improved bounds for the transition temperature of directed polymers in a finite dimensional random medium we consider the problem of directed polymers in a random medium of a finitedimensional lattice in the high temperature phase of this system it is known that the annealed and quenched free energies coincide upper bounds on the transition temperature to a low temperature phase had previously been obtained by calculating the first two moments z and z2 of the partition function we improve these bounds by estimating noninteger moments z for 1 2

oriented distance function and its evolution equation for initial sets with thin boundary the central result of this paper is a new nonlinear equation which describes the evolution of the oriented distance function b omega of a set omega with thin boundary under the influence of a velocity field we relate it to equations and constructions used in the context of level set methods we further introduce a new moving narrow band method which not only can be readily implemented to solve our evolution equation but could also be used for equations of motion by curvatures in the process we review and sharpen the characterization of smooth sets and manifolds and sets of positive reach e g local semiconvexity in an extended sense of the oriented distance function of the closure of the set for w2 p sobolev domains a new characterization and a compactness theorem are given in terms of the laplacian of the oriented distance function rather than its whole hessian matrix

bipartitioning of random graphs of fixed extensive valence the procedure of fu and anderson ibid vol 19 p 1605 1986 for the application of statistical mechanics to the problem of bipartitioning random graphs is extended to graphs of fixed extensive valence the cost function is shown to be independent of whether the valence is locally or globally constrained

optimal approximation of sparse hessians and its equivalence to a graph coloring problem we consider the problem of approximating the hessian matrix of a smooth non linear function using a minimum number of gradient evaluations particularly in the case that the hessian has a known fixed sparsity pattern we study the class of direct methods for this problem and propose two new ways of classifying direct methods examples are given that show the relationships among optimal methods from each class the problem of finding a non overlapping direct cover is shown to be equivalent to a generalized graph coloring problem the distance 2 graph coloring problem a theorem is proved showing that the general distance k graph coloring problem is np complete for all fixedk 2 and hence that the optimal non overlapping direct cover problem is also np complete some worst case bounds on the performance of a simple coloring heuristic are given an appendix proves a well known folklore result which gives lower bounds on the number of gradient evaluations needed in any possible approximation method

on the statistical physics of directed polymers in a random medium and their relation to tree codes using well known results from statistical physics concerning the almost sure behavior of the free energy of directed polymers in a random medium we prove that random tree codes achieve the distortion rate function not only on the average but moreover almost surely under a certain symmetry condition

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

the distribution function of the effective field of the ising spin glass on the bethe lattice for the coordination number z 4 5 6 abstract the integral equation for the effective fields of the spin glass is solved analytically for the j ising model on the bether lattice with coordination number z 4 5 and 6 and symmetric distribution functions are obtained the paramagnetic distribution spin glass distribution with three delta functions and spin glass distributions with a continuous distribution are obtained the effect of the truncation of the algebraic equation for solving the integral equation is discussed

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

relaxation in graph coloring and satisfiability problems using t 0 monte carlo simulation we study the relaxation of graph coloring k col and satisfiability k sat two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied a change from exponentially fast to power law relaxation and a transition to freezing behavior are found these changes take place for smaller values of the parameter than the solvability transition results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented

directed polymers on trees a martingale approach the authors use martingale methods and simple convexity arguments to compute rigorously the limiting free energy in the problem of directed polymers on a tree the limit is a degenerate random variable and convergence holds almost surely the only assumption on the common distribution of the random potentials attached to the bonds of the tree is that its laplace transform exists everywhere in 0 infinity

convergence of a brinkman type penalization for compressible fluid flows abstract we show convergence of a brinkman type penalization of the compressible navier stokes equation in particular the existence of weak solutions for the system in domains with boundaries varying in time is established

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

optimization of output load in a junction model an optimization problem of minimizing the output load of a junction model with multiple input and output channels is solved by the replica method simulation results show that for the case of extensive connectivity the replica solution obtained by assuming a vanishing solution space is satisfactory however for the case of intensive connectivity the degeneracy of the solution space causes the simulation results to deviate from the replica solution of vanishing volume

random 2 sat results and problems in the random 2 sat problem we are given a set c of m disjunctions of two literals chosen at random within the 2n2 pairs of distinct literals coming from n logical variables the basic problem is to nd out for which values of the ratio m n the disjunctions in c are almost surely simultaneously satisfiable or almost surely not simultaneously satisfiable as n tends to infinity the purpose of this paper is to review the main steps in the solution of this problem starting with the location of the asymptotic critical ratio around 8 years ago and ending with the recent almost complete solution due to bollob4as et al thus this paper is not a review in the usual sense of the word i e it does not include all the known results about random 2 sat we will also make a few comments concerning the behaviour of the number of satisfying assignments of random instances of 2 sat below the critical ratio a problem relevant to theoretical physics

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

directed polymers in a random medium abstract the problem of a polymer in a random medium can be formulated as follows consider a regular lattice with on each bond ij a random energy ij the energy e w of a self avoiding walk w on this lattice is the sum of the energies of all the bonds visited by the walk then the problem is to compute the partition function z w exp e w t several recent results will be presented 1 the mean field version of this problem can be solved exactly and one observes a low temperature phase similar to the spin glass phase 2 for finite dimensional lattices one can derive bounds on the transition temperature in high enough dimension 3 for hierarchical lattices the problem can be reduced to finding stable laws when one combines random variables in a non linear way for example z z 1 z 2 z 3 z 4

polynomial iterative algorithms for coloring and analyzing random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find with a one step replica symmetry breaking approximation the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c in c d c q in which ground states spontaneously divide into an exponential number of clusters furthermore we extended our considerations to the case of single instances showing consistent results this leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region i e when c in c d c q

possible sets of autocorrelations and the simplex algorithm the problem of imposing a set of correlations of any order on binary sequences is addressed the entropy of infinitely long sequences obeying such a given set was calculated in previous works using the saddle point method and it was observed that a finite fraction of sets are characterized by a non extensive entropy in this paper the region of finite entropy the allowed region of sets of correlations is found to be a convex hyper polygon in the space of correlation sets using the simplex algorithm outside of this region the simplex solution indicates that sequences obeying the correlations cannot be found therefore the entropy is in particular the boundaries of the allowed region for c1 cm are presented at the boundaries the entropy drops in a first order phase transition fashion and this drop can be explained from a combinatorial point of view finally we observe that the fraction of the volume occupied by allowed correlation sets drops exponentially with the number of correlations imposed and a qualitative explanation of this scaling phenomenon is provided

statistical mechanics of an np complete problem subset sum we study the statistical properties of an np complete problem the subset sum using the methods and concepts of statistical mechanics the problem is a generalization of the number partitioning problem which is also an npcomplete problem and has been studied in the physics literature the asymptotic expressions for the number of solutions are obtained these results applied to the number partitioning problem as a special case are compared with those which were previously obtained by a different method we discuss the limit of applicability of the techniques of statistical mechanics to the present problem

finite size effects in random energy models and in the problem of polymers in a random medium by the use of traveling wave equations we calculate the finite size corrections to the free energy of random energy models in their low temperature phases and in the neighborhood of the transition temperature we find that although the extensive part of the free energy does not show any critical behavior when the temperature approaches its transition value the finite size corrections signal the transition by becoming singular we obtain a scaling form for these finite size corrections valid in the limitn andt t c by considering a generalized random energy model in the limit of a very large number of steps we obtain results for the finite size corrections in the problem of a polymer in a random medium

an identity of chernoff bounds with an interpretation in statistical physics and applications in information theory an identity between two versions of the rate function of the probability a certain large deviations event is established this identity has an interpretation in statistical physics namely an isothermal equilibrium of a composite system that consists of multiple subsystems of particles several information theoretic application examples where the analysis of this large deviations probability naturally arises are then described from the viewpoint of this statistical mechanical interpretation this results in several relationships between information theory and statistical physics which we hope the reader will find insightful

