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Let G=G(n,m) be a random graph whose average degree d=2m/n is below the k-colorability threshold. If we sample a k-coloring Sigma of G uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart? According to a prediction from statistical physics, for average degrees below the so-called condensation threshold d_c, the colors assigned to far away vertices are asymptotically independent [Krzakala et al: PNAS 2007]. We prove this conjecture for k exceeding a certain constant k_0. More generally, we determine the joint distribution of the k-colorings that Sigma induces locally on the bounded-depth neighborhoods of a fixed number of vertices.

We introduce a parametric family for random convex polytopes in Rd which allows for an easy generation of samples for further use, e. g., as random particles in materials modelling and simulation. The basic idea consists in weighting the Poisson cell, which is the typical cell of the stationary and isotropic Poisson hyperplane tessellation, by suitable geometric characteristics. Since this approach results in an exponential family, parameters can be efficiently estimated by maximum likelihood. This work has been motivated by the desire for a flexible model for random convex particles as can be found in many composite materials such as concrete or refractory castables.

Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a KacZwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M-1/2) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small O(M -1/2) friction and o(M-1/2) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuationdissipation relation., QC 20111103

Motivated by the autologistic model for the analysis of spatial binary data on the two-dimensional lattice, we develop efficient computational methods for calculating the normalizing constant for models for discrete data defined on the cylinder and lattice. Because the normalizing constant is generally unknown analytically, statisticians have developed various ad hoc methods to overcome this difficulty. Our aim is to provide computationally and statistically efficient methods for calculating the normalizing constant so that efficient likelihood-based statistical methods are then available for inference. We extend the so-called transition method to find a feasible computational method of obtaining the normalizing constant for the cylinder boundary condition. To extend the result to the free-boundary condition on the lattice we use an efficient path sampling Markov chain Monte Carlo scheme. The methods are generally applicable to association patterns other than spatial, such as clustered binary data, and to variables taking three or more values described by, for example, Potts models.

It is expected that nano-scale devices and interconnections will introduce unprecedented level of defects in the substrates, and architectural designs need to accommodate the uncertainty inherent at such scales. This consideration motivates the search for new architectural paradigms based on redundancy based defect-tolerant designs. However, redundancy is not always a solution to the reliability problem, and often too much or too little redundancy may cause degradation in reliability. The key challenge is in determining the granularity at which defect tolerance is designed, and the level of redundancy to achieve a specific level of reliability. Analytical probabilistic models to evaluate such reliability-redundancy trade-offs are error prone and cumbersome, and do not scalewell for complex networks of gates. In this thesiswe develop different tools and techniques that can evaluate the reliability measures of combinational circuits, and can be used to analyze reliability-redundancy trade-offs for different defect-tolerant architectural configurations. In particular, we have developed two tools, one of which is based on probabilistic model checking and is named NANOPRISM, and another MATLAB based tool called NANOLAB. We also illustrate the effectiveness of our reliability analysis tools by pointing out certain anomalies which are counter-intuitive but can be easily discovered by these tools, thereby providing better insight into defecttolerant design decisions. We believe that these tools will help furthering research and pedagogical interests in this area, expedite the reliability analysis process and enhance the accuracy of establishing reliability-redundancy trade-off points.

We consider sampling methods for multidimensional random fields. We study the convergence of a sampler process generated by the methods stated in Introduction. We can apply convergence theorem to the restoration of degraded images in image processing. (Author)

This report presents a new statistical approach to the image segmentation problem. By modelling image data as a Markov random field characterized by a Gibbs distribution, a dynamic programming algorithm is developed. The primary contribution of the paper is this new near optimal method for processing scenes described by the non-causal Gibbs model. The report is organized as follows. Section 2 defines the segmentation problem in a statistical framework, introduces some notation, and presents some background on Markov random fields. Section 3 then presents the dynamic programming algorithm in detail for the case of segmenting images consisting of uniform intensity regions in high levels of additive white Gaussian noise. Section 4 presents results of applying the algorithms to some experimentally generated images consistent with this model as well as some synthetic aperture radar images which are clearly inconsistent with the assumed model. These results clearly demonstrate the applicability of the technique to realistic data as well as the robustness of the algorithm with respect to modelling assumptions. In Section 5, some comments and concluding remarks are given, and extensions to this work which are in progress are briefly outlined.