graphical representations and cluster algorithms i discrete spin systems abstract graphical representations similar to the fk representation are developed for a variety of spin systems in several cases it is established that these representations have fkg monotonicity properties which enables characterization theorems for the uniqueness phase and the low temperature phase of the spin system certain systems with intermediate phases and or first order transitions are also described in terms of the percolation properties of the representations in all cases these representations lead in a natural fashion to swendsen wang type algorithms hence at least in the above mentioned instances these algorithms realize the program described by kandel and domany phys rev b 43 1991 8539 8548 all of the algorithms are shown to satisfy a li sokal bound which at least for systems with a divergent specific heat implies critical slowing down however the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points

nodal line densities of chaotic quantum billiard modes satisfying mixed boundary conditions statistics of nodal lines for eigenmodes u in the stadium are computed and compared with previously derived formulae for monochromatic boundary adapted gaussian random waves in the plane these modes and random waves satisfy the helmholtz equation and mixed boundary conditions in which a linear combination of u and its normal derivative must vanish for the density of nodal lines and the excess density of nodal lines arising from the boundary the gaussian model accurately describes the statistics of the billiard eigenfunctions

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

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

percolation transitions in two dimensions we investigate bond and site percolation models on several two dimensional lattices numerically by means of transfer matrix calculations and monte carlo simulations the lattices include the square triangular honeycomb kagome and diced lattices with nearest neighbor bonds and the square lattice with nearest and next nearest neighbor bonds results are presented for the bond percolation thresholds of the kagome and diced lattices and the site percolation thresholds of the square honeycomb and diced lattices we also include the bond and site percolation thresholds for the square lattice with nearest and next nearest neighbor bonds we find that corrections to scaling behave according to the second temperature dimension xt2 4 predicted by the coulomb gas theory and the theory of conformal invariance in several cases there is evidence for an additional term with the same exponent but modified by a logarithmic factor only for the site percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent the amplitude of the power law correction associated with xt2 4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems

nonconvergence of formal integrals ii improved estimates for the optimal order of truncation we investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 dof hamiltonian systems in particular we study the dependence of the optimal order of truncation nopt on the distance from the elliptic equilibrium by numerical and analytical means the function nopt determines the region of nekhoroshev stability of the orbits and the time of practical stability we find that the function nopt decreases by abrupt steps the decrease is roughly approximated with an average power law nopt o a with a 1 we find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via lie series and in the direct construction of first integrals precisely we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity our results agree with those obtained by servizi g et al 1983 phys lett a 95 11 for a conservative map of the plane moreover our analytical considerations allow us to explain the results of our previous paper contopoulos g et al 2003 j phys a math gen 36 8639 including in particular the different behaviour observed for low order and higher order resonances

the steep nekhoroshev s theorem revising nekhoroshev s geometry of resonances we provide a fully constructive and quantitative proof of nekhoroshev s theorem for steep hamiltonian systems proving in particular that the exponential stability exponent can be taken to be 1 2n alpha 1 cdots alpha n 2 alpha i s being nekhoroshev s steepness indices and n ge 3 the number of degrees of freedom on the base of a heuristic argument we conjecture that the new stability exponent is optimal

critical behaviour of two ising models with multispin interactions abstract we have performed extensive monte carlo simulations on the two dimensional ising models with n spin interactions described recently by debierre and turban results for n 3 models with sizes up to 128 128 are analyzed by means of finite size scaling this yields a value of the magnetic exponent y h close to 15 8 direct estimates of the temperature exponent y t do not converge convincingly however assuming the presence of logarithmic corrections such as in the 4 state potts model we obtain an improved estimate of the temperature exponent close to 3 2 in agreement with 4 state potts universality this result is further supported by an exact mapping between the n 3 model and the 4 state potts model in an anisotropic limit for the n 4 model we confirm that the phase transition is first order and we estimate the discontinuities in the energy and the magnetization