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

coloring random graphs and maximizing local diversity we study a variation of the graph coloring problem on random graphs of finite average connectivity given the number of colors we aim to maximize the number of different colors at neighboring vertices i e one edge distance of any vertex two efficient algorithms belief propagation and walksat are adapted to carry out this task we present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution the problem and the suggested algorithms have practical relevance since various applications such as distributed storage can be mapped onto this problem

a mapping approach to rate distortion computation and analysis in rate distortion theory results are often derived and stated in terms of the optimizing density over the reproduction space in this paper the problem is reformulated in terms of the optimal mapping from the unit interval with lebesgue measure that would induce the desired reproduction probability density this results in optimality conditions that are random relatives of the known lloyd 1982 optimality conditions for deterministic quantizers the validity of the mapping approach is assured by fundamental isomorphism theorems for measure spaces we show that for the squared error distortion the optimal reproduction random variable is purely discrete at supercritical distortion where the shannon 1948 lower bound is not tight the gaussian source is thus the only source that produces continuous reproduction variables for the entire range of positive rate to analyze the evolution of the optimal reproduction distribution we use the mapping formulation and establish an analogy to statistical mechanics the solutions are given by the distribution at isothermal statistical equilibrium and are parameterized by the temperature in direct correspondence to the parametric solution of the variational equations in rate distortion theory the analysis of an annealing process shows how the number of symbols grows as the system undergoes phase transitions thus an algorithm based on the mapping approach often needs but a few variables to find the exact solution while the blahut 1972 algorithm would only approach it at the limit of infinite resolution finally a quick deterministic annealing algorithm to generate the rate distortion curve is suggested the resulting curve is exact as long as continuous phase transitions in the process are accurately followed

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

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

weak solutions to the barotropic navier stokes system with slip boundary conditions in time dependent domains abstract we consider the compressible barotropic navier stokes system on time dependent domains supplemented with slip boundary conditions our approach is based on penalization of the boundary behavior viscosity and the pressure in the weak formulation global in time weak solutions are obtained

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

mean field theory of directed polymers with random complex weights we show that for the problem of directed polymers on a tree with i i d random complex weights on each bond three possible phases can exist the phase of a particular system is determined by the distribution of the random weights for each of these three phases we give the expression of the free energy per unit length in the limit of infinitely long polymers our proofs require several hypotheses on the distribution most importantly that the amplitude and the phase of each complex weight be statistically independent the main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy we illustrate our results by some examples and discuss possible generalizations to a larger class of distributions to random energy models and to the finite dimensional case we note that our results are not in agreement with the predictions of a recent replica approach to a similar problem

a brinkman penalization method for compressible flows in complex geometries to simulate flows around solid obstacles of complex geometries various immersed boundary methods had been developed their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non body conformal cartesian grids the brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations its main idea is to model solid obstacles as porous media with porosity f and viscous permeability approaching zero it has the pronounced advantages of mathematical proof of error bound strong convergence and ease of numerical implementation with the volume penalization technique in this paper it is extended to compressible flows the straightforward extension of penalizing momentum and energy equations using brinkman penalization with respective normalized viscous h and thermal h t permeabilities produces unsatisfactory results mostly due to nonphysical wave transmissions into obstacles resulting in considerable energy and mass losses in reflected waves the objective of this paper is to extend the brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media in addition to penalizing momentum and energy equations the continuity equation for porous media is considered inside obstacles in this model the penalized porous region acts as a high impedance medium resulting in negligible wave transmissions the asymptotic analysis reveals that the proposed brinkman penalization technique results in the amplitude and phase errors of order o email protected 1 2 and o h h t 1 email protected 3 4 when the boundary layer within the porous media is respectively resolved or unresolved the proposed method is tested using 1 and 2 d benchmark problems the results of direct numerical simulation are in excellent agreement with the analytical solutions the numerical simulations verify the accuracy and convergence rates

critical weight statistics of the random energy model and of the directed polymer on the cayley tree we consider the critical point of two mean field disordered models i the random energy model rem introduced by derrida as a mean field spin glass model of n spins and ii the directed polymer of length n on a cayley tree dpct with random bond energies both models are known to exhibit a freezing transition between a high temperature phase where the entropy is extensive and a low temperature phase of finite entropy where the weight statistics coincides with the weight statistics of l evy sums with index ensuremath mu t t c l1 in this paper we study the weight statistics at criticality via the entropy s ensuremath ensuremath sum w i phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex w i and the generalized moments y k ensuremath sum w i k where the w i are the boltzmann weights of the 2 n configurations in the rem we find that the critical weight statistics is governed by the finite size exponent ensuremath nu 2 the entropy scales as overline s n t c ensuremath sim n 1 2 the typical values e overline mathrm ln phantom rule 0 2em 0ex y k decay as n ensuremath k 2 and the disorder averaged values overline y k are governed by rare events and decay as n ensuremath 1 2 for any kg1 for the dpct we find that the entropy scales similarly as overline s n t c ensuremath sim n 1 2 whereas another exponent ensuremath nu ensuremath 1 governs the y k statistics the typical values e overline mathrm ln phantom rule 0 2em 0ex y k decay as n ensuremath k and the disorder averaged values overline y k decay as n ensuremath 1 for any kg1 as a consequence the asymptotic probability distribution overline ensuremath pi n ensuremath infty q of the overlap q in addition to the delta function ensuremath delta q which bears the whole normalization contains an isolated point at q 1 as a memory of the delta peak 1 ensuremath t t c ensuremath delta q ensuremath 1 of the low temperature phase tl t c the associated value overline ensuremath pi n ensuremath infty q 1 is finite for the dpct and diverges as overline ensuremath pi n ensuremath infty q 1 ensuremath sim n 1 2 for the rem

the free energy at t 0 of the spin glass in the case of the continuous distribution of the effective field on the bethe lattice for z 4 5 6 abstract we evaluated the free energy at t 0 of the j random ising model on the bethe lattice of the coordination number z 4 5 6 in the case where the distribution function of the effective field has a continuous distribution by using the method of the integral equation of the effective field

threshold values stability analysis and high q asymptotics for the coloring problem on random graphs we consider the problem of coloring erdos renyi and regular random graphs of finite connectivity using q colors it has been studied so far using the cavity approach within the so called one step replica symmetry breaking 1rsb ansatz we derive a general criterion for the validity of this ansatz and applying it to the ground state we provide evidence that the 1rsb solution gives exact threshold values c q for the transition from the colorable to the uncolorable phase with q colors we also study the asymptotic thresholds for q 1 finding c q 2q ln q ln q 1 o 1 in perfect agreement with rigorous mathematical bounds as well as the nature of excited states and give a global phase diagram of the problem

a characteristic based volume penalization method for general evolution problems applied to compressible viscous flows in order to introduce solid obstacles into flows several different methods are used including volume penalization methods which prescribe appropriate boundary conditions by applying local forcing to the constitutive equations one well known method is brinkman penalization which models solid obstacles as porous media while it has been adapted for compressible incompressible viscous and inviscid flows it is limited in the types of boundary conditions that it imposes as are most volume penalization methods typically approaches are limited to dirichlet boundary conditions in this paper brinkman penalization is extended for generalized neumann and robin boundary conditions by introducing hyperbolic penalization terms with characteristics pointing inward on solid obstacles this characteristic based volume penalization cbvp method is a comprehensive approach to conditions on immersed boundaries providing for homogeneous and inhomogeneous dirichlet neumann and robin boundary conditions on hyperbolic and parabolic equations this cbvp method can be used to impose boundary conditions for both integrated and non integrated variables in a systematic manner that parallels the prescription of exact boundary conditions furthermore the method does not depend upon a physical model as with porous media approach for brinkman penalization and is therefore flexible for various physical regimes and general evolutionary equations here the method is applied to scalar diffusion and to direct numerical simulation of compressible viscous flows with the navier stokes equations both homogeneous and inhomogeneous neumann boundary conditions are demonstrated through external flow around an adiabatic and heated cylinder theoretical and numerical examination shows that the error from penalized neumann and robin boundary conditions can be rigorously controlled through an a priori penalization parameter h the error on a transient boundary is found to converge as o h which is more favorable than the error convergence of the already established dirichlet boundary condition

minimizing unsatisfaction in colourful neighbourhoods colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods given a predetermined number of colours in the analytical framework of a tree approximation carried out at both zero and finite temperatures solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition which are consistent with those of existing algorithms the nature of the transition as well as the validity of the tree approximation are investigated

