2009/09/22 by Yvan Saint-Aubin, Louis‐Pierre Arguin, Louis-Pierre Arguin +4
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.0909.4041
32 pages
arxiv created 2009/09/22 · openalex publication_date 2009/09/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ki'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.