study of the iterations of a mapping associated to a spin glass model we study the iterations of the mapping mathcal n f s frac f s 2 f 0 2 s f 0 2 with the constraintsf 1 1 f s ansn an 0 and find that except iff s s n f s approaches either 0 or 1 for s 1 ask

hierarchical pinning models quadratic maps and quenched disorder we consider a hierarchical model of polymer pinning in presence of quenched disorder introduced by derrida et al j stat phys 66 1189 1213 1992 which can be re interpreted as an infinite dimensional dynamical system with random initial condition the disorder it is defined through a recurrence relation for the law of a random variable r n n 1 2 which in absence of disorder i e when the initial condition is degenerate reduces to a particular case of the well known logistic map the large n limit of the sequence of random variables 2 n log r n a non random quantity which is naturally interpreted as a free energy plays a central role in our analysis the model depends on a parameter 0 1 related to the geometry of the hierarchical lattice and has a phase transition in the sense that the free energy is positive if the expectation of r 0 is larger than a certain threshold value and it is zero otherwise it was conjectured in derrida et al j stat phys 66 1189 1213 1992 that disorder is relevant respectively irrelevant or marginally relevant if 1 2 1 2 we find the correct scaling form for weak disorder of the critical point shift

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

differential migdal kadanoff renormalization group for disordered systems phase transitions in quenched random systems are studied within the migdal kadanoff approximation exact differential recursion relations for the distribution of coupling are derived the q state potts model with random ferromagnetic couplings is studied in two and three dimensions in three dimensions there is a non perturbative random fixed point which exists when the specific heat exponent is either positive or slightly negative if the specific heat exponent is slightly negative a multicritical point separates strong and weak disorder regimes the analytic results are confirmed by numerical simulations of the recursion relations

the migdal kadanoff approximation optimisation generalisation ising model in external fields and the migdal kadanoff hierarchies the simplest migdal kadanoff approximation is optimised on hypercubic lattices generalisation is made to all regular lattices leading to recursion relations continuous in lattice coordination one variety of these relations shows universality whilst another variety shows duality on planar lattices the ising model in external fields both normal and staggered is investigated phase diagrams and exponents are found under a complete treatment of the ambiguity in the handling of external fields by bond moving finally the underlying exactly solved family of hierarchical lattices is considered it is shown that these lattices exhibit a complete breakdown in the concept of universality

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

locally self similar phase diagram of the disordered potts model on the hierarchical lattice we study the critical behavior of the random q state potts model in the large q limit on the diamond hierarchical lattice with an effective dimensionality d eff 2 by varying the temperature and the strength of the frustration the system has a phase transition line between the paramagnetic and the ferromagnetic phases which is controlled by four different fixed points according to our renormalization group study the phase boundary in the vicinity of the multicritical point is self similar it is well represented by a logarithmic spiral we expect an infinite number of reentrances in the thermodynamic limit consequently one cannot define standard thermodynamic phases in this region

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

pure and dilute z n spin and generalised gauge lattice systems duality and criticality considers the pure z n spin systems including the standard ising and potts models as well as generalised gauge systems plaquettes or more complex simplex in d dimensional hypercubic lattices these models are self dual and the authors show how this duality can be thought of as a series parallel transformation the simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems including some particular asymptotic behaviours which are exact as an illustration the d 2 diluted z 4 spin system is discussed in some detail for those regions where exact results are available the agreement is satisfactory

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

critical behavior of the random bond ashkin teller model a monte carlo study the critical behaviour of a bond disordered ashkin teller model on a square lattice is investigated by intensive monte carlo simulations a duality transformation is used to locate a critical plane of the disordered model this critical plane corresponds to the line of critical points of the pure model along which critical exponents vary continuously along this line the scaling exponent corresponding to randomness varies continuously and is positive so that randomness is relevant and different critical behaviour is expected for the disordered model we use a cluster algorithm for the monte carlo simulations based on the wolff embedding idea and perform a finite size scaling study of several critical models extrapolating between the critical bond disordered ising and bond disordered four state potts models the critical behaviour of the disordered model is compared with the critical behaviour of an anisotropic ashkin teller model which is used as a refference pure model we find no essential change in the order parameters critical exponents with respect to those of the pure model the divergence of the specific heat c is changed dramatically our results favor a logarithmic type divergence at tc c logl for the random bond ashkin teller and four state potts models and c log logl for the random bond ising model

