tricriticality and the blume capel model a monte carlo study within the microcanonical ensemble the microcanonical partition function ensuremath omega ensuremath alpha n e of a three dimensional ferromagnetic simple cubic blume capel model is calculated for several coupling ratios ensuremath alpha close to and on the first order side of the tricritical point to this end a single monte carlo simulation of a suitably extended partition function for systems of l ifmmode times else texttimes fi l ifmmode times else texttimes fi l spins with l ensuremath in 8 10 12 14 18 was performed a finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well an empirical scaling of its coordinates to the thermodynamic limit yields ensuremath alpha t 2 84479 ifmmode pm else textpm fi 0 00030 for the tricritical coupling ratio and k b t t j 1 4182 ifmmode pm else textpm fi 0 0055 for the tricritical transition temperature the results are compared with values obtained on fcc lattices

cluster method for the ashkin teller model a cluster monte carlo algorithm for the ashkin teller at model is constructed according to the guidelines of a general scheme for such algorithms its dynamical behaviour is tested for the square lattice at model we perform simulations on the line of critical points along which the exponents vary continuously and find that critical slowing down is significantly reduced we find continuous variation of the dynamical exponent z along the line following the variation of the ratio alpha nu in a manner which satisfies the li sokal bound z cluster geq alpha nu that was so far proved only for potts models

logarithmic corrections and finite size scaling in the two dimensional 4 state potts model we analyze the scaling and finite size scaling behavior of the two dimensional 4 state potts model we find new multiplicative logarithmic corrections for the susceptibility in addition to the already known ones for the specific heat we also find additive logarithmic corrections to scaling some of which are universal we have checked the theoretical predictions at criticality and off criticality by means of high precision monte carlo data

microcanonical finite size scaling in the microcanonical ensemble suitably defined observables show nonanalyticities and power law behavior even for finite systems for these observables a microcanonical finite size scaling theory is established and combined with the experimentally observed power law behavior scaling laws are obtained which relate exponents of the finite system and critical exponents of the infinite system to the system size dependence of the affiliated microcanonical observables

on the application of the critical minimum energy subspace method to disordered systems abstract we discuss the recent application to strongly disordered systems of the critical minimum energy subspace crmes method used to limit the energy subspace of the wang landau sampling we compare with our results on the 3d random field ising model obtained by a multi range wang landau simulation over the whole energy range we point out some problems that may arise when applying the crmes scheme to models having a complex free energy landscape

a rigorous theory of finite size scaling at first order phase transitions a large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered the dependence of the finite volume magnetizationm per h l in cubes of sizel d under periodic boundary conditions on the external fieldh is analyzed for the case where two phases coexist at the infinite volume transition pointh t we prove that independent of the details of the model the finite volume magnetization per lattice site behaves likem per h t m tanh ml d h ht withm per h denoting the infinite volume magnetization and m 1 2 m per h t 0 m per h t 0 introducing the finite size transition pointh m l as the point where the finite volume susceptibility attains the maximum we show that in the case of asymmetric field driven transitions its shift ish t h m l o l 2d in contrast to claims in the literature starting from the obvious observation that the number of stable phases has a local maximum at the transition point we propose a new way of determining the pointh t from finite size data with a shift that is exponentially small inl finally the finite size effects are discussed also in the case where more than two phases coexist

finite size scaling for potts models recently borgs and koteck developed a rigorous theory of finite size effects near first order phase transitions here we apply this theory to the ferromagneticq state potts model which forq large andd 2 undergoes a first order phase transition as the inverse temperature is varied we prove a formula for the internal energy in a periodic cube of side lengthl which describes the rounding of the infinite volume jump e in terms of a hyperbolic tangent and show that the position of the maximum of the specific heat is shifted by m l inq e l d o l 2d with respect to the infinite volume transition point t we also propose an alternative definition of the finite volume transition temperature t l which might be useful for numerical calculations because it differs only by exponentially small corrections from t

density of states of classical spin systems with continuous degrees of freedom in the last years different studies have revealed the usefulness of a microcanonical analysis of finite systems when dealing with phase transitions in this approach the quantities of interest are exclusively expressed as derivatives of the entropy s ln omega where omega is the density of states obviously the density of states has to be known with very high accuracy for this kind of analysis important progress has been achieved recently in the computation of the density of states of classical systems as new types of algorithms have been developed here we extend one of these methods originally formulated for systems with discrete degrees of freedom to systems with continuous degrees of freedom as an application we compute the density of states of the three dimensional xy model and demonstrate that critical quantities can directly be determined from the density of states of finite systems in cases where the degrees of freedom take continuous values

renormalization of correlations in a quasiperiodically forced two level system quadratic irrationals generalizing from the case of golden mean frequency to a wider class of quadratic irrationals we extend our renormalization analysis of the self similarity of correlation functions in a quasiperiodically forced two level system we give a description of all piecewise constant periodic orbits of an additive functional recurrence generalizing that present in the golden mean case we establish a criterion for periodic orbits to be globally bounded and also calculate the asymptotic height of the main peaks in the correlation function

critical behavior of the q 3 4 potts model on quasiperiodic decagonal lattices abstract in this study we performed monte carlo simulations of the q 3 4 potts model on quasiperiodic decagonal lattices qdl to assess the critical behavior of these systems using the single histogram technique in conjunction with the finite size scaling analysis we estimate the infinite lattice critical temperatures and the leading critical exponents for q 3 and q 4 states our estimates for the critical exponents on qdl are in good agreement with the exact values on 2d periodic lattices supporting the claim that both the q 3 and q 4 potts model on quasiperiodic lattices belong to the same universality class as those on 2d periodic lattices

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

logarithmic corrections for spin glasses percolation and lee yang singularities in six dimensions we study analytically the logarithmic corrections to the critical exponents of the critical behaviour of correlation length susceptibility and specific heat for the temperature and the finite size scaling behaviour for a generic theory at its upper critical dimension six we have also computed the leading correction to scaling as a function of the lattice size we distinguish the obtained formulae to the following special cases percolation lee yang singularities and m component spin glasses we have compared our results for the ising spin glass case with numerical simulations finding a very good agreement finally and using the results obtained for the lee yang singularities in six dimensions we have computed the logarithmic corrections to the singular part of the free energy for lattice animals in eight dimensions

dynamic critical behavior of the swendsen wang algorithm the two dimensional three state potts model revisited we have performed a high precision monte carlo study of the dynamic critical behavior of the sweden wang algorithm for the two dimensional three state potts model we find that the li sokal bound int const cn is almost but not quite sharp the ratio int cn seems to diverge either as a small power 0 08 or as a logarithm

determining the density of states for classical statistical models a random walk algorithm to produce a flat histogram we describe an efficient monte carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models the density of states is modified at each step when the energy level is visited to produce a flat histogram by carefully controlling the modification factor we allow the density of states to converge to the true value very quickly even for large systems from the density of states at the end of the random walk we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature using this method we not only can avoid repeating simulations at multiple temperatures but we can also estimate the free energy and entropy quantities that are not directly accessible by conventional monte carlo simulations this algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability as with the multicanonical monte carlo technique our method overcomes the tunneling barrier between coexisting phases at first order phase transitions in this paper we apply our algorithm to both first and second order phase transitions to demonstrate its efficiency and accuracy we obtained direct simulational estimates for the density of states for two dimensional ten state potts models on lattices up to 200 x 200 and ising models on lattices up to 256 x 256 our simulational results are compared to both exact solutions and existing numerical data obtained using other methods applying this approach to a three dimensional j spin glass model we estimate the internal energy and entropy at zero temperature and using a two dimensional random walk in energy and order parameter space we obtain the rough canonical distribution and energy landscape in order parameter space preliminary data suggest that the glass transition temperature is about 1 2 and that better estimates can be obtained with more extensive application of the method this simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

