Short abstract, isn't it? P.A.C.S. numbers 05.20, 02.50, 87.10 1 Introduction Large Numbers "...the optimal tour displayed (see Figure 6) is the possible unique tour having one arc fixed from among 10 655 tours that are possible among 318 points and have one arc fixed. Assuming that one could possibly enumerate 10 9 tours per second on a computer it would thus take roughly 10 639 years of computing to establish the optimality of this tour by exhaustive enumeration." This quote shows the real difficulty of a combinatorial optimization problem. The huge number of configurations is the primary difficulty when dealing with one of these problems. The quote belongs to M.W Padberg and M. Grotschel, Chap. 9., "Polyhedral computations", from the book The Traveling Salesman Problem: A Guided tour of Combinatorial Optimization [124]. It is interesting to compare the number of configurations of real-world problems in combinatorial optimization with those large numbers arising in Cosmol...

Abstract not found

Abstract. The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the random-cluster representation. This systematic summary of random-cluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinite-volume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for two-dimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the cluster-weighting factor q, and the problem of proving that the critical random-cluster model in two

Abstract: We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete picture in the regime of “small α”; a particularly satisfactory result concerns a non-trivial regime of parameters in which we prove 1) the convergence of the local “mean fields ” to gaussian random variables with constant variance and random mean; the random means are from site to site independent gaussians themselves; 2) “propagation of chaos”, i.e. factorization of the extremal infinite volume Gibbs measures, and 3) the correctness of the “replica symmetric solution ” of Amit, Gutfreund and Sompolinsky [AGS]. This last result was first proven by M. Talagrand [T4], using different techniques.

We study threshold phenomena pertaining to the colourability of random graphs and the satisfiability of random formulas. Consider a random graph G(n, p) on n vertices formed by including each of the possible edges independently of all others with probability p. For a fixed integer k, let f k (n, d) = Pr[G(n, d/n) is k-colourable]. Erdos asked the following fundamental question: for k 3, is there a constant c k such that for any # > 0, #) = 1 , and lim f k (n, c k + #) = 0 ? (1) We prove that for all k 3, there exists a function t k (n) such that (1) holds upon replacing c k by t k (n), thus establishing that indeed k-colourability has a sharp threshold. Let d k = sup{d lim n## f k (n, d) = 1}. Note that if c k exists then, by definition, c k = d k . For the basic and most studied case k = 3 we prove 3.84 < d 3 < 5.05 . These are the best

We consider zero-temperature, stochastic Ising models t on Z d with nearest-neighbor interactions and an initial spin configuration 0 chosen from a symmetric Bernoulli distribution (corresponding physically to a deep quench). Whether 1 exists | i.e., whether each spin flips only finitely many times as t ! 1 (for almost every 0 and realization ! of the dynamics) | depends on the nature of the couplings, in particular the presence of continuous disorder. For the homogeneous (nondisordered) ferromagnet, we prove that 1 does not exist, at least for d 2 (although it does exist for some lattices other than Z d ). For continuous disorder, under mild conditions, we show that 1 does exist. We also analyze a dynamical order parameter that measures how much 1 depends on 0 and how much on !.

Abstract—This paper studies randomly spread code-division multiple access (CDMA) and multiuser detection in the large-system limit using the replica method developed in statistical physics. Arbitrary input distributions and flat fading are considered. A generic multiuser detector in the form of the posterior mean estimator is applied before single-user decoding. The generic detector can be particularized to the matched filter, decorrelator, linear minimum mean-square error (MMSE) detector, the jointly or the individually optimal detector, and others. It is found that the detection output for each user, although in general asymptotically non-Gaussian conditioned on the transmitted symbol, converges as the number of users go to infinity to a deterministic function of a “hidden ” Gaussian statistic independent of the interferers. Thus, the multiuser channel can be decoupled: Each user experiences an equivalent single-user Gaussian channel, whose signal-to-noise ratio (SNR) suffers a degradation due to the multiple-access interference (MAI). The uncoded error performance (e.g., symbol error rate) and the mutual information can then be fully characterized using the degradation factor, also known as the multiuser efficiency, which can be obtained by solving a pair of coupled fixed-point equations identified in this paper. Based on a general linear vector channel model, the results are also applicable to multiple-input multiple-output (MIMO) channels such as in multiantenna systems. Index Terms—Channel capacity, code-division multiple access (CDMA), free energy, multiple-input multiple-output (MIMO) channel, multiuser detection, multiuser efficiency, replica method, statistical mechanics. I.

Abstract. We discuss the Fortuin–Kasteleyn (FK) random cluster representation for Ising models with no external field and with pair interactions which need not be ferromagnetic. In the ferromagnetic case, the close connections between FK percolation and Ising spontaneous magnetization and the availability of comparison inequalities to independent percolation have been applied to certain disordered systems, such as dilute Ising ferromagnets and quantum Ising models in random environments; we review some of these applications. For non-ferromagnetic disordered systems, such as spin glasses, the state of the art is much more primitive. We discuss some of the many open problems for spin glasses and show how the FK representation leads to one small result, that there is uniqueness of the spin glass Gibbs distribution above the critical temperature of the associated ferromagnet. Key words: FK representations, spin glasses, disordered Ising models, percolation. 1. The FK Random Cluster Representation

This paper presents an approach, recently introduced by the authors and based on the notion of \metastates", to the chaotic size dependence expected in systems with many competing pure states, and applies it to the Edwards-Anderson (EA) spin glass model. We begin by reviewing the standard picture of the EA model based on the Sherrington-Kirkpatrick (SK) model and why that standard SK picture is untenable. We then introduce metastates, which are the analogues of the invariant probability measures describing chaotic dynamical systems and discuss how they should appear in several models simpler than the EA spin glass. Finally, we consider possibilities for the nature of the EA metastate, including one which is a nonstandard SK picture, and speculate on their prospects. An appendix contains proofs used in our construction of metastates and in the earlier construction by Aizenman and Wehr. Research supported in part by NSF Grant DMS-9500868 y Research supported in part by DOE Grant DE-...

In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given. 1 1