The renormalization group approach is used to study the effects of a “canonical” constraint (e.g., a fixed number of occupied bonds) on critical quenched disordered systems. The constraint is found to be always irrelevant, even near the “random” fixed point. This proves that α<0, or that ν>2/d. “Canonical” and “grand canonical” averages thus belong to the same universality class. Related predictions concerning the universality of non-self-averaging distributions are tested by Monte Carlo simulations of the site-diluted Ising model on the cubic lattice. In this case, the approach to the asymptotic distribution for “canonical” averaging is slow, resulting in effectively smaller fluctuations.
The renormalization group approach is used to study the effects of a ``canonical'' constraint (e.g., a fixed number of occupied bonds) on critical quenched disordered systems. The constraint is found to be always irrelevant, even near the ``random'' fixed point. This proves that $\ensuremath{\alpha}<0$, or that $\ensuremath{\nu}>2/d$. ``Canonical'' and ``grand canonical'' averages thus belong to the same universality class. Related predictions concerning the universality of non-self-averaging distributions are tested by Monte Carlo simulations of the site-diluted Ising model on the cubic lattice. In this case, the approach to the asymptotic distribution for ``canonical'' averaging is slow, resulting in effectively smaller fluctuations.
The equivalence between canonical and grand canonical constraints near a random fixed point in a critical disordered system is confirmed by means of Monte Carlo simulations. The slow approach to the asymptotic distribution for canonical averaging given by the ${L}^{(\ensuremath{\alpha}/\ensuremath{\nu}{)}_{\mathrm{random}}}$ term is overcome by simulating long range correlated diluted Ising systems with $(\ensuremath{\alpha}/\ensuremath{\nu}{)}_{\mathrm{random}}=(a\ensuremath{-}d)$ for the particular values
a=2 (linear defects) and
d=3 (three dimensional systems).
We evaluate by Monte Carlo simulations various singular thermodynamic quantities
X for ensembles of quenched random Ising and Ashkin-Teller models. The measurements are taken at
Tc and we study how the distributions
P(X) (and, in particular, their relative squared width,
RX) over the ensemble depend on the system size
l. The Ashkin-Teller model was studied in the regime where bond randomness is irrelevant and we found weak self-averaging; ${R}_{X}\ensuremath{\sim}{l}^{\ensuremath{\alpha}/\ensuremath{\nu}}\ensuremath{\rightarrow}0,$ where $\ensuremath{\alpha}<0$ and $\ensuremath{\nu}$ are the exponents (of the pure model fixed point) governing the transition. For the site-dilute Ising model on a cubic lattice, known to be governed by a random fixed point, we find that
RX tends to a constant, as predicted by Aharony and Harris. We tested whether this constant is universal. However, this constant is different for canonical and grand canonical disorder. We identify the pseudocritical temperature of each sample
i, Tc(i,l), as the temperature at which the susceptibility reaches its maximal value ${\ensuremath{\chi}}^{\mathrm{max}}.$ The distribution of these sample dependent
Tc(i,l) was investigated; we found that its variance scales as $[\ensuremath{\delta}{T}_{c}(l){]}^{2}\ensuremath{\sim}{l}^{\ensuremath{-}2/\ensuremath{\nu}}.$ Our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold. We did observe deviations from a single function, which imply that sample dependent scaling functions are needed. These deviations are, however, relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ${\ensuremath{\chi}}^{\mathrm{max}}$ than the commonly used $\ensuremath{\chi}{(T}_{c}^{\ensuremath{\infty}}).$
We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent y can = 2 y t − d both in energy-like and magnetic quantities, where y t = 1/ν is the thermal renormalization exponent and d is the spatial dimension. Furthermore, we find that while most of the universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has y can = −1/2. This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.
Universal dimensionless quantities, such as Binder ratios and wrapping probabilities, play an important role in the study of critical phenomena. We study the finite-size scaling behavior of the wrapping probability for the Potts model in the random-cluster representation, under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We derive that, in the limit L→∞, the critical values of the wrapping probability are different from those of the unconstrained model, i.e. the model in the grand-canonical ensemble, but still universal, for systems with 2yt−d>0 where yt=1/ν is the thermal renormalization exponent and d is the spatial dimension. Similar modifications apply to other dimensionless quantities, such as Binder ratios. For systems with 2yt−d≤0, these quantities share same critical universal values in the two ensembles. It is also derived that new finite-size corrections are induced. These findings apply more generally to systems in the canonical ensemble, e.g. the dilute Potts model with a fixed total number of vacancies. Finally, we formulate an efficient cluster-type algorithm for the canonical ensemble, and confirm these predictions by extensive simulations.
We consider the two-dimensional randomly site diluted Ising model and the random-bond +/-J Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.