stability properties of the riemann ellipsoids the riemann ellipsoids are steady motions of an ideal incompressible self gravitating fluid that retain an ellipsoidal shape the existence properties and stability of these steady motions have been investigated since newton s time most of the stability results are due to riemann who studied lyapunov stability and to chandrasekhar who studied primarily numerically spectral stability thus obtaining lyapunov instability results this article addresses the nekhoroshev stability stability for finite but very long time scales of those riemann ellipsoids that are spectrally stable but of unknown lyapunov stability we base our analysis on a hamiltonian formulation of the problem derived from riemann s original formulation which we interpret here as a formulation on a covering space using recent results from hamiltonian perturbation theory given the complexity of the system we resort to numerical calculations at certain steps of the stability analysis as a prerequisite to our analysis we repeat the lyapunov and spectral stability analyses finding important discrepancies with chandrasekhar s findings we provide numerical evidence that

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

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

supporting random wave models a quantum mechanical approach we show how two point correlation functions recently derived within non isotropic random wave models can be obtained in the appropriate limit in terms of the exact green function of the quantum system since no statistical model is required for this derivation this shows that taking the wavefunctions as gaussian processes is the only assumption of those models we also show how for clean systems the two point correlation function based on an energy average defines a gaussian theory which substantially reduces the spurious contributions coming from the normalization problem

super exponential stability for generic real analytic elliptic equilibrium points we consider the dynamics in a neighborhood of an elliptic equilibrium point with a diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability generically both in a topological and measure theoretical sense any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point

estimates on invariant tori near an elliptic equilibrium point of a hamiltonian system abstract we give a precise statement for the kam theorem in a neighbourhood of an elliptic equilibrium point of a hamiltonian system if the frequencies of the elliptic point are nonresonant up to a certain order k 4 and a nondegeneracy condition is fulfilled we get an estimate for the measure of the complement of the kam tori in a neighbourhood of given radius moreover if the frequencies satisfy a diophantine condition with exponent we show that in a neighbourhood of radius r the measure of the complement is exponentially small in 1 r 1 1 we also give a related result for quasi diophantine frequencies which is more useful for practical purposes the results are obtained by putting the system in birkhoff normal form up to an appropiate order and the key point relies on giving accurate bounds for its terms

percolation and critical o n loop configurations we study a percolation problem based on critical loop configurations of the text o n loop model on the honeycomb lattice we define dual clusters as groups of sites on the dual triangular lattice that are not separated by a loop and investigate the bond percolation properties of these dual clusters the universal properties at the percolation threshold are argued to match those of kasteleyn fortuin random clusters in the critical potts model this relation is checked numerically by means of cluster simulations of several text o n models in the range 1 ensuremath le n ensuremath le 2 the simulation results include the percolation threshold for several values of n as well as the universal exponents associated with bond dilution and the size distribution of the diluted clusters at the percolation threshold our numerical results for the exponents are in agreement with existing coulomb gas results for the random cluster model which confirms the relation between both models we discuss the renormalization flow of the bond dilution parameter p as a function of n and provide an expression that accurately describes a line of unstable fixed points as a function of n corresponding with the percolation threshold furthermore the renormalization scenario indicates the existence in a p versus n diagram of another line of fixed points at p 1 which is stable with respect to p

nodal densities of gaussian random waves satisfying mixed boundary conditions nodal statistics are derived for boundary adapted gaussian random waves u i v in th ep lane satisfying the helmholtz equation and mixed boundary conditions in which a linear combination of and its normal derivative characterized by a parameter a v anishes on aline for both the density of nodal lines for re u and the density of nodal points for complex e ffects of the boundary persist surprisingly infinitely far from the boundary and also surprisingly are independent of a a sa increases from the dirichlet value a 0 nodal line and point structures migrate from the boundary and can be described analytically

nekhoroshev type stability results for hamiltonian systems with an additional transversal component abstract we prove exponential stability theorems of nekhoroshev type for motion in the neighbourhood of an elliptic fixed point in hamiltonian systems having an additional transverse component of arbitrary dimension under certain conditions on this transverse component we consider both the cases i of a strongly constrained motion and ii of a weak perturbation

