We carry out a study of the two-dimensional Blume-Capel model using the Wang-Landau Monte Carlo method which estimates the density of states g(E) directly. This work validates the applicability of this method to multiparametric systems, since only one computer run is needed for all range of macroscopic parameters (temperature, anisotropy, etc.). The location of the tricritical point is determined as kBTt/J=0.609(3), Dt/J=1.966(2), and is in excellent agreement with previous estimates. The free energy and the entropy, which are not directly accessible by conventional Monte Carlo simulations, are obtained simply using g(E).

The phase diagram of the spin-3/2 Blume-Capel model in two dimensions is explored by conventional finite-size scaling, conformal invariance and Monte Carlo simulations. The model in its τ -continuum Hamiltonian version is also considered and compared with others spin-3/2 quantum chains. Our results indicate that differently from the standard spin-1 Blume-Capel model there is no multicritical point along the order-disorder transition line. This is in qualitative agreement with mean field prediction but in disagreement with previous approximate renormalization group calculations.

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Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the nite size scaling hypothesis, the approximate critical behavior of model on in nite lattice is obtained through the exact computation of some thermal quantities of the model on nite clusters. In this work some of these methods are reviewed, namely the mean eld renormalization group, the e ective eld renormalization group and the nite size scaling renormalization group procedures. Although special emphasis is given to the mean eld renormalization group since it has been, up to now, much more used to study a wide variety of di erent systems a discussion of their potentialities and interrelations to other methods is also presented. I IntroductionThe renormalization group formalism introduced by Wilson in the early 70's 1, 2 is by n o w one of the basic strategies to solve fundamental problems in statistical mechanics. It is also a very useful tool to tackle problems in several elds of theoretical physics such as the study of nonlinear dynamics and transitions to chaos 3 , disorder surface growth 4 , earthquakes 5 , among others. The conceptual foundation of the method, rst laid by Kadano 6 to qualitatively predict scaling behavior at a second-order phase transition, is to reduce, in a step-by-step way, the degrees of freedom of the system leaving unchanged the underlying physics of the problem. This reduction, carried out repeatedly through a renormalization recursion relation, leads the original system with a large correlation length the range at which p h ysical microscopic operators are correlated to one with correlation length of unity order, where well-known methods as perturbation theory can, at least in principle, be used to treat the problem. Depending on the mathematical technique, such thinning of the degrees of freedom can be implemented directly in the reciprocal momentum space or in the real position space. The former approach makes use of mathematical tools from quantum eld theory with the crystalline system being replaced by its continuous limit. As a result, the so-called ,expansion proposed by Wilson and Fisher 7 and further developed by using techniques of renormalized perturbation theory 8, 9 provides analytical and quite well controlled asymptotically exact results for critical exponents despite being unable to predict values of critical points, critical lines and phase diagrams. On the other hand, the more intuitive real space version of the renormalization group works directly in the position space. It was introduced by Niemeijer and van Leeuwen 10 and several di erent techniques have been proposed and applied to a great variety of statistical models 11 . The real space renormalization group RSRG has since become an important apparatus in studying critical phenomena.The main feature of the renormalization group is to obtain, from the renormalization recursion relations, ow diagrams in the parameter space of the...

In this work we apply a highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff to treat percolation problems. The site and bond percolations are studied on a number of lattices in two and three dimensions. Quite good results for the wrapping probabilities, correlation length critical exponent, and critical concentration are obtained for the square, simple cubic, hexagonal close packed, and hexagonal lattices by using relatively small systems. We also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.