Potts models with magnetic field: Arithmetic, geometry, and computation

JOURNAL OF GEOMETRY AND PHYSICS

卷:97

Article

Dasu, Shival¹  Marcolli, Matilde¹ 

[1] CALTECH, Dept Math, Pasadena, CA 91125 USA

关键词: Potts models with magnetic field;    Constructible sheaves;    Points over finite fields;    Grothendieck ring of varieties;    Euler characteristic;    Computational complexity;   

DOI  :  10.1016/j.geomphys.2015.06.018

来源: Elsevier

【 摘 要 】

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming. (C) 2015 Elsevier B.V. All rights reserved.

【 授权许可】

Free