Typical configurations for one-dimensional random field Kac model
type: Item
instance of: scholarly article
title: Typical configurations for one-dimensional random field Kac model
zbMATH Open document ID: 0983.60091
author: Marzio Cassandro
author: Enza Orlandi
author: Pierre Picco
published in: The Annals of Probability
publication date: 2002-04-02T00:00:00Z
review text: Typical interfaces in one-dimensional Kac systems (at vanishing external field) are rather well understood, see the recent results by \textit{M. Cassandro, E. Orlandi} and \textit{E. Presutti} [Probab. Theory Relat. Fields 96, No. 1, 57-96 (1993; Zbl 0791.60096)] or \textit{T. Bodineau} [Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 5, 559-590 (1997; Zbl 0893.60014)]. Here, the authors study new phenomena arising in such systems after adding a zero-mean external random field given by symmetric i.i.d. Bernoulli process. From the authors' summary: We study the typical profiles of a random field Kac model. We give upper and lower bounds of the space scale where the profiles are constant. The results hold almost surely with respect to the realizations of the random field. The analysis is based on a block-spin construction, deviation techniques for the local empirical order parameters and concentration inequalities for the realizations of the random magnetic field. For the upper bound, we exhibit a scale related to the law of the iterated logarithm, where the random field makes an almost sure fluctuation that obliges the system to break its rigidity. For the lower bound, we prove that on a smaller scale the fluctuations are not strong enough to allow this transition.
reviewed by: Ostap Hryniv
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 1462558
zbMATH Keywords: Kac potentials
zbMATH Keywords: large deviations
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: local central limit theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677454
OpenAlex ID: W2081129800
cites work: Rounding effects of quenched randomness on first-order phase transitions
cites work: Some properties of random Ising models
cites work: One-dimensional random-field Ising model: Gibbs states and structure of ground states.
cites work: Interface in a one-dimensional Ising spin system
cites work: Distribution of overlap profiles in the one-dimensional Kac-Hopfield model
cites work: Universality in blow-up for nonlinear heat equations
cites work: Interfaces and typical Gibbs configurations for one-dimensional Kac potentials
cites work: Q4183968
cites work: The ground state of the three-dimensional random-field Ising model
cites work: On the van der Waals Theory of the Vapor-Liquid Equilibrium. II. Discussion of the Distribution Functions
cites work: Metastates in disordered mean-field models: Random field and Hopfield models
cites work: Rigorous Treatment of the Van Der Waals-Maxwell Theory of the Liquid-Vapor Transition
cites work: On Talagrand's deviation inequalities for product measures
cites work: Q3997990
cites work: Metastability and convergence to equilibrium for the random field Curie-Weiss model
cites work: A Remark on Stirling's Formula
cites work: Random walks in a random environment
cites work: The Limiting Behavior of a One-Dimensional Random Walk in a Random Medium
cites work: Q4084474
cites work: Concentration of measure and isoperimetric inequalities in product spaces
cites work: Q5656203
cites work: Logarithmic Sobolev inequality for lattice gases mixing conditions
cites work: Exponential inequalities for sums of random vectors
DOI: 10.1214/AOP/1022677454
description: scientific article; zbMATH DE number 1462558