red bond exponents of the critical and the tricritical ising model in three dimensions using the wolff and geometric cluster algorithms and finite size scaling analysis we investigate the critical ising and the tricritical blume capel models with nearest neighbor interactions on the simple cubic lattice the sampling procedure involves the decomposition of the ising configuration into geometric clusters each of which consists of a set of nearest neighboring spins of the same sign connected with bond probability p these clusters include the well known kasteleyn fortuin clusters as a special case for p 1 exp 2k where k is the ising spin spin coupling along the critical line k kc the size distribution of geometric clusters is investigated as a function of p we observe that unlike in the case of two dimensional tricriticality the percolation threshold in both models lies at pc 1 exp 2kc further we determine the corresponding red bond exponents as yr 0 757 2 and 0 501 5 for the critical ising and the tricritical blume capel models respectively on this basis we conjecture yr 1 2 for the latter model

continuous phase transitions with a convex dip in the microcanonical entropy the appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition however a convex dip also shows up in some systems with a continuous transition as for example in the baxter wu model and in the four state potts model in two dimensions we demonstrate that the appearance of a convex dip in those cases can be traced back to a finite size effect the properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite size scaling theory for continuous phase transitions results obtained from numerical simulations corroborate the predictions of the scaling theory

monte carlo study of the triangular blume capel model under bond randomness the effects of bond randomness on the universality aspects of a two dimensional d 2 blume capel model embedded in the triangular lattice are discussed the system is studied numerically in both its first and second order phase transition regimes by a comprehensive finite size scaling analysis for a particularly suitable value of the disorder strength we find that our data for the second order phase transition emerging under random bonds from the second order regime of the pure model are compatible with the universality class of the two dimensional 2d random ising model furthermore we find evidence that the second order transition emerging under bond randomness from the first order regime of the pure model belongs again to the same universality class although the first finding reinforces the scenario of strong universality in the 2d ising model with quenched disorder the second is in difference from the critical behavior emerging under randomness in the cases of the ex first order transitions of the potts model finally our results verify previous renormalization group calculations on the blume capel model with disorder in the crystal field coupling

finite size scaling analysis of the critical behavior of the baxter wu model abstract we use the recently developed critical minimum energy subspace crmes approximation scheme to study the critical behavior of the baxter wu model this scheme uses only a properly determined part of the energy spectrum and allows us to obtain high accuracy for relatively large systems with considerably reduced computational effort the density of states is constructed by a multi range wang landau sampling and from this we obtain the critical properties of specific heat and of the reduced binder cummulant the good agreement of our results with existing exact solutions demonstrates the accuracy of our approximation technique

word problems in elliott monoids abstract algorithmic issues concerning elliott local semigroups are seldom considered in the literature although these combinatorial structures completely classify af algebras in general the addition operation of an elliott local semigroup is partial but for every af algebra b whose murray von neumann order of projections is a lattice this operation is uniquely extendible to the addition of an involutive monoid e b let m 1 be the farey af algebra introduced by the present author in 1988 and rediscovered by f boca in 2008 the freeness properties of the involutive monoid e m 1 yield a natural word problem for every af algebra b with singly generated e b because b is automatically a quotient of m 1 given two formulas and in the language of involutive monoids the problem asks to decide whether and code the same equivalence of projections of b this mimics the classical definition of the word problem of a group presented by generators and relations we show that the word problem of m 1 is solvable in polynomial time and so is the word problem of the behncke leptin algebras a n k and of the effros shen algebras f for 0 1 q a real algebraic number or 1 e we construct a quotient of m 1 having a godel incomplete word problem and show that no primitive quotient of m 1 is godel incomplete

on first order phase transitions in microcanonical and canonical non extensive systems abstract two examples of microcanonical potts models two dimensional nearest neighbor and mean field are considered via exact enumeration of states and analytical asymptotic methods in the interval of energies corresponding to a first order phase transition both of these models exhibit a convex dip in the entropy vs energy plot and a region with negative specific heat within the dip it is observed that in the nearest neighbor model the dip flattens and disappears as the lattice size grows while in the mean field model the dip persists even in the limit of an infinite system if formal transitions from microcanonical to canonical ensembles and back are performed for an infinite but non extensive system the convex dip in the microcanonical entropy plot disappears

