Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Weakly Gibbsian representations for joint measures of quenched lattice spin models
Download PDF
Download PDF
  • Published: January 2001

Weakly Gibbsian representations for joint measures of quenched lattice spin models

  • Christof Külske1 

Probability Theory and Related Fields volume 119, pages 1–30 (2001)Cite this article

  • 79 Accesses

  • 12 Citations

  • Metrics details

Abstract.

Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spin-space and disorder-space be represented as (suitably generalized) Gibbs measures of an “annealed system”? - We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive” result is surprising when contrasted with the results of a previous paper [K6], where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinite-volume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov's constructions.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. WIAS, Mohrenstrasse 39, 10117 Berlin, Germany. e-mail: kuelske@wias-berlin.de, , , , , , DE

    Christof Külske

Authors
  1. Christof Külske
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 11 November 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000

RID="*"

ID="*" Work supported by the DFG Schwerpunkt `Wechselwirkende stochastische Systeme hoher Komplexität'

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Külske, C. Weakly Gibbsian representations for joint measures of quenched lattice spin models. Probab Theory Relat Fields 119, 1–30 (2001). https://doi.org/10.1007/PL00012737

Download citation

  • Issue Date: January 2001

  • DOI: https://doi.org/10.1007/PL00012737

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

  • Mathematics Subject Classification (2000): 82B44, 82B26, 82B20
  • Key words and phrases: Disordered systems – Gibbs-measures – Non-Gibbsian measures – Ising model – Random field model – Random bond model – Dilute Ising model
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature