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

TL;DR

This paper analyzes the boundary behavior of the 2D Ising model on a half-infinite cylinder across different lattices and regimes, deriving probabilities of spinflips and confirming Gaussian distribution of their count.

Contribution

It provides rigorous, explicit calculations of boundary spinflip probabilities for various lattices and regimes, extending Onsager's solution to boundary phenomena.

Findings

Probability of 2n spinflips computed for different lattices and regimes.

Limit of the mesh going to zero derived.

Distribution of spinflips is Gaussian, confirming previous results.

Abstract

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.