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

susceptibility amplitudes for the three and four state potts models abstract the finite lattice method of series expansions is used to calculate a number of extended series for susceptibilities of the three and four state potts models on the square lattice we analyse all these series and estimate their critical amplitudes we resolve an uncertainty in amplitude ratios for high to low temperature susceptibilities and thus provide confirmation of field theoretic predictions

fractal weyl law for chaotic microcavities fresnel s laws imply multifractal scattering we demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities as an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium we show that the conjectured fractal weyl law for open chaotic systems lu et al phys rev lett 91 154101 2003 is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating fresnel s laws

finite size behaviour of the microcanonical specific heat for models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non monotonic behaviour of the microcanonical specific heat as a function of the system size this is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system a phenomenological theory is developed which permits us to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents

the square lattice f model revisited a loop cluster update scaling study the six vertex f model on the square lattice constitutes the unique example of an exactly solved model exhibiting an infinite order phase transition of the kosterlitz thouless type as one of the few non trivial exactly solved models it provides a welcome gauge for new numerical simulation methods and scaling techniques in view of the notorious problems of clearly resolving the kosterlitz thouless scenario in the two dimensional xy model numerically the f model in particular constitutes an instructive reference case for the simulational description of this type of phase transition we present a loop cluster update monte carlo study of the square lattice f model with a focus on the properties not exactly known such as the polarizability or the scaling dimension in the critical phase for the analysis of the simulation data finite size scaling is explicitly derived from the exact solution and plausible assumptions guided by the available exact results the careful inclusion of correction terms in the scaling formulae allows for a reliable determination of the asymptotic behaviour

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

renormalization of correlations in a quasiperiodically forced two level system for a general class of modulation function we provide a renormalization analysis of correlations in a quasi periodically forced two level system in a time dependent field with periodic kicks whose amplitude is given by a general class of discontinuous modulation function for certain intensities of modulation we give a complete understanding of the autocorrelation function furthermore once the locations of the discontinuities of the modulation function are known aperiodic orbits lead to correlations on renormalization strange sets which are determined by two specified features of the modulation function of which there are only a finite number of variations

high temperature series analysis of the free energy and susceptibility of the 2d random bond ising model abstract we derive high temperature series expansions for the free energy and susceptibility of the two dimensional random bond ising model with a symmetric bimodal distribution of two positive coupling strengths j1 and j2 and study the influence of the quenched random bond disorder on the critical behavior of the model by analysing the series expansions over a wide range of coupling ratios j2 j1 covering the crossover from weak to strong disorder we obtain for the susceptibility with two different methods compelling evidence for a singularity of the form t 7 4 ln t 7 8 as predicted theoretically by shalaev shankar and ludwig for the specific heat our results are less convincing but still compatible with the theoretically predicted log log singularity

exact finite size scaling corrections to the critical two dimensional ising model on a torus ii triangular and hexagonal lattices we compute the finite size corrections to the free energy internal energy and specific heat of the critical two dimensional spin 1 2 ising model on triangular and hexagonal lattices wrapped on a torus we find the general form of the finite size corrections to these quantities as well as explicit formulae for the first coefficients of each expansion we analyse the implications of these findings for the renormalization group description of the model

the three state square lattice potts antiferromagnet at zero temperature we study the 3 state square lattice potts antiferromagnet at zero temperature by a monte carlo simulation using the wang swendsen koteck cluster algorithm on lattices up to 1024 1024 we confirm the critical exponents predicted by burton and henley based on the height representation of this model