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

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

remarks on disorder and aperiodicity in a model for interacting polymers abstract we present a comparative study of the effects of random and aperiodically distributed interactions on the critical behavior of a model for two interacting polymers on a diamond hierarchical lattice the problem is formulated in terms of exact renormalization group rg recursion relations in the disordered case it is possible to develop a perturbative treatment in order to obtain the fixed points of the moments associated with the random distribution of interactions fully uncorrelated disorder may become relevant driving the system away from a homogeneous fixed point layered disorder may lead to a breakdown of the perturbative treatment in the case of aperiodic interactions we also show some examples of relevance and irrelevance of geometric fluctuations and further investigate the models by resorting to an independent transfer matrix tm analysis which fully corroborates the scaling results

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

accelerated non linear finite volume method for diffusion a non linear finite volume method with monotone matrix for the diffusion equation is presented it does not extrapolate the primary variable to neumann boundaries as this was previously done in similar methods this change results in faster convergence computation time is significantly shortened further using the reduced rank extrapolation method rre and imposing an upper limit on the number of linear iterations per non linear step second order accuracy and performance improvement are demonstrated by numerical examples

potts model on directed small world voronoi delaunay lattices abstract the critical properties of the potts model with q 3 and 4 states in two dimensions on directed small world voronoi delaunay random lattices with quenched connectivity disorder are investigated this disordered system is simulated by applying the monte carlo update heat bath algorithm the potts model on these directed small world random lattices presents in fact a second order phase transition with new critical exponents for q 3 and value of the rewiring probability p 0 01 but for q 4 the system exhibits only a first order phase transition independent of p 0 p 1

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

knots with small lattice stick numbers the lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice in this paper we mathematically prove that the trefoil knot 31 and in figure 8 knot 41 are the only knot types of lattice stick number less than 15 which verifies the result from previous numerical estimations on this quantity

upper bounds on the minimum length of cubic lattice knots knots have been considered to be useful models for simulating molecular chains such as dna and proteins one quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot in this paper we consider every knot in the cubic lattice in particular the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice diao introduced this term he used minimum edge number instead and proved that the minimum length of the trefoil knot 31 is 24 also the minimum lengths of the knots 41 and 51 are known to be 30 and 34 respectively in this paper we find a general upper bound of the minimum length of a nontrivial knot k except the trefoil knot in terms of the minimum crossing number c k the upper bound is moreover if k is a non alternating prime knot then the upper bound is our work can be considered a direct consequence of the results obtained by the authors in hong et al 2012 arxiv 1209 0048 furthermore if k is n 1 n torus knot then the upper bound is

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

effect of random impurities on fluctuation driven first order transitions we analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of n coupled two dimensional ising models for n 2 the pure model exhibits a fluctuation driven first order transition characterized by runaway renormalization group behaviour we show that the addition of weak randomness acts to stabilize these flows in such a way that the trajectories ultimately flow back towards the pure decoupled ising fixed point with the usual critical exponents apart from logarithmic corrections we also show by examples that in higher dimensions such transitions may either become continuous or remain first order in the presence of randomness

critical behavior induced by quenched disorder abstract domain arguments and renormalization group calculations indicate that all temperature driven symmetry breaking first order phase transitions are converted to second order by quenched bond randomness this occurs for infinitesimal randomness in d 2 or d 4 respectively for discrete or continuous n 1 or n 2 component microscopic degrees of freedom even strongly first order transitions undergo this conversion to second order above these dimensions this conversion still occurs but requires a threshold bond randomness presumably with an intervening new tricritical point for example q state potts transitions can be made second order for any q in any d via bond randomness non symmetry breaking temperature driven first order transitions are eliminated under the above conditions these quenched fluctuation induced second order phase transitions suggest the possibility of new universality classes of criticality and tricriticality

spin models on random lattices under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a dierent behaviour than in the corresponding pure system here we discuss a series of monte carlo studies of a special type of such disordered systems namely spin models dened on quenched random lattices exhibiting geometrical disorder in the connectivity of the lattice sites in two dimensions we present results for the q state potts model on random tri valent 3 planar graphs which appear quite naturally in the dynamically triangulated random surface dtrs approach to quantum gravity as well as on poissonian random lattices of voronoi delaunay type both cases q64 and 4 are discussed which in the pure model without disorder give rise to second and rst order phase transitions respectively in three dimensions results for the ising model on poissonian random lattices are briey described we conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder and a discussion of the distinguishing dierences c 2000 elsevier science b v all rights reserved

