Behavior of the two-dimensional Ising model at the boundary of a half-infinite cylinder

Details Behavior of the two-dimensional Ising model at the boundary of a half-infinite cylinder Behavior of the two-dimensional Ising model at the boundary of a half-infinite cylinder

2009-09-22

arxiv.org/abs/0909.4041v1

Physics

Condensed Matter

Statistical Mechanics

32 pages

Scientific paper

The two-dimensional Ising model is studied at the boundary of a half-infinite cylinder. The three regular lattices (square, triangular and hexagonal) and the three regimes (sub-, super- and critical) are discussed. The probability of having precisely 2n spinflips at the boundary is computed as a function of the positions k_i's, i=1,..., 2n, of the spinflips. The limit when the mesh goes to zero is obtained. For the square lattice, the probability of having 2n spinflips, independently of their position, is also computed. As a byproduct we recover a result of De Coninck showing that the limiting distribution of the number of spinflips is Gaussian. The results are obtained as consequences of Onsager's solution and are rigorous.

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