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

coloring random graphs we study the graph coloring problem over random graphs of finite average connectivity c given a number q of available colors we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable depending on q we find the precise value of the critical average connectivity c q moreover we show that below c q there exists a clustering phase c e c d c q in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms

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

random graph coloring statistical physics approach the problem of vertex coloring in random graphs is studied using methods of statistical physics and probability our analytical results are compared to those obtained by exact enumeration and monte carlo simulations we critically discuss the merits and shortcomings of the various methods and interpret the results obtained we present an exact analytical expression for the two coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and k body edges

phase coexistence and finite size scaling in random combinatorial problems we study an exactly solvable version of the well known random boolean satisfiability sat problem the so called random xor sat problem rare events are shown to affect the combinatorial phase diagram leading to a coexistence of solvable and unsolvable instances of the combinatorial problem in a certain region of the parameters characterizing the model such instances differ by a non extensive quantity in the ground state energy of the associated diluted spin glass model we also show that the critical exponent controlling the size of the critical window where the probability of having solutions vanishes depends on the model parameters shedding light on the link between random hyper graph topology and universality classes in the case of random sat a similar behaviour was conjectured to be connected to the onset of computational intractability

a convergent finite element finite volume scheme for the compressible stokes problem part i the isothermal case in this paper we propose a discretization for the nonlinearized compressible stokes problem with a linear equation of state rho p based on crouzeix raviart elements the approximation of the momentum balance is obtained by usual finite element techniques since the pressure is piecewise constant the discrete mass balance takes the form of a finite volume scheme in which we introduce an upwinding of the density together with two additional stabilization terms we prove em a priori estimates for the discrete solution which yields its existence by a topological degree argument and then the convergence of the scheme to a solution of the continuous problem

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

statistical mechanics of the random k satisfiability model the random k satisfiability problem consisting in verifying the existence of an assignment of n boolean variables that satisfy a set of m ensuremath alpha n random logical clauses containing k variables each is studied using the replica symmetric framework of diluted disordered systems we present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest k 2 kg 3 and k ensuremath gg 1 the calculation of the number of solutions which allowed us phys rev lett 76 3881 1996 to predict a first order jump at the threshold where the boolean expressions become unsatisfiable with probability one is thoroughly displayed in the case k 2 the rigorously known critical value ensuremath alpha 1 of the number of clauses per boolean variable is recovered while for kg 3 we show that the system exhibits a replica symmetry breaking transition the annealed approximation is proven to be exact for large k

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

q partitioning of graphs with finite coordination number the np complete optimisation problem of q partitioning of graphs with finite connectivity is discussed using the effective local field method the authors obtain the local field distributions with and without the continuous part and use them to calculate the ground state energies of the three state potts spin glass on lattices with finite coordination number equal to three this in turn gives analytic results for the optimal cost of the 3 partitioning of graphs they perform simulations of actual 3 partitions of random graphs to compare with their theoretical results and obtain very good agreement ways for further improvement of the estimates are discussed

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

the graph partitioning and the spin glass comments on papers by sherrington and his group the recent theory of graph partitioning proposed by sherrington and his group 1987 1988 used the integral equation to determine the field distribution this integral equation and its solution are pointed out to be the same as those used by katsura and co workers 1979 1980 for spin glasses

finitely connected vector spin systems with random matrix interactions we use finite connectivity equilibrium replica theory to solve models of finitely connected unit length vectorial spins with random pair interactions which are of the orthogonal matrix type finitely connected spin models although still of a mean field nature can be regarded as a convenient level of description in between fully connected and finite dimensional ones since the spins are continuous and the connectivity c remains finite in the thermodynamic limit the replica symmetric order parameter is a functional the general theory is developed for arbitrary values of the dimension d of the spins and arbitrary choices of the ensemble of random orthogonal matrices we calculate phase diagrams and the values of moments of the order parameter explicitly for d 2 finitely connected xy spins with random chiral interactions and for d 3 finitely connected classical heisenberg spins with random chiral interactions numerical simulations are shown to support our predictions quite satisfactorily

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well