graph optimisation problems and the potts glass the np complete problems of partitioning and colouring of random graphs with p partitions and colours respectively are mapped onto the statistical mechanical problem of p state potts glasses an estimate of the cost functions of these optimisation problems has been derived using the potts glass mean field theory this estimate applies to dense graphs in the thermodynamic limit an exact expression for the cost function in the large p limit is given

the mean field traveling salesman and related problems the edges of a complete graph on n vertices are assigned i i d random costs from a distribution for which the interval 0 t has probability asymptotic to t as t 0 through positive values in this so called pseudo dimension 1 mean field model we study several optimization problems of which the traveling salesman is the best known we prove that as n the cost of the minimum traveling salesman tour converges in probability to a certain number approximately 2 0415 which is characterized analytically

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

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

numerical approximations for stochastic differential games the ergodic case the markov chain approximation method is a widely used relatively easy to use and efficient family of methods for the bulk of stochastic control problems in continuous time for reflected jump diffusion type models it has been shown to converge under broad conditions and there are good algorithms for solving the numerical problems if the dimension is not too high we consider a class of stochastic differential games with a reflected diffusion system model and ergodic cost criterion and where the controls for the two players are separated in the dynamics and cost function it is shown that the value of the game exists and that the numerical method converges to this value as the discretization parameter goes to zero the actual numerical method solves a stochastic game for a finite state markov chain and ergodic cost criterion the essential conditions are nondegeneracy and that a weak local consistency condition hold almost everywhere for the numerical approximations just as for the control problem

stochastic monotonicity and stationary distributions for dynamic economies the existence and stability of invariant distributions for stochastically monotone processes is studied the knaster tarski fixed point theorem is applied to establish existence of fixed points of mappings on compact sets of measures that are increasing with respect to a stochastic ordering global convergence of a monotone markov process to its unique invariant distribution is established under an easily verified assumption topkis theory of supermodular functions is applied to stochastic dynamic optimization providing conditions under which optimal stationary decisions are monotone functions of the state and induce a monotone markov process applications of these results to investment theory stochastic growth and industry equilibrium dynamics are given

measurable selection theorems for minimax stochastic optimization problems this paper resolves the measurability questions which arise in the analysis of various minimax stochastic optimization models posed in borel spaces for a large class of such models we provide a set of sufficient conditions which ensure that the questions have positive answers these conditions are essentially weaker than those described in the existing literature moreover they allow us to conclude a fundamental theorem yielding some new results on the existence of universally measurable selections of extrema for both zero sum stochastic games and minimax stochastic optimal control problems in particular a random version of the well known minimax theorem of fan is in this way established the fundamental theorem yields also a descriptive set theoretic fact concerning the projections of borel sets the paper indicates also some counterexamples to possibly more general problems

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

on the optimality equation for average cost markov control processes with feller transition probabilities we consider markov control processes with borel state space and feller transition probabilities satisfying some generalized geometric ergodicity conditions we provide a new theorem on the existence of a solution to the average cost optimality equation

zero sum stochastic games in borel spaces average payoff criteria this paper is concerned with two person zero sum dynamic stochastic games in borel spaces with possibly unbounded payoff function and several average or ergodic payoff criteria we give conditions under which the long run expected average payoff criterion the sample path average criterion the existence of solutions to the average payoff shapley equations and a certain martingale condition are all equivalent

minimax control of discrete time stochastic systems this paper gives a unified self contained presentation of minimax control problems for discrete time stochastic systems on borel spaces with possibly unbounded costs the main results include conditions for the existence of minimax strategies for finite horizon problems and infinite horizon discounted and undiscounted average cost criteria the results are specialized to control systems with unknown disturbance distributions also known as games against nature two examples illustrate the theory one of them on the mold level control problem which is a key problem in the steelmaking industry

borel stochastic games with lim sup payoff we consider two person zero sum stochastic games with limit superior payoff function and borel measurable state and action spaces the games are shown to have a value and the value function is calculated by transfinite iteration of an operator and proved to be upper analytic the paper extends results of our earlier article 17 in which the same class of games was considered for countable state spaces and finite action sets