stability of some waves in the dissipative boussinesq system abstract in this paper we study analytically a class of waves in the variant of the classical dissipative boussinesq system given by t u x v x x x v x x t u x u v t v x u c x x x u x x t v v x v where c 0 is a small parameter and 0 1 this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every c and almost every 1 it contains solutions that are defined for large values of time and they are very close of order o to a linear torus for long times of order o 1 the proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

knotted and linked phase singularities in monochromatic waves exact solutions of the helmholtz equation are constructed possessing wavefront dislocation lines phase singularities in the form of knots or links where the wave function vanishes knotted nothings the construction proceeds by making a nongeneric structure with a strength n dislocation loop threaded by a strength m dislocation line and then perturbing this in the resulting unfolded stable structure the dislocation loop becomes an m n torus knot if m and n are coprime and n linked rings or knots if m and n have a common factor n the loop or rings are threaded by an m stranded helix in our explicit implementation the wave is a superposition of bessel beams accessible to experiment paraxially the construction fails

nodal structure of chaotic eigenfunctions we investigate the distribution of nodes at and beyond the classical turning point of idealized chaotic quantum eigenfunctions a formula for the density of nodes is derived in the semiclassical limit and the rate at which this density falls off as one moves into the forbidden region is also studied the discussion is supported by numerical results corrections to the bessel function correlation in the classically allowed region are necessary for finite and are given here

random cluster representation of the ashkin teller model we show that a class of spin models containing the ashkin teller model admits a generalized random cluster grc representation moreover we show that basic properties of the usual representation such as fkg inequalities and comparison inequalities still hold for this generalized random cluster model some elementary consequences are given we also consider the duality transformations in the spin representation and in the grc model and show that they commute

quantum billiards and constrained random wave correlations we study chaotic eigenfunctions in wedge shaped and rectangular regions using a generalization of berry s conjecture an expression for the two point correlation function is derived and verified numerically

statistics of nodal lines and points in chaotic quantum billiards perimeter corrections fluctuations curvature for real time reversal symmetric quantum billiards the mean length l of nodal line is calculated for the nth mode n 1 with wavenumber k using a gaussian random wave model adapted locally to satisfy dirichlet or neumann boundary conditions the leading term is of order k i e n and the first perimeter correction dominated by an unanticipated long range boundary effect is of order log k i e log n with the same sign negative for both boundary conditions the leading order state to state fluctuations l are of order log k for the curvature of nodal lines and 2 are of order k but 3 and higher moments diverge for complex e g aharonov bohm billiards the mean number n of nodal points phase singularities in the mode has a leading term of order k2 i e n the perimeter correction again a long range effect is of order klog k i e nlog n and positive notwithstanding nodal depletion near the boundary and the fluctuations n are of order k log k generalizations of the results for mixed robin boundary conditions are stated

curved boundary corrections to nodal line statistics in chaotic billiards a gaussian random wavefunction that satisfies dirichlet and neumann conditions locally on a convex circular boundary is introduced the average of the square of the wavefunction and its derivatives are computed and their asymptotics studied in the semi classical limit the mean nodal line length l is calculated and the first order boundary effect shown to be of order log k where k is the wavenumber in the limit of vanishing boundary curvature large boundary radius these results are shown to approach those for a straight wall boundary

density and correlation functions of vortex and saddle points in open billiard systems we present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random wave model rwm in particular due to an improvement of the experimental setup we determine experimentally the spatial correlation of saddle points of the current field an asymptotic expression for the vortex saddle and saddle saddle correlation functions based on the rwm is derived with experiment and theory agreeing well we also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the rwm and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying dirichlet conditions

long term stability of proper rotations of the perturbed euler rigid body we study the long term stability of the proper rotations of the perturbed euler rigid body in the framework of nekhoroshev theory for simplicity we treat here in detail only the kinetically symmetric case the potential needs not to be symmetric but we indicate how to extend the results to the triaxial case we show that the proper rotations around the symmetry axis are nekhoroshev stable more precisely if the initial datum is sufficiently close to a proper rotation then for