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I Introduction

[page 68, §1]   
[68.1.1.1] A quantity of central importance for finite-size scaling analysis of critical phenomena is the order parameter distribution [1, 2]. [68.1.1.2] Despite many years of research there remain open questions even for the much studied case of the Ising universality class  [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

[68.1.2.1] Most properties of the critical order parameter distribution p(m) are known from computer simulations [13, 14, 6, 11]. Analytical information comes from field theoretic renormalization group calculations [15, 16, 7], from conformal field theory [17] and also from a generalized classification theory of phase transitions [18, 19, 20, 21, 22, 23]. [68.1.2.2] In Refs. [23, 3] some of the analytical predictions seem to have been corroborated by numerical simulations. [68.1.2.3] On the other hand the simulations in Refs. [23, 3] were not able to corroborate the predictions for the tails of the critical order parameter distribution. [68.1.2.4] Recording of the very small probabilities in the tails requires special techniques such as multicanonical simulations. [68.1.2.5] Even in a multicanonical simulation it is necessary to accumulate sufficient statistics in order to probe the tails and to be able to distinguish different theoretical predictions. [68.1.2.6] Many different simulations [3, 4, 9, 10] in recent times have attempted this, but failed in establishing the true behavior at the tails of the critical order parameter distribution.

[68.1.3.1] Detailed investigations of the tails were carried out in one of the early multi-canonical Monte Carlo simulation  [4] for the critical two dimensional Ising model (square lattice of size L=32 and L=64). [68.1.3.2] Even though this work measured extremely small tail probabilities with remarkably high precision, no power law behavior was observed for large magnetization. [68.1.3.3] In addition this simulation could not establish convincingly the agreement of the finite-size scaling predictions in the far tail regime.

[68.1.4.1] Given the observations of non-Gaussian “fat tails” in many other physical phenomena, and following the predictions of the generalized classification theory [23, 3], a recent work [9] has tried to ascertain the behavior in the tails of critical order-parameter distribution through high precision Monte Carlo simulation (Swendsen-Wang cluster flip algorithms) of square and simple cubic Ising models at T=T_{c} with a mixture of free and helical boundary conditions. [68.2.0.1] The work concludes that in two and three dimensions the tails of the distribution are consistent with Gaussian behaviour even at the critical point.

[68.2.1.1] A more recent high-precision Monte Carlo study [10] of the probability distribution of the order parameter for the three-dimensional Ising model (L=12 to 58) at T=T_{c} presents a phenomenological formula (different from a plain Gaussian distribution) that describes well the main peak of the measured distribution but excludes the far tail regime. [68.2.1.2] This simulation based on Swendsen-Wang cluster flip algorithms with periodic boundary condition also reports some discrepancy with earlier estimates [9].

[68.2.2.1] In the present paper we report results of high precision multicanonical Monte Carlo (MCMC) simulation for the Ising model on square lattices with periodic and fixed (i.e. all boundary spins fixed to +1) boundary conditions. [68.2.2.2] Our central objective is to study whether the order parameter distribution obtained from the simulation can be considered to be asymptotic with respect to the number of Monte-Carlo steps (MCS-convergence) and system size (L-convergence). [68.2.2.3] A secondary objective is to study fixed (symmetry breaking) boundary conditions because the asymmetry of the order parameter distribution should give rise to an asymmetry in the far tail behaviour. [68.2.2.4] Different boundary conditions are important for the study of critical finite-size scaling functions. We study the two-dimensional Ising model, firstly because exact analytical results are available, and secondly because we expect the true tail behaviour to emerge more quickly in this case.

[68.2.3.1] The paper is organized in the following manner. [page 69, §0]    [69.1.0.1] In Sec. II we recall the basic quantities and assumptions from finite-size scaling. [69.1.0.2] In Sec. III the MCMC simulation method is described and convergence with respect to system size and number of Monte Carlo steps is discussed briefly. [69.1.0.3] The data analysis, results, and the discussion are presented in Sec. IV.