Typical Gibbs configurations for the 1d Random Field Ising Model with long range interaction

We study a one--dimensional Ising spin systems with ferromagnetic, long--range interaction decaying as n^{-2+\a}, \a \in [0,\frac 12], in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, gaussian or subgaussian with variance $\theta$. We show that for temperature and variance of the randomness small enough, with an overwhelming probability with respect to the random fields, the typical configurations, within volumes centered at the origin whose size grow faster than any power of $\th^{-1}$, % {\bf around the origin} are intervals of $+$ spins followed by intervals of $-$ spins whose typical length is \simeq \th^{-\frac{2}{(1-2\a)}} for 0\le \a<1/2 and $\simeq e^{\frac 1 {\th^{2}}}$ for \a=1/2

Geometry of contours and Peierls estimates in d=1 Ising models

Following Fr\"ohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as $|x-y|^{-2+\alpha}$, $0\leq \alpha\leq 1/2$. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well known result by Dyson about phase transitions at low temperatures.Comment: 28 pages, 3 figure

Genetic parameters for milk mineral content and acidity predicted by mid-infrared spectroscopy in Holstein–Friesian cows

peer-reviewedThis article was first published in animal, Volume 9, Issue 05, May 2015, pp 775-780 © The Animal Consortium 2015The aim of the present study was to estimate genetic parameters for calcium (Ca), phosphorus (P) and titratable acidity (TA) in bovine milk predicted by mid-IR spectroscopy (MIRS). Data consisted of 2458 Italian Holstein−Friesian cows sampled once in 220 farms. Information per sample on protein and fat percentage, pH and somatic cell count, as well as test-day milk yield, was also available. (Co)variance components were estimated using univariate and bivariate animal linear mixed models. Fixed effects considered in the analyses were herd of sampling, parity, lactation stage and a two-way interaction between parity and lactation stage; an additive genetic and residual term were included in the models as random effects. Estimates of heritability for Ca, P and TA were 0.10, 0.12 and 0.26, respectively. Positive moderate to strong phenotypic correlations (0.33 to 0.82) existed between Ca, P and TA, whereas phenotypic weak to moderate correlations (0.00 to 0.45) existed between these traits with both milk quality and yield. Moderate to strong genetic correlations (0.28 to 0.92) existed between Ca, P and TA, and between these predicted traits with both fat and protein percentage (0.35 to 0.91). The existence of heritable genetic variation for Ca, P and TA, coupled with the potential to predict these components for routine cow milk testing, imply that genetic gain in these traits is indeed possible

Phase Transition in the 1d Random Field ising model with long range interaction

We study the one dimensional Ising model with ferromagnetic, long range interaction which decays as |i-j|^{-2+a}, 1/2< a<1, in the presence of an external random filed. we assume that the random field is given by a collection of independent identically distributed random variables, subgaussian with mean zero. We show that for temperature and strength of the randomness (variance) small enough with P=1 with respect to the distribution of the random fields there are at least two distinct extremal Gibbs measures

The low-temperature phase of Kac-Ising models

We analyse the low temperature phase of ferromagnetic Kac-Ising models in dimensions $d\geq 2$. We show that if the range of interactions is \g^{-1}, then two disjoint translation invariant Gibbs states exist, if the inverse temperature \b satisfies \b -1\geq \g^\k where \k=\frac {d(1-\e)}{(2d+1)(d+1)}, for any \e>0. The prove involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous spin system which is suitable for the use of a variant of the Peierls argument.Comment: 19pp, Plain Te

One-sided versus two-sided stochastic descriptions

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.Comment: 13 pages, Contribution for "Statistical Mechanics of Classical and Disordered Systems

Metastability for reversible probabilistic cellular automata with self--interaction

The problem of metastability for a stochastic dynamics with a parallel updating rule is addressed in the Freidlin--Wentzel regime, namely, finite volume, small magnetic field, and small temperature. The model is characterized by the existence of many fixed points and cyclic pairs of the zero temperature dynamics, in which the system can be trapped in its way to the stable phase. %The characterization of the metastable behavior %of a system in the context of parallel dynamics is a very difficult task, %since all the jumps in the configuration space are allowed. Our strategy is based on recent powerful approaches, not needing a complete description of the fixed points of the dynamics, but relying on few model dependent results. We compute the exit time, in the sense of logarithmic equivalence, and characterize the critical droplet that is necessarily visited by the system during its excursion from the metastable to the stable state. We need to supply two model dependent inputs: (1) the communication energy, that is the minimal energy barrier that the system must overcome to reach the stable state starting from the metastable one; (2) a recurrence property stating that for any configuration different from the metastable state there exists a path, starting from such a configuration and reaching a lower energy state, such that its maximal energy is lower than the communication energy

Renormalization Group Approach to Interacting Crumpled Surfaces: The hierarchical recursion

We study the scaling limit of a model of a tethered crumpled D-dimensional random surface interacting through an exclusion condition with a fixed impurity in d-dimensional Euclidean space by the methods of Wilson's renormalization group. In this paper we consider a hierarchical version of the model and we prove rigorously the existence of the scaling limit and convergence to a non-Gaussian fixed point for $1 \leq D0$ sufficiently small, where $\epsilon = D - (2-D) {d\over 2}$.Comment: 47 pages in simple Latex, PAR-LPTHE 934

Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus

We consider the "thermodynamic limit"of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling. The proof is based on symbolic dynamics techniques that allow us to map the SRB measure into a Gibbs measure for a spin system on a (d+1)-dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual "cluster expansion" techniques.Comment: 28 pages, 2 figure

Temperature dependent fluctuations in the two-dimensional XY model

We present a detailed investigation of the probability density function (PDF) of order parameter fluctuations in the finite two-dimensional XY (2dXY) model. In the low temperature critical phase of this model, the PDF approaches a universal non-Gaussian limit distribution in the limit T-->0. Our analysis resolves the question of temperature dependence of the PDF in this regime, for which conflicting results have been reported. We show analytically that a weak temperature dependence results from the inclusion of multiple loop graphs in a previously-derived graphical expansion. This is confirmed by numerical simulations on two controlled approximations to the 2dXY model: the Harmonic and ``Harmonic XY'' models. The Harmonic model has no Kosterlitz-Thouless-Berezinskii (KTB) transition and the PDF becomes progressively less skewed with increasing temperature until it closely approximates a Gaussian function above T ~ 4\pi. Near to that temperature we find some evidence of a phase transition, although our observations appear to exclude a thermodynamic singularity.Comment: 15 pages, 5 figures and 1 tabl