The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work investigates various spectral properties of the RG map for Ising-type classical lattice systems. It consists of four parts. The first part carries out some explicit calculations of the spectrum of the linearization of the RG at infinite temperature, and discovers that it is of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, but only residual spectrum. The second part presents a rigorous justification of the existence and differentiability of the RG map in the infinite volume limit at high temperature by a cluster expansion approach. The third part continues the theme of the third part, and shows that the matrix of partial derivatives of the RG map displays an approximate band property for finite-range and translation-invariant Hamiltonians at high temperature. The last part justifies the differentiability of the RG map in the infinite volume limit at the critical temperature under a certain condition. In summary, the first part deals with special cases where exact computations can be done, whereas the remaining parts are concerned with a general theory and provide a mathematically sound base.

We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice $Z^{d} : X = (X_{i}(t), i ∈ Z^{d}, t ∈ [0, T], 0 < T < +∞)$. In a first part, these processes are characterized as Gibbs states on path spaces of the form $C([0, T],R)Z^{d}$. In a second part, we study the Gibbsian character on $R^{Z}^{d}$ of $v^{t}$, the law at time t of the infinite-dimensional diffusion X(t), when the initial law $v = v^{0}$ is Gibbsian.

We study the (strong-)Gibbsian character on RZd of the law at time t of an infinitedimensional gradient Brownian diffusion / when the initial distribution is Gibbsian.

The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. <br><br> AMS Classifications: 60G15 / 60G60 / 60H10 / 60J60

By means of the cluster expansion method we show that a recent result of Poghosyan and Ueltschi (2009) combined with a result of Nehring (2012) yields a construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in R^d of Ginibre's Fermi-Dirac gas of such loops. The latter will be identified as a Gibbs perturbation of the ideal Fermi gas. On generalizing these considerations we will obtain the existence of a large class of Gibbs perturbations of the so-called KMM-processes as they were introduced by Nehring (2012). Moreover, it is shown that certain "limiting Gibbs processes" are Gibbs in the sense of Dobrushin, Lanford and Ruelle if the underlying potential is positive. And finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.

We study the (strong-)Gibbsian character on R Z d of the law at time t of an infinitedimensional gradient Brownian diffusion / when the initial distribution is Gibbsian.

We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.

The steady march of new technology depends crucially on our ability to discover and design new, advanced materials. Partially due to increases in computing power, computational methods are now having an increased role in this discovery process. Advances in this area speed the discovery and development of advanced materials by guiding experimental work down fruitful paths. Density functional theory (DFT)has proven to be a highly accurate tool for computing material properties. However, due to its computational cost and complexity, DFT is unsuited to performing exhaustive searches over many candidate materials or for extracting thermodynamic information. To perform these types of searches requires that we construct a fast, yet accurate model. One model commonly used in materials science is the cluster expansion, which can compute the energy, or another relevant physical property, of millions of derivative superstructures quickly and accurately. This model has been used in materials research for many years with great success. Currently the construction of a cluster expansion model presents several noteworthy challenges. While these challenges have obviously not prevented the method from being useful, addressing them will result in a big payoff in speed and accuracy. Two of the most glaring challenges encountered when constructing a cluster expansion model include:(i) determining which of the infinite number of clusters to include in the expansion, and (ii) deciding which atomic configurations to use for training data. Compressive sensing, a recently-developed technique in the signal processing community, is uniquely suited to address both of these challenges. Compressive sensing (CS) allows essentially all possible basis (cluster) functions to be included in the analysis and offers a specific recipe for choosing atomic configurations to be used for training data. We show that cluster expansion models constructed using CS predict more accurately than current state-of-the art methods, require little user intervention during the construction process, and are orders-of-magnitude faster than current methods. A Bayesian implementation of CS is found to be even faster than the typical constrained optimization approach, is free of any user-optimized parameters, and naturally produces error bars on the predictions made. The speed and hands-off nature of Bayesian compressive sensing (BCS) makes it a valuable tool for automatically constructing models for many different materials. Combining BCS with high-throughput data sets of binary alloy data, we automatically construct CE models for all binary alloy systems. This work represents a major stride in materials science and advanced materials development.

We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.

We propose a new construction of point processes, which generalizes the class of infinitely divisible point processes. Examples are the quantum Boson and Fermion gases as well as the classical Gibbs point processes, where the interaction is given by a stable and regular pair potential.