A computer simulation modeled the change of attitudes in a population resulting from the interac-tive, reciprocal, and reeursive operation of Latan~'s (198 I) theory of social impact, which specifies principles underlying how individuals are affected by their social environment. Surprisingly, several macrolevel phenomena emerged from the simple operation of this microlevei theory, including an incomplete polarization of opinions reaching a stable equilibrium, with coherent minority sub-groups managing to exist near the margins of the whole population. Computer simulations, neglected in group dynamics for 20 years, may, as in modern physics, help determine the extent to which group-level phenomena result from individual-level processes. Writing about social phenomena, social scientists have pro-duced empirical generalizations and theoretical analyses of so-cial processes representing differing levels of social reality. Some analyses concern the cognitions, feelings, and behavior of individuals; others deal with small, medium, or large groups, collectivities, and organizations; still others involve such large-scale human aggregates and systems as nations, societies, or cul-

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Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers,...This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm-Loewner Evolution.

In this thesis, we develop image processing algorithms and applications for a particular class of multiscale stochastic models. First, we provide background on the model class, including a discussion of its relationship to wavelet transforms and the details of a two-sweep algorithm for estimation. A multiscale model for the error process associated with this algorithm is derived. Next, we illustrate how the multiscale models can be used in the context of regularizing ill-posed inverse problems and demonstrate the substantial computational savings that such an approach offers. Several novel features of the approach are developed including a technique for choosing the optimal resolution at which to recover the object of interest. Next, we show that this class of models contains other widely used classes of statistical models including 1-D Markov processes and 2-D Markov random fields, and we propose a class of multiscale models for approximately representing Gaussian Markov random fields...

Abstract. The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results which have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing

This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NP-complete. We provide rigorous solutions to several working conjectures in the statistical mechanics literature, such as the Crossed-Bonds conjecture, and the impossibility to compute effectively the partition functions for any three-dimensional lattice Ising model � these conjectures apply to the Onsager algebraic method, the Fermion operators method, and the combinatorial method based on Pfaffians. The fundamental results of the area, including those of Onsager, Kac, Feynman, Fisher, Kasteleyn, Temperley, Green, Hurst and more recently Barahona: for every Planar crystal lattice the partition functions for the nite sublattices can be computed in polynomial-time, paired with the results of this paper: for every Non-Planar crystal lattice computing the parition functions for the finite sublattices is NP-complete, provide an exact characterization for several of the most studied Ising models. Our results settle at once, for several models, (1) the 2D non-planar vs. 2D planar, (2) the next-nearest neighbour

Abstract. We consider the covariance matrix G mn (x − y) = 〈qδ(σx, m) qδ(σy, n) 〉 − 〈qδ(σx, m) 〉 〈qδ(σy, n)〉 wir of the d-dimensional q-states Potts model, rewriting it in terms of the connectivity, the finite-cluster connectivity and the infinite-cluster covariance in the random cluster representation of Fortuin and Kasteleyn. In any of the q ordered phases, we show that – in addition to the trivial eigenvalue 0 – the matrix Gmn (x − y) has one simple eigenvalue G (1) (x − y) and one (q − 2)-fold degenerate eigenvalue G(2) (x − y). Furthermore, we identify the eigenvalues both in terms of representations of the unbroken symmetry group of the model, and in terms of connectivities and cluster covariances, thereby attributing algebraic significance to these stochastic geometric quantities. In addition to establishing the existence of the correlation lengths ξ (1) wir and ξ(2)

We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn. 1

: We study the 2D Ising model in a rectangular box L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization P t2 L oe(t) when L ! 1 for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m is strictly positive. We study in particular boundary effects due to an arbitrary real-- valued boundary magnetic field. Using the self--duality of the model a large part of the analysis consists in deriving properties of the covariance function h oe(0)oe(t) i, as ktk !1, at dual values of the parameters of the model. To do this analysis we establish new results about the high--temperature representation of the model. These results are valid for dimensions D 2 and up to the critical temperature. We then study the Gibbs measure conditionned by f j P t2 L oe(t) \Gamma mj L j j j L jL \Gammac g, with 0 ! c ! 1=4 and \Gammam ! m ! m . We construct the continuum limit of t...

We develop a non-perturbative version of the Dobrushin-Kotecky-Shlosman theory of phase separation in the canonical 2D Ising ensemble. The results are valid for all temperatures below critical.