and the kagome lattices as next nearest neighbor bonds braces are randomly added to the system in particular we focus on the differences between regular lattices which are perfectly periodic and generic lattices with the same topology of bonds but whose sites are at random positions in space we find that the regular square and kagome lattices exhibit a rigidity percolation transition when the number of braces is llnl where l is the linear size of the lattice this transition exhibits features of both first order and second order transitions the whole lattice becomes rigid at the transition and a diverging length scale also exists in contrast we find that the rigidity percolation transition in the generic lattices occur when the number of braces is very close to the number obtained from maxwell s law for floppy modes which is l the transition in generic lattices is a very sharp first order like transition at which the addition of one brace connects all small rigid regions in the bulk of the lattice leaving only floppy modes on the edge we characterize these transitions using numerical simulations and develop analytic theories capturing each transition our results relate to other interesting problems including jamming and bootstrap percolation

fluctuations in the random link matching problem using the replica approach and the cavity method we study the fluctuations of the optimal cost in the random link matching problem by means of replica arguments we derive the exact expression of its variance moreover we study the large deviation function deriving its expression in two different ways namely using both the replica method and the cavity method

scalar and vectorial percolation in compressed expanded graphite dc electrical conductivity and elastic moduli of cubic samples made of two kinds of compressed expanded graphite are measured as a function of their apparent density different percolation thresholds at which the physical properties under study are found to vanish are determined the accuracy of the values is confirmed by the application of the usual scaling laws above but near these thresholds the corresponding critical exponents are indeed found to be very close to their universal values the relationships between the connectivity dc i e scalar and rigidity dr i e vectorial critical densities are discussed it is shown that the ratio dr dc is always almost equal to 8 5 which fact may be accounted by the so called kirkwood keating kk model to our knowledge it is the first time that such a constant value is observed in real materials the kk model is also consistent with critical exponents of elasticity found in both kind of samples hence compressed expanded graphite appears to behave like a disordered heterogeneous medium in which elastic forces are mostly of central nature but include also bond bending contributions

rigid backbone a new geometry for percolation understood in both cases from the nodes linksblobs picture of percolation clusters in fact it is fair to say that in the vast majority of percolation problems encountered to date even continuum percolation 6 the basic geometric features of the percolation clusters such as the scaling of the number of singly connected bonds are identical and the differences arise only at the level of the dynamics for the class of models of interest here however geometric connection the existence of a bond does not imply complete elastic connection because the number of constraints associated with the existence of a bond is smaller than the number of degrees of freedom associated with the sites it connects the percolation threshold is higher8 than that of the corresponding standard percolation problem thorpe6 has suggested the name rigidity percolation for this class of models distinguishing it from the elasticity percolation universality class mentioned above in this paper we study the simplest rigidity percolation model namely the two dimensional central force model 8 ex

combinatorial models of rigidity and renormalization we first introduce the percolation problems associated with the graph theoretical concepts of k l sparsity and make contact with the physical concepts of ordinary and rigidity percolation we then devise a renormalization transformation for k l percolation problems and investigate its domain of validity in particular we show that it allows an exact solution of k l percolation problems on hierarchical graphs for k l 2k we introduce and solve by renormalization such a model which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions depending on the values of the parameters

possible canonical distributions for finite systems with nonadditive energy region des pays de la loire of france 2007 6088 science research fund 07bs105 huaqiao university people s republic of china

central limit behavior of deterministic dynamical systems we investigate the probability density of rescaled sums of iterates of deterministic dynamical systems a problem relevant for many complex physical systems consisting of dependent random variables a central limit theorem clt is valid only if the dynamical system under consideration is sufficiently mixing for the fully developed logistic map and a cubic map we analytically calculate the leading order corrections to the clt if only a finite number of iterates is added and rescaled and find excellent agreement with numerical experiments at the critical point of period doubling accumulation a clt is not valid anymore due to strong temporal correlations between the iterates nevertheless we provide numerical evidence that in this case the probability density converges to a q gaussian thus leading to a power law generalization of the clt the above behavior is universal and independent of the order of the maximum of the map considered i e relevant for large classes of critical dynamical systems

