A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising Model

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model are convex combinations of the two pure phases. We present here a new approach to this result, with a number of advantages: (i) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (ii) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (iii) this new approach might be applicable to systems for which the classical method fails.Comment: A couple of typos corrected. To appear in Probab. Theory Relat. Field

On the Gibbs states of the noncritical Potts model on Z^2

We prove that all Gibbs states of the q-state nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature.Comment: Minor typos corrected after proofreading. Final version, to appear in Probab. Theory Relat. Field

Random-cluster representation of the ashkin-teller model

We show that a class of spin models, containing the Ashkin-Teller model, admits a generalized random-cluster (GRC) representation. Moreover, we show that basic properties of the usual representation, such as FKG inequalities and comparison inequalities, still hold for this generalized random-cluster model. Some elementary consequences are given. We also consider the duality transformations in the spin representation and in the GRC model and show that they commute

Macroscopic description of phase separation in the 2D Ising model

We review recent results about the macroscopic description of phase separation in the 2D Ising model, with special emphasis on boundary effects and related surface phase transitions. In particular, after having recalled some facts about the wetting transition, we describe two situations in which this transition has major consequences at the macroscopic scale. We also briefly describe a more general situation for which it is possible to derive the thermodynamical variational problem characterizing the interfaces of the equilibrium state

Mathematical Theory Of The Wetting Phenomenon In The 2D Ising Model

. We give a mathematical theory of the wetting phenomenon in the 2D Ising model using the formalism of Gibbs states. We treat the grand canonical and canonical ensembles. 1 Introduction We study the wetting phenomenon in the 2D Ising model, starting from basic principles of Statistical Mechanics. The results of section 3 are based on [10], [11] and [12] and those of section 5 follow from recent results on the large deviations of the magnetization [26]. We shall in general refer to these papers for proofs. Our purpose is to give a global view of the mathematical results, which are now fairly complete. Since the results about large deviations are valid in the 2D case only we restrict the whole discussion to this case. i Supported by Fonds National Suisse Grant 2000-041806.94/1 Let us suppose that we have a binary mixture and that the physical parameters are chosen so that we have coexistence of the two phases, called + phase and \Gamma phase. The system is inside a box; the horizont..

Large deviations and continuum limit in the 2D Ising model

We study the 2D Ising model in a rectangular box Λ L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑ t∈ΛL σ(t) when L→∞ 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 , as |t|→∞, 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. They give a complete non-perturbative exposition of the high-temperature representation. We then study the Gibbs measure conditioned by {|∑ t∈Λ_L σ(t) −m|Λ_L| |≤|Λ_L| L^{−c} }, with 0<c<1/4 and −m *<m<m *. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type

Macroscopic Description of Phase Separation in the 2D Ising Model

: We review recent results about the macroscopic description of phase separation in the 2D Ising model, with special emphasis on boundary effects and related surface phase transitions. In particular, after having recalled some facts about the wetting transition, we describe two situations in which this transition has major consequences at the macroscopic scale. We also briefly describe a more general situation for which it is possible to derive the thermodynamical variational problem characterizing the interfaces of the equilibrium state. Keywords: Phase separation, Wulff droplet, Winterbottom droplet, interface pinning, wetting transition. 1 Introduction The aim of these notes is to give a non-technical account of recent results about the macroscopic description of phase separation in the 2D Ising model. They are based on a series of works by the authors [9, 10, 11, 12]. Consider some two-dimensional container Q, filled with some substance in the phase coexistence regime (we suppos..

Prevention of apoptotic neuronal death by controlling procaspases? A point of view

In various animal models of neurodegenerative diseases the long-lasting control of cell death by anti-apoptotic therapies is not successful. We present here our view on the control of procaspase expression in a model of cerebral stroke. We have investigated how Hu-Bcl-2 overexpression modifies cell death protein activation in a model of cerebral ischemia induced by permanent middle cerebral artery occlusion (MCAO). In wild type mice MCAO induced release of cytochrome c from the mitochondria, and activation of caspases 9 and 3. In parallel with caspases activation, procaspase 9 and procaspase 3 were, respectively, increased and decreased. In Hu-Bcl-2 transgenic mice cytochrome c release and caspases 9 and 3 activation were blocked. However procaspase 9 increased, like in wt mice, but procaspase 3 remained unchanged. By 2 weeks after MCAO caspases were no longer blocked in Hu-Bcl-2 transgenic mice. Procaspase 9 increase could represent a time bomb in Hu-Bcl-2 mice where caspase 9 activation is blocked. Indeed, cellular accumulation of procaspase 9 is a potentially harmful event able to overcome anti-apoptotic protection by Bcl-2 and threaten cells with rapid destruction. Through understanding of the upstream regulation of procaspase 9, early targets for the pharmacological control of apoptotic cell death may be revealed