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Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential
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  • Published: April 2001

Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential

  • Franz Merkl1 &
  • Mario V. Wüthrich2 

Probability Theory and Related Fields volume 119, pages 475–507 (2001)Cite this article

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  • 15 Citations

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Abstract.

We consider d-dimensional Brownian motion in a scaled Poissonian potential and the principal Dirichlet eigenvalue (ground state energy) of the corresponding Schrödinger operator. The scaling is chosen to be of critical order, i.e. it is determined by the typical size of large holes in the Poissonian cloud. We prove existence of a phase transition in dimensions d≥ 4: There exists a critical scaling constant for the potential. Below this constant the scaled infinite volume limit of the corresponding principal Dirichlet eigenvalue is linear in the scale. On the other hand, for large values of the scaling constant this limit is strictly smaller than the linear bound. For d > 4 we prove that this phase transition does not take place on that scale. Further we show that the analogous picture holds true for the partition sum of the underlying motion process.

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Authors and Affiliations

  1. Eurandom, PO Box 513, 5600 MB Eindhoven, The Netherlands. e-mail: merkl@eurandom.tue.nl, , , , , , NL

    Franz Merkl

  2. Winterthur Insurance, Römerstrasse 17, P.O. Box 357, 8401 Winterthur, Switzerland. e-mail: mario.wuethrich@winterthur.ch, , , , , , CH

    Mario V. Wüthrich

Authors
  1. Franz Merkl
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  2. Mario V. Wüthrich
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Received: 10 December 1999 / Revised version: 14 July 2000/¶Published online: 15 February 2001

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Merkl, F., Wüthrich, M. Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential. Probab Theory Relat Fields 119, 475–507 (2001). https://doi.org/10.1007/PL00008768

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  • Issue Date: April 2001

  • DOI: https://doi.org/10.1007/PL00008768

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