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

rigidity percolation in a field rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted erdos renyi graphs with average connectivity gamma in the presence of a field h in the gamma h plane the rigid and flexible phases are separated by a line of first order transitions whose location is determined exactly this line ends at a critical point with classical critical exponents analytic expressions are given for the densities n f of uncanceled degrees of freedom and gamma r of redundant bonds upon crossing the coexistence line gamma r and n f are continuous although their first derivatives are discontinuous we extend for the case of nonzero field a recently proposed hypothesis namely that the density of uncanceled degrees of freedom is a free energy for rigidity percolation analytic expressions are obtained for the energy entropy and specific heat some analogies with a liquid vapor transition are discussed particularizing to zero field we find that the existence of a g 1 core is a necessary condition for rigidity percolation with g degrees of freedom at the transition point gamma c maxwell counting of degrees of freedom is exact on the rigid cluster and on the g 1 rigid core i e the average coordination of these subgraphs is exactly 2g although gamma c the average coordination of the whole system is smaller than 2g gamma c is found to converge to 2g for large g i e in this limit maxwell counting is exact globally as well

1 d expansion for k core percolation the physics of k core percolation pertains to those systems whose constituents require a minimum number of k connections to each other in order to participate in any clustering phenomenon examples of such a phenomenon range from orientational ordering in solid ortho para mathrm h 2 mixtures to the onset of rigidity in bar joint networks to dynamical arrest in glass forming liquids unlike ordinary k 1 and biconnected k 2 percolation the mean field k ensuremath geqslant 3 core percolation transition is both continuous and discontinuous i e there is a jump in the order parameter accompanied with a diverging length scale to determine whether or not this hybrid transition survives in finite dimensions we present a 1 d expansion for k core percolation on the d dimensional hypercubic lattice we show that to order 1 d 3 the singularity in the order parameter and in the susceptibility occur at the same value of the occupation probability this result suggests that the unusual hybrid nature of the mean field k core transition survives in high dimensions

is sp bp the survey propagation sp algorithm for solving k sat problems has been shown recently as an instance of the belief propagation bp algorithm in this paper we show that for general constraint satisfaction problems sp may not be reducible from bp we also establish the conditions under which such a reduction is possible along our development we present a unification of the existing sp algorithms in terms of a probabilistically interpretable iterative procedure weighted probabilistic token passing

inference by replication in densely connected systems an efficient bayesian inference method for problems that can be mapped onto dense graphs is presented the approach is based on message passing where messages are averaged over a large number of replicated variable systems exposed to the same evidential nodes an assumption about the symmetry of the solutions is required for carrying out the averages here we extend the previous derivation based on a replica symmetric rs like structure to include a more complex one step replica symmetry breaking like 1rsb like ansatz to demonstrate the potential of the approach it is employed for studying critical properties of the ising linear perceptron and for multiuser detection in code division multiple access cdma under different noise models results obtained under the rs assumption in the noncritical regime give rise to a highly efficient signal detection algorithm in the context of cdma while in the critical regime one observes a first order transition line that ends in a continuous phase transition point finite size effects are also observed while the 1rsb ansatz is not required for the original problems it was applied to the cdma signal detection problem with a more complex noise model that exhibits rsb behavior resulting in an improvement in performance

numerical calculation of the energy relative fluctuation for a system in contact with a finite heat bath abstract we have recently shown that the dispersion and the energy square relative fluctuations of a finite system in contact with a finite heat bath depends on the number of degrees of freedom of the system and of the bath in this paper we present numerical simulations which support this result

dissipative dynamics in a finite chaotic environment relationship between damping rate and lyapunov exponent we consider the energy flow between a classical one dimensional harmonic oscillator and a set of n two dimensional chaotic oscillators which represents the finite environment using linear response theory we obtain an analytical effective equation for the system harmonic oscillator which includes a frequency dependent dissipation a shift and memory effects the damping rate is expressed in terms of the environment mean lyapunov exponent a good agreement is shown by comparing theoretical and numerical results even for environments with mixed regular and chaotic motion resonance between system and environment frequencies is shown to be more efficient to generate dissipation than larger mean lyapunov exponents or a larger number of bath chaotic oscillators