position space renormalization group study of the spin 1 random semi infinite blume capel model abstract we study the spin 1 blume capel model under a random crystal field in the tridimensional semi infinite case this has been done by using the real space renormalization group approximation and specifically the migdal kadanoff technique interesting results are obtained which tell us that the randomness destroys the first order phase transitions and only those of the second order occur we give the list of nine fixed points and their topology describing the surface critical behavior five new types of phase diagram are found with a rich variety of phase transitions in accordance with the values of the bulk and surface probabilities and the ratios linking bulk and surface interactions

new monte carlo algorithm entropic sampling we present a new monte carlo sampling algorithm with which one can obtain any desired distribution of the sampling in one monte carlo simulation the free energy and the entropy of a system can thus be obtained from a simple exercise of this algorithm the main idea is to sample directly the entropy of a system at infinite temperature importance sampling is shown to be a particular case of the new algorithm the algorithm is tested against the exact partition function of the ital l 4 simple cubic ising model a comparison with the multicanonical ensemble for the ital l 12 ital q 10 potts model shows that the new algorithm is more general and more efficient

chaotic correlations in barrier billiards with arbitrary barriers we study autocorrelation functions in symmetric barrier billiards for golden mean trajectories with arbitrary barriers renormalization analysis reveals the presence of a chaotic invariant set and thus that for a typical barrier there are chaotic correlations the chaotic renormalization set is the analogue of the so called orchid that arises in a generalized harper equation

enriched stone type dualities abstract a common feature of many duality results is that the involved equivalence functors are liftings of hom functors into the two element space resp lattice due to this fact we can only expect dualities for categories cogenerated by the two element set with an appropriate structure a prime example of such a situation is stone s duality theorem for boolean algebras and boolean spaces the latter being precisely those compact hausdorff spaces which are cogenerated by the two element discrete space in this paper we aim for a systematic way of extending this duality theorem to categories including all compact hausdorff spaces to achieve this goal we combine duality theory and quantale enriched category theory our main idea is that when passing from the two element discrete space to a cogenerator of the category of compact hausdorff spaces all other involved structures should be substituted by corresponding enriched versions accordingly we work with the unit interval 0 1 and present duality theory for ordered and metric compact hausdorff spaces and suitably defined finitely cocomplete categories enriched in 0 1

scaling relations for logarithmic corrections multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical mechanical systems here a lee yang zero approach is used to systematically analyze the exponents of such logarithms and to propose scaling relations between them these proposed relations are then confronted with a variety of results from the literature

self consistent scaling theory for logarithmic correction exponents multiplicative logarithmic corrections frequently characterize critical behavior in statistical physics here a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific heat critical exponent vanishes also the theory is widened to encompass the correlation function the new relations are then confronted with results from the literature and some new predictions for logarithmic corrections in certain models are made

the number of link and cluster states the core of the 2d q state potts model due to fortuin and kastelyin the q state potts model has a representation as a sum over random graphs generalizing the potts model to arbitrary q is based on this representation a key element of the random cluster representation is the combinatorial factor which is the number of ways to form distinct clusters consisting of totally edges we have devised a method to calculate from monte carlo simulations

ensemble inequivalence in systems with long range interactions ensemble inequivalence has been observed in several systems in particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long range interactions we display a connection between such behaviour and a mean field like structure of the partition function since short range models cannot display this kind of behaviour this strongly suggests that such systems are necessarily non mean field in the sense indicated here we illustrate our results showing an application to the blume emery griffiths model we further show that a broad class of systems with non integrable interactions are indeed of mean field type in the sense specified so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble

monte carlo study of the pure and dilute baxter wu model we studied the pure and dilute baxter wu bw models using the wang landau wl sampling method to calculate the density of states dos we first used the exact result for the dos of the ising model to test our code then we calculated the dos of the dilute ising model to obtain a phase diagram in good agreement with previous studies we calculated the energy distribution together with its first second and fourth moments to give the specific heat and the energy fourth order cumulant better known as the binder parameter for the pure bw model for small samples the energy distribution displayed a doubly peaked shape and finite size scaling analysis showed the expected reciprocal scaling of the positions of the peaks with l the energy distribution yielded the expected bw 2 3 critical exponent for the specific heat the binder parameter minimum appeared to scale with lattice size l with an exponent b equal to the specific heat exponent its location temperature showed a large correction to scaling term 1 0 248 0 025 for the dilute bw model we found a clear crossover to a single peak in the energy distribution even for small sizes and the expected 0 was recovered

exact finite size scaling corrections to the critical two dimensional ising model on a torus we analyse the finite size corrections to the energy and specific heat of the critical two dimensional spin ising model on a torus we extend the analysis of ferdinand and fisher to compute the correction of order l 3 to the energy and the corrections of order l 2 and l 3 to the specific heat we also obtain general results on the form of the finite size corrections to these quantities only integer powers of l 1 occur unmodified by logarithms except of course for the leading log l term in the specific heat and the energy expansion contains only odd powers of l 1 in the specific heat expansion any power of l 1 can appear but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion

consistency of microcanonical and canonical finite size scaling typically in order to obtain finite size scaling laws for quantities in the microcanonical ensemble an assumption is taken as a starting point in this paper consistency of such a microcanonical finite size scaling assumption with its commonly accepted canonical counterpart is shown which puts microcanonical finite size scaling on a firmer footing

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

critical exponents of the three dimensional blume capel model on a cellular automaton abstract the static critical exponents of the three dimensional blume capel model which has a tricritical point at d j 2 82 value are estimated for the standard and the cooling algorithms which improved from creutz cellular automaton the analyses of the data using the finite size scaling and power law relations reproduce their well established values in the d j 3 and d j 2 8 parameter region at standard and cooling algorithms respectively for the cooling algorithm at d j 2 8 value of single ion anisotropy parameter the static critical exponents are estimated as 0 31 1 6 0 32 and 0 87 these values are different from 0 31 1 25 0 12 and 0 64 universal values this case indicated that the bc model exhibits an ununiversal critical behavior at the d j 2 8 parameter value near the tricritical point d j 2 82 the simulations were carried out on a simple cubic lattice with periodic boundary conditions

antiferromagnetic potts models on the square lattice a high precision monte carlo study we study the antiferromagnetic q state potts model on the square lattice for q 3 and q 4 using the wang swendsen koteck wsk monte carlo algorithm and a powerful finite size scaling extrapolation method for q 3 we obtain good control up to correlation length 5000 the data are consistent with ae2 p 1 a1e as with p 1 the staggered susceptibility behaves as stagg 5 3 for q 4 the model is disordered 2 even at zero temperature in appendices we prove a correlation inequality for potts antiferromagnets on a bipartite lattice and we prove ergodicity of the wsk algorithm at zero temperature for potts antiferromagnets on a bipartite lattice

non self averaging in autocorrelations for potts models on quenched random gravity graphs we investigate the non self averaging properties of the dynamics of ising 4 state potts and 10 state potts models in single cluster monte carlo simulations on quenched ensembles of planar trivalent 3 random graphs which we use as an example of relevant quenched connectivity disorder we employ a novel application of scaling techniques to the cumulative probability distribution of the autocorrelation times for both the energy and magnetization in order to discern non self averaging although the specific results discussed here are for quenched random graphs the method has quite general applicability

microcanonical finite size scaling in second order phase transitions with diverging specific heat a microcanonical finite size ansatz in terms of quantities measurable in a finite lattice allows extending phenomenological renormalization the so called quotients method to the microcanonical ensemble the ansatz is tested numerically in two models where the canonical specific heat diverges at criticality thus implying fisher renormalization of the critical exponents the three dimensional ferromagnetic ising model and the two dimensional four state potts model where large logarithmic corrections are known to occur in the canonical ensemble a recently proposed microcanonical cluster method allows simulating systems as large as l 1024 potts or l 128 ising the quotients method provides accurate determinations of the anomalous dimension ensuremath eta and of the fisher renormalized thermal ensuremath nu exponent while in the ising model the numerical agreement with our theoretical expectations is very good in the potts case we need to carefully incorporate logarithmic corrections to the microcanonical ansatz in order to rationalize our data

first order transition features of the triangular ising model with nearest and next nearest neighbor antiferromagnetic interactions abstract we implement a new and accurate numerical entropic scheme to investigate the first order transition features of the triangular ising model with nearest neighbor j nn and next nearest neighbor j nnn antiferromagnetic interactions in ratio r j nn j nnn 1 important aspects of the existing theories of first order transitions are briefly reviewed tested on this model and compared with previous work on the potts model using lattices with linear sizes l 30 40 100 120 140 160 200 240 360 and 480 we estimate the thermal characteristics of the present weak first order transition our results improve the original estimates of rastelli et al and verify all the generally accepted predictions of the finite size scaling theory of first order transitions including transition point shifts thermal and magnetic anomalies however two of our findings are not compatible with current phenomenological expectations the behavior of transition points derived from the number of phases parameter is not in accordance with the theoretically conjectured exponentially small shift behavior and the well known double gaussian approximation does not correctly describe higher correction terms of the energy cumulants it is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections

singular continuous spectra in a pseudointegrable billiard the pseudointegrable barrier billiard invented by hannay and mccraw j phys a 23 887 1990 a rectangular billiard with line segment barrier placed on a symmetry axis is generalized it is demonstrated that the fourier spectrum of a typical function on the phase space exhibits a singular continuous component

resolving the order of phase transitions in monte carlo simulations the authors made a monte carlo simulation of the two dimensional potts model with q 3 4 and 5 to examine the question whether numerical methods can distinguish the order of a phase transition for the subtle cases that this model exemplifies they found that the finite size scaling test for susceptibilities has sufficient power for this purpose whereas the simple method of detecting metastability signals often fails

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

stone duality above dimension zero axiomatising the algebraic theory of c x abstract it has been known since the work of duskin and pelletier four decades ago that k op the opposite of the category of compact hausdorff spaces and continuous maps is monadic over the category of sets it follows that k op is equivalent to a possibly infinitary variety of algebras in the sense of slominski and linton isbell showed in 1982 that the lawvere linton algebraic theory of can be generated using a finite number of finitary operations together with a single operation of countably infinite arity in 1983 banaschewski and rosick independently proved a conjecture of bankston establishing a strong negative result on the axiomatisability of k op in particular is not a finitary variety isbell s result is best possible the problem of axiomatising by equations has remained open using the theory of chang s mv algebras as a key tool along with isbell s fundamental insight on the semantic nature of the infinitary operation we provide a finite axiomatisation of

percolation of the site random cluster model by monte carlo method we propose a site random cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation to simulate the model on a square lattice we combine the color assignation and the swendsen wang methods to design a highly efficient cluster algorithm with a small critical slowing down phenomenon to verify whether or not it is consistent with the bond random cluster model we measure several quantities such as the wrapping probability re the percolating cluster density p and the magnetic susceptibility per site p as well as two exponents such as the thermal exponent yt and the fractal dimension yh of the percolating cluster we find that for different exponents of cluster weight q 1 5 2 2 5 3 3 5 and 4 the numerical estimation of the exponents yt and yh are consistent with the theoretical values the universalities of the site random cluster model and the bond random cluster model are completely identical for larger values of q we find obvious signatures of the first order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site our results are helpful for the understanding of the percolation of traditional statistical models

dynamic critical behavior of the chayes machta algorithm for the random cluster model i two dimensions we study via monte carlo simulation the dynamic critical behavior of the chayes machta dynamics for the fortuin kasteleyn random cluster model which generalizes the swendsen wang dynamics for the q state potts ferromagnet to non integer q 1 we consider spatial dimension d 2 and 1 25 q 4 in steps of 0 25 on lattices up to 10242 and obtain estimates for the dynamic critical exponent z cm we present evidence that when 1 q 1 95 the ossola sokal conjecture z cm is violated though we also present plausible fits compatible with this conjecture we show that the li sokal bound z cm is close to being sharp over the entire range 1 q 4 but is probably non sharp by a power as a byproduct of our work we also obtain evidence concerning the corrections to scaling in static observables

universality of the ising and the s 1 model on archimedean lattices a monte carlo determination the ising models s 1 2 and s 1 are studied by efficient monte carlo schemes on the 3 4 6 4 and the 3 3 3 3 6 archimedean lattices the algorithms used a hybrid metropolis wolff algorithm and a parallel tempering protocol are briefly described and compared with the simple metropolis algorithm accurate monte carlo data are produced at the exact critical temperatures of the ising model for these lattices their finite size analysis provide with high accuracy all critical exponents which as expected are the same with the well known 2d ising model exact values a detailed finite size scaling analysis of our monte carlo data for the s 1 model on the same lattices provides very clear evidence that this model obeys also very well the 2d ising model critical exponents as a result we find that recent monte carlo simulations and attempts to define effective dimensionality for the s 1 model on these lattices are misleading accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two archimedean lattices

renormalisation of correlations in a barrier billiard quadratic irrational trajectories abstract we present an analysis of autocorrelation functions in symmetric barrier billiards using a renormalisation approach for quadratic irrational trajectories depending on the nature of the barrier this leads to either self similar or chaotic behaviour in the self similar case we give an analysis of the half barrier and present a detailed calculation of the locations asymptotic heights and signs of the main peaks in the autocorrelation function then we consider arbitrary barriers illustrating that typically these give rise to chaotic correlations of the autocorrelation function which we further represent by showing the invariant sets associated with these correlations our main ingredient here is a functional recurrence which has been previously derived and used in work on the harper equation strange non chaotic attractors and a quasi periodically forced two level system

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we study the dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we find that the li sokal bound on the autocorrelation time int const xc h holds along the self dual curve of the symmetric ashkin teller model and is almost but not quite sharp the ratio int c h appears to tend to infinity either as a logarithm or as a small power 0 05 p 0 12 in an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time

potts and percolation models on bowtie lattices we give the exact critical frontier of the potts model on bowtie lattices for the case of q 1 the critical frontier yields the thresholds of bond percolation on these lattices which are exactly consistent with the results given by ziff et al j phys a 39 15083 2006 for the q 2 potts model on a bowtie a lattice the critical point is in agreement with that of the ising model on this lattice which has been exactly solved furthermore we do extensive monte carlo simulations of the potts model on a bowtie a lattice with noninteger q our numerical results which are accurate up to seven significant digits are consistent with the theoretical predictions we also simulate the site percolation on a bowtie a lattice and the threshold is s c 0 5479148 7 in the simulations of bond percolation and site percolation we find that the shape dependent properties of the percolation model on a bowtie a lattice are somewhat different from those of an isotropic lattice which may be caused by the anisotropy of the lattice

conformal invariance studies of the baxter wu model and a related site colouring problem the partition function of the baxter wu model is exactly related to the generating function of a site colouring problem on a hexagonal lattice we extend the original bethe ansatz solution of these models in order to obtain the eigenspectra of their transfer matrices in finite geometries and general toroidal boundary conditions the operator content of these models is studied by solving numerically the bethe ansatz equations and by exploring conformal invariance since the eigenspectra are calculated for large lattices the corrections to finite size scaling are also calculated

kronecker s double series and exact asymptotic expansions for free models of statistical mechanics on torus for the free models of statistical mechanics on torus including the ising model dimer model and gaussian model exact asymptotic expansions of free energy internal energy and specific heat in the vicinity of the critical point are found it is shown that there is a direct relation between the terms of the expansion and kronecker s double series the latter can be expressed in terms of the elliptic functions in all orders of the asymptotic expansion