The tails of the critical order-parameter distribution of the two-dimensional Ising model are investigated through extensive multicanonical Monte Carlo simulations. Results for fixed boundary conditions are reported here, and compared with known results for periodic boundary conditions. Clear numerical evidence for "fat" stretched exponential tails exists below the critical temperature, indicating the possible presence of fat tails at the critical temperature. Our work suggests that the true order-parameter distribution at the critical temperature must be considered to be unknown at present.The tails of the critical order-parameter distribution of the two-dimensional Ising model are investigated through extensive multicanonical Monte Carlo simulations. Results for fixed boundary conditions are reported here, and compared with known results for periodic boundary conditions. Clear numerical evidence for "fat" stretched exponential tails exists below the critical temperature, indicating the possible presence of fat tails at the critical temperature. Our work suggests that the true order-parameter distribution at the critical temperature must be considered to be unknown at present.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsOct 22, 200319 citations

Phase transition of the two-dimensional Ising model

Tomoyasu Tanaka

Cambridge University Press eBooksMar 14, 20020 citations

Large-deviation tails of critical order-parameter distributions

Jinhong Zhu, Yihao Xu, Abbas Ali Saberi +1 more

Large-deviation tails of critical probability distributions provide a sensitive probe of universality beyond standard finite-size scaling. We study these tails for critical percolation and Fortuin--Kasteleyn Ising models on two-dimensional lattices, three-dimensional lattices, and complete graphs. We consider two rescaled order parameters: the magnetization-like variable $x_m=|M|/\langle |M|\rangle$ , including a signed cluster-mass analogue for percolation, and the largest-cluster variable $x_C=C_1/\langle C_1\rangle$ . For $x_m$ , we test the expected stretched-exponential large-deviation tail and show that the same form applies to the percolation analogue. For $x_C$ , guided by the exact complete-graph result and scaling arguments, we propose universal scaling forms for both tails of the cumulative distribution and test them by extensive Monte Carlo simulations. In the complete-graph FK-Ising model, the left tail is governed by rare configurations with percolation-like scaling rather than by the typical Ising scaling. Our results show that the tails of order-parameter distributions reveal universal features of critical fluctuations that are not captured by averaged observables alone.Large-deviation tails of critical probability distributions provide a sensitive probe of universality beyond standard finite-size scaling. We study these tails for critical percolation and Fortuin--Kasteleyn Ising models on two-dimensional lattices, three-dimensional lattices, and complete graphs. We consider two rescaled order parameters: the magnetization-like variable $x_m=|M|/\langle |M|\rangle$, including a signed cluster-mass analogue for percolation, and the largest-cluster variable $x_C=C_1/\langle C_1\rangle$. For $x_m$, we test the expected stretched-exponential large-deviation tail and show that the same form applies to the percolation analogue. For $x_C$, guided by the exact complete-graph result and scaling arguments, we propose universal scaling forms for both tails of the cumulative distribution and test them by extensive Monte Carlo simulations. In the complete-graph FK-Ising model, the left tail is governed by rare configurations with percolation-like scaling rather than by the typical Ising scaling. Our results show that the tails of order-parameter distributions reveal universal features of critical fluctuations that are not captured by averaged observables alone.

arXiv:2606.19030Jun 17, 20260 citations

Statistical mechanics of finite three-dimensional Ising models

Kurt Binder

PhysicaDec 1, 1972173 citations

Partition function zeros and leading-order scaling correction of the 3D Ising model from multicanonical simulations

Nelson A. Alves, J. R. Drugowich de Felício, Ulrich H. E. Hansmann

The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.

arXiv:cond-mat/0004199Journal of Physics A Mathematical and GeneralOct 11, 200013 citations

x_m=|M|/\langle |M|\rangle

, including a signed cluster-mass analogue for percolation, and the largest-cluster variable

x_C=C_1/\langle C_1\rangle

. For

x_m

, we test the expected stretched-exponential large-deviation tail and show that the same form applies to the percolation analogue. For

x_C

, guided by the exact complete-graph result and scaling arguments, we propose universal scaling forms for both tails of the cumulative distribution and test them by extensive Monte Carlo simulations. In the complete-graph FK-Ising model, the left tail is governed by rare configurations with percolation-like scaling rather than by the typical Ising scaling. Our results show that the tails of order-parameter distributions reveal universal features of critical fluctuations that are not captured by averaged observables alone.