quasicanonical gibbs distribution and tsallis non extensive statistics abstract we derive and study quasicanonical gibbs distribution function which is characterized by the thermostat with finite number of particles quasithermostat we show that this naturally leads to tsallis non extensive statistics and thermodynamics with tsallis parameter q is found to be related to the number of particles in the quasithermostat we show that the chi square distribution of fluctuating temperature used recently by beck can be partially understood in terms of normal random momenta of particles in the quasithermostat also we discuss on the importance of the time scale hierarchy and fluctuating probability distribution functions in understanding of tsallis distribution within the framework of kinetics of dilute gas and weakly inhomogeneous systems

a modified belief propagation algorithm for code word quantization modern coding advances including dirty paper coding and information hiding require quantizing a given binary word to a code word a good solution would approach the rate distortion bound in lossy source compression here we propose a simple variant on belief propagation which is observed to converge to a solution giving respectable rate distortion performance comparisons with other recently proposed source quantization methods reveal that the proposed algorithm holds particular interest in short block length applications as encountered in packet based communication systems

application of logical transport to determine the directed and isotropic percolation thresholds the authors present a monte carlo algorithm based on logical transport which determines the effective isotropic percolation and directed percolation thresholds of finite size lattices they apply the algorithm to bond and site directed percolation in two dimensions and determine the asymptotic percolation thresholds by finite size scaling

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

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

the bond bending model in three dimensions the author discusses the bond bending model in three dimensions he argues that for this model loops are very important in all length scales and the rigidity percolation is non local exhibiting similar behaviour to the central force model

statistical mechanics of quartic oscillators we study statistical mechanics of quartic oscillator with two degrees of freedom which is known to be chaotic almost everywhere except in a few regions of the parameter range we obtain exact expressions for temperature entropy and distribution functions temperature is also obtained numerically by time averaging the kinetic energy and using equipartition theorem and agrees with our expressions when the system is almost chaotic we further generalize our model to quartic oscillators with n degrees of freedom and exact expressions for thermodynamic quantities are obtained as n standard statistical mechanics results are recovered we also discuss pressure density and equation of state of this system

nonlocal percolation and partial ordering in the dilute baxter wu model the zero temperature limit of the dilute baxter wu model is studied in the context of percolation theory the percolation model describing the propagation of order is identified and shown to possess a nonlocal mechanism for the propagation of that order a monte carlo study of this model is performed using the nonlocal cluster identification algorithm developed by fried and schick 1992 the author also shows that the mechanism responsible for nonlocal percolation allows for the possibility of the propagation of partial order at intermediate impurity concentration a partially ordered phase is found and its structure is discussed

how many clusters an information theoretic perspective clustering provides a common means of identifying structure in complex data and there is renewed interest in clustering as a tool for the analysis of large data sets in many fields a natural question is how many clusters are appropriate for the description of a given system traditional approaches to this problem are based on either a framework in which clusters of a particular shape are assumed as a model of the system or on a two step procedure in which a clustering criterion determines the optimal assignments for a given number of clusters and a separate criterion measures the goodness of the classification to determine the number of clusters in a statistical mechanics approach clustering can be seen as a trade off between energy and entropy like terms with lower temperature driving the proliferation of clusters to provide a more detailed description of the data for finite data sets we expect that there is a limit to the meaningful structure that can be resolved and therefore a minimum temperature beyond which we will capture sampling noise this suggests that correcting the clustering criterion for the bias that arises due to sampling errors will allow us to find a clustering solution at a temperature that is optimal in the sense that we capture maximal meaningful structure without having to define an external criterion for the goodness or stability of the clustering we show that in a general information theoretic framework the finite size of a data set determines an optimal temperature and we introduce a method for finding the maximal number of clusters that can be resolved from the data in the hard clustering limit