critical dynamics of cluster algorithms in the dilute ising model autocorrelation times for thermodynamic quantities att c are calculated from monte carlo simulations of the site diluted simple cubic ising model using the swendsen wang and wolff cluster algorithms our results show that for these algorithms the autocorrelation timesdecrease when reducing the concentration of magnetic sites from 100 down to 40 this is of crucial importance when estimating static properties of the model since the variances of these estimators increase with autocorrelation time the dynamical critical exponents are calculated for both algorithms observing pronounced finite size effects in the energy autocorrelation data for the algorithm of wolff we conclude that when applied to the dilute ising model cluster algorithms become even more effective than local algorithms for whichincreasing autocorrelation times are expected

hierarchical lattices some examples with a comparison of intrinsic dimension and connectivity and ising model exponents the recently recognised class of hierarchical lattices is examined through a number of examples definitions of length intrinsic dimension and connectivity are made and used to discuss variations of exponents calculated on the ising model generalisations of regular lattice results are found for exponents at discontinuity fixed points and in 1 epsilon dimensions

second order accurate finite volume method for well driven flows we consider a finite volume method for a well driven fluid flow in a porous medium due to the singularity of the well modeling in the near well region with standard numerical schemes results in a completely wrong total well flux and an inaccurate hydraulic head local grid refinement can help but it comes at computational cost in this article we propose two methods to address the well singularity in the first method the flux through well faces is corrected using a logarithmic function in a way related to the peaceman model coupling this correction with a non linear second order accurate two point scheme gives a greatly improved total well flux but the resulting scheme is still inconsistent in the second method fluxes in the near well region are corrected by representing the hydraulic head as a sum of a logarithmic and a linear function this scheme is second order accurate

quenched randomness at first order transitions abstract a rigorous theorem due to aizenman and wehr asserts that there can be no latent heat in a two dimensional system with quenched random impurities we examine this result and its possible extensions to higher dimensions in the context of several models for systems whose pure versions undergo a strong first order transition we show that there is an asymptotically exact mapping to the random field ising model at the level of the interface between the ordered and disordered phases this provides a physical explanation for the above result and also implies a correspondence between the problems in higher dimensions including scaling relations between their exponents the particular example of the q state potts model in two dimensions has been considered in detail by various authors and we review the numerical results obtained for this case turning to weak fluctuation driven first order transitions we describe analytic renormalization group calculations which show how the continuous nature of the transition is restored by randomness in two dimensions

bounds for the minimum step number of knots in the simple cubic lattice knots are found in dna as well as in proteins and they have been shown to be good tools for structural analysis of these molecules an important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot here we address this problem by characterizing both analytically and numerically the minimum length also called minimum step number needed to form a particular knot in the simple cubic lattice our analytical work is based on improvement of a method introduced by diao to enumerate conformations of a given knot type for a fixed length this method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 which is 24 to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34 respectively using an independent method based on the bfacf algorithm we provide a complete list of numerical estimates upper bounds of the minimum step numbers for prime knots up to ten crossings which are improvements over current published numerical results we enumerate all minimum lattice knots of a given type and partition them into classes defined by bfacf type 0 moves

critical behavior of spin and polymer models with aperiodic interactions we review and extend some recent investigations of the effects of aperiodic interactions on the critical behavior of ferromagnetic q state potts models by considering suitable diamond or necklace hierarchical lattices and assuming a distribution of interactions according to a class of two letter substitution rules the problem can be formulated in terms of recursion relations in parameter space the analysis of stability of the fixed points leads to an exact criterion to gauge the relevance of geometric fluctuations for irrelevant fluctuations the critical behavior remains unchanged with respect to the uniform systems for relevant fluctuations there appears a two cycle of saddle point character in parameter space a scaling analysis supported by direct numerical thermodynamic calculations shows the existence of novel critical universality classes associated with relevant geometric fluctuations also we show that similar qualitative results are displayed by a simple model of two directed polymers on a diamond hierarchical structure with aperiodic bond interactions

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities