Document Detail


Dimer packings with gaps and electrostatics.
MedLine Citation:
PMID:  18292229     Owner:  NLM     Status:  In-Data-Review    
Abstract/OtherAbstract:
Fisher and Stephenson conjectured in 1963 that the correlation function (defined by dimer packings) of two unit holes on the square lattice is rotationally invariant in the limit of large separation between the holes. We consider the same problem on the hexagonal lattice, extend it to an arbitrary finite collection of holes, and present an explicit conjectural answer. In recent work we managed to prove this conjecture in two fairly general cases. The quantity giving the answer can be regarded as the exponential of the negative of the two-dimensional electrostatic energy of a system of charges naturally associated with the holes. We further develop this analogy to electrostatics by presenting two different natural ways to define a field in our setup, and showing that both lead to the electric field, in the limit of large separations between the holes. For one of the fields, this is also stated as a limit shape theorem for random surfaces, with the continuum limit being a sum of helicoids. We conclude by explaining the relationship of our results to previous results in the physics literature on spin correlations in the Ising model.
Authors:
Mihai Ciucu
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Publication Detail:
Type:  Journal Article; Research Support, U.S. Gov't, Non-P.H.S.     Date:  2008-02-21
Journal Detail:
Title:  Proceedings of the National Academy of Sciences of the United States of America     Volume:  105     ISSN:  1091-6490     ISO Abbreviation:  Proc. Natl. Acad. Sci. U.S.A.     Publication Date:  2008 Feb 
Date Detail:
Created Date:  2008-02-28     Completed Date:  -     Revised Date:  -    
Medline Journal Info:
Nlm Unique ID:  7505876     Medline TA:  Proc Natl Acad Sci U S A     Country:  United States    
Other Details:
Languages:  eng     Pagination:  2766-72     Citation Subset:  IM    
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA. mciucu@indiana.edu
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