elastic property of the self avoiding random walk calculates the end to end elastic susceptibility for the bond bending model on the self avoiding random walk the numerical result shows clearly that the central force term is irrelevant when the bond bending term is present the result also indicates that when the elastic constant of the bond bending term is set equal to infinity the elastic network is governed by the conductivity exponent of the random resistor network

exact relations between elastic and electrical response of d dimensional percolating networks with angle bending forces arguments are presented to demonstrate that exact equality relations exist between the critical exponents which characterize the macroscopic conductivity e and the macroscopic elastic stiffness moduli c e of percolating systems of any dimensionality using the notation e p t c e p t for the critical behavior of a randomly diluted system slightly above the percolation threshold p c p p p c 0 and e p s c e p s for the critical behavior of a random mixture of normal and perfectly conducting or normal and perfectly rigid constituents slightly below that threshold p p p c 0 we show that t t 2 and s s where is the percolation correlation length critical exponent p p 0

a mean field theory of the elastic granular disc model in two dimensions the author has constructed a stephen type mean field theory for the elastic granular disc model in two dimensions and calculated the mean field value of two crossover exponents phi 1 and phi 2 which describe the ways in which the two point elastic susceptibility and the angle angle correlation function scale with the distance respectively the author has found that phi 1 2 and phi 2 1 the possible relation of phi 1 and phi 2 to the bulk modulus exponent f is discussed

generalized entropies from first principles a derivation of power law canonical distributions from first principle statistical mechanics including the exponential distribution as a particular case is presented it is shown that these distributions arise naturally and that the heat capacity of the heat bath is the condition that determines its type as a consequence a physical interpretation for the parameter q of the generalized entropy is given

scaling analysis of affinity propagation we analyze and exploit some scaling properties of the em affinity propagation ap clustering algorithm proposed by frey and dueck 2007 following a divide and conquer strategy we setup an exact renormalization based approach to address the question of clustering consistency in particular how many cluster are present in a given data set we first observe that the divide and conquer strategy used on a large data set hierarchically reduces the complexity cal o n 2 to cal o n h 2 h 1 for a data set of size n and a depth h of the hierarchical strategy for a data set embedded in a d dimensional space we show that this is obtained without notably damaging the precision except in dimension d 2 in fact for d larger than 2 the relative loss in precision scales like n 2 d h 1 d finally under some conditions we observe that there is a value s of the penalty coefficient a free parameter used to fix the number of clusters which separates a fragmentation phase for ss of the underlying hidden cluster structure at this precise point holds a self similarity property which can be exploited by the hierarchical strategy to actually locate its position as a result of an exact decimation procedure from this observation a strategy based on ap can be defined to find out how many clusters are present in a given dataset

hiding quiet solutions in random constraint satisfaction problems we study constraint satisfaction problems on the so called planted random ensemble we show that for a certain class of problems e g graph coloring many of the properties of the usual random ensemble are quantitatively identical in the planted random ensemble we study the structural phase transitions and the easy hard easy pattern in the average computational complexity we also discuss the finite temperature phase diagram finding a close connection with the liquid glass solid phenomenology

inference algorithm for finite dimensional spin glasses belief propagation on the dual lattice starting from a cluster variational method and inspired by the correctness of the paramagnetic ansatz at high temperatures in general and at any temperature in the two dimensional 2d edwards anderson ea model we propose a message passing algorithm the dual algorithm to estimate the marginal probabilities of spin glasses on finite dimensional lattices we use the ea models in 2d and 3d as benchmarks the dual algorithm improves the bethe approximation and we show that in a wide range of temperatures compared to the bethe critical temperature our algorithm compares very well with monte carlo simulations with the double loop algorithm and with exact calculation of the ground state of 2d systems with bimodal and gaussian interactions moreover it is usually 100 times faster than other provably convergent methods as the double loop algorithm in 2d and 3d the quality of the inference deteriorates only where the correlation length becomes very large i e at low temperatures in 2d and close to the critical temperature in 3d