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

zero sum semi markov games this paper deals with borel state and action spaces zero sum semi markov games under the expected long run average payoff criterion the transition probabilities are assumed to satisfy some generalized geometric ergodicity conditions the main result states that the optimality equation has a solution which is approximated by the solutions of some varepsilon perturbed semi markov games as a corollary the existence of value and average optimal strategies for the players is established

stationary markov equilibria conditions are established under which there are equilibria in various settings described by ergodic markov processes with a borel state space s let rho s denote the probability measures on s and let s approaching g s is a subset of rho s be a possibly empty valued correspondence with closed graph a non empty measurable set j is a subset of s is self justified if g s intersects rho j is not empty for all s an element of j a time homogeneous markov equilibrium thme for g is a self justified set j and a measurable selection ii j approaching rho j from the restriction of g to j the paper gives sufficient conditions for existence of compact self justified sets and applies the theorem if g is convex valued and has a compact self justified set then g has an thme with an ergodic measure coauthors are j geanakoplos a mas colell and a mclennan copyright 1994 by the econometric society

zero sum ergodic stochastic games with feller transition probabilities we consider zero sum stochastic games with borel state spaces satisfying a generalized geometric ergodicity condition the main objective of this paper is to establish a minimax theorem for a class of ergodic stochastic games with the feller transition probability function all previous results on ergodic stochastic games are based on a strong continuity of the transition probabilities in the actions of the players

average optimality in dynamic programming with general state space a markovian decision model with general state space compact action space and the average cost as criterion is considered the existence of an optimal policy is shown via an optimality inequality in terms of the minimal average cost g and a relative value function w the existence of some w is usually shown via relative compactness in a space of real valued functions on the state space here it shall be shown that one can instead do with pointwise relative compactness in the set of real numbers if one makes use of a generalized lower limit of functions an application to an inventory model is given

finite size corrections in the random assignment problem we analytically derive in the context of the replica formalism the first finite size corrections to the average optimal cost in the random assignment problem for a quite generic distribution law for the costs we show that when moving from a power law distribution to a mathrm ensuremath gamma distribution the leading correction changes both in sign and in its scaling properties we also examine the behavior of the corrections when approaching a ensuremath delta function distribution by using a numerical solution of the saddle point equations we provide predictions that are confirmed by numerical simulations

one loop diagrams in the random euclidean matching problem the matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community here we analyze the euclidean version of the problem i e the optimal matching problem between points randomly distributed on a d dimensional euclidean space where the cost to minimize depends on the points pairwise distances using mayer s cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation we give the diagrammatic rules for each term of the expansion and we analyze in detail the one loop diagrams a characteristic feature of the theory when diagrams are perturbatively computed around the mean field part of the action is the vanishing of the mass at zero momentum in the non euclidean case of uncorrelated costs instead we predict and numerically verify an anomalous scaling for the sub sub leading correction to the asymptotic average cost

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

zero sum average semi markov games fixed point solutions of the shapley equation this paper deals with zero sum average semi markov games with borel state and action spaces unbounded payoffs and mean holding times a solution to the shapley equation is obtained via the banach fixed point theorem assuming that the model satisfies a lyapunov like condition a growth hypothesis on the payoff function and the mean holding times besides standard continuity and compactness requirements

the fuchsian differential equation of the square lattice ising model χ 3 susceptibility using an expansion method in the variables xi that appear in n 1 dimensional integrals representing the n particle contribution to the ising square lattice model susceptibility we generate a long series of coefficients for the three particle contribution 3 using an n4 polynomial time algorithm we give the fuchsian differential equation of order 7 for 3 that reproduces all the terms of our long series an analysis of the properties of this fuchsian differential equation is performed

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

semicontinuous nonstationary stochastic games ii abstract in this paper we introduce a borel space framework for zero sum discrete time stochastic games that is a game theoretic extension of some nonstationary dynamic programming models in the sense of hinderer our game model which allows for all of the primitive data to be nonstationary contains a large class of markov games at the same time it constitutes a considerable generalization of the game model introduced by sengupta to ensure the existence of a value of a nonstationary stochastic game and the existence of universally measurable optimal or e optimal strategies for the players we impose some asymmetric semicontinuity and compactness conditions on the primitive data that are weaker than those described in the existing literature on the subject a certain special case in which the players can restrict themselves to borel measurable strategies is also studied