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
pm 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=Tc 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=Tc 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.