survey propagation at finite temperature application to a sourlas code as a toy model in this paper we investigate a finite temperature generalization of survey propagation by applying it to the problem of finite temperature decoding of a biased finite connectivity sourlas code for temperatures lower than the nishimori temperature we observe that the result is a shift of the location of the dynamical critical channel noise to larger values than the corresponding dynamical transition for belief propagation as suggested recently by migliorini and saad for ldpc codes we show how the finite temperature 1rsb sp gives accurate results in the regime where competing approaches fail to converge or fail to recover the retrieval state

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

thermodynamical relations for systems in contact with finite heat baths abstract we obtain the relations between macroscopic thermodynamical properties and microscopic ensemble averages of a system in contact with a heat bath which has a finite number of degrees of freedom our approach is based on the geometry of the accessible phase space of the system by using a suitable definition of temperature for such kind of system energy equipartition theorem it is shown that the relations obtained here have the same dependence on the system s partition function z 1 as in boltzmann gibbs statistics

percolation in media with columnar disorder we study a generalization of site percolation on a simple cubic lattice where not only single sites are removed randomly but also entire parallel columns of sites we show that typical clusters near the percolation transition are very anisotropic with different scaling exponents for the sizes parallel and perpendicular to the columns below the critical point there is a griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power law tails with continuously varying non universal powers this region is very similar to the griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction and the proof follows essentially the same arguments as in that case but in contrast to directed percolation in disordered media the number of active growth sites in a growing cluster at criticality shows a power law while the probability of a cluster to continue to grow shows logarithmic behavior

bond diluted rigid animals for central force model an exact enumeration of rigid clusters is analysed using pade approximants the number of bond diluted rigid clusters with n b bonds is found to grow like n theta b b 2 165 or 0 005 n b with theta b 0 988 a value different from that of site diluted rigid animals

tsallis power laws and finite baths with negative heat capacity it is often stated that heat baths with finite degrees of freedom i e finite baths are sources of tsallis distributions for classical hamiltonian systems by using well known fundamental statistical mechanics expressions we rigorously show that tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails however the correspondence between tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium we comment on the implications of the finite bath for the recent attempts towards a q generalized central limit theorem

superparamagnetic clustering of data the physical aspects of a recently introduced method for data clustering are considered in detail this method is based on an inhomogeneous potts model no assumption concerning the underlying distribution of the data is made a potts spin is assigned to each data point and short range interactions between neighboring points are introduced spin spin correlations measured by monte carlo computations serve to partition the data points into clusters in this paper we examine the effects of varying different details of the method such as the definition of neighbors the type of interaction and the number of potts states q in addition we present and solve a granular mean field potts model relevant to the clustering method the model consists of strongly coupled groups of spins coupled to noise spins which are themselves weakly coupled the phase diagram is computed by solving analytically the model in various limits our main result is that in the range of parameters of interest the existence of the superparamagnetic phase is independent of the ordering process of the noise spins next we use the known properties of regular and inhomogeneous potts models in finite dimensions to discuss the performance of the clustering method in particular the spatial resolution of the clustering method is argued to be connected to the correlation length of spin fluctuations the behavior of the method as more and more data points are sampled is also investigated

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

on the satisfiability threshold of formulas with three literals per clause in this paper we present a new upper bound for randomly chosen 3 cnf formulas in particular we show that any random formula over n variables with a clauses to variables ratio of at least 4 4898 is as n grows large asymptotically almost surely unsatisfiable the previous best such bound due to dubois in 1999 was 4 506 the first such bound independently discovered by many groups of researchers since 1983 was 5 19 several decreasing values between 5 19 and 4 506 were published in the years between we believe that the probabilistic techniques we use for the proof are of independent interest

rigid clusters enumeration animal counting techniques suggest that the fraction of rigid animals among all animals with s sites on a triangular lattice varied roughly as s1 2 0 46 or 0 01 s

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb