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
Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials
Download PDF
Download PDF
  • Published: June 2003

Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials

  • Franz Merkl1 

Probability Theory and Related Fields volume 126, pages 307–338 (2003)Cite this article

Abstract.

 This article describes the almost sure infinite volume asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials. The random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered Gibbs measure with a superstable pair interaction. There is no restriction on the strength of the pair interaction: it may be taken, e.g., at a critical point. The potential is scaled with the box size in a critical way, i.e. the scale is determined by the typical size of large deviations in the Gibbsian cloud. The almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities. The qualitative behaviour of the ground state energy asymptotics is analysed: Depending on the dimension and on the Hölder exponents of the free energy density, it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands. e-mail: merkl@math.leidenuniv.nl, , , , , , NL

    Franz Merkl

Authors
  1. Franz Merkl
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 24 June 2002 / Revised version: 17 February 2003 Published online: 12 May 2003

Mathematics Subject Classification (2000): Primary 82B44; Secondary 60K35

Key words or phrases: Gibbs measure – Hölder exponents – Random Schrödinger operator – Ground state – Large deviations

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Merkl, F. Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials. Probab. Theory Relat. Fields 126, 307–338 (2003). https://doi.org/10.1007/s00440-003-0266-2

Download citation

  • Issue Date: June 2003

  • DOI: https://doi.org/10.1007/s00440-003-0266-2

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

Keywords

  • Phase Transition
  • Free Energy
  • Energy Density
  • Asymptotic Behaviour
  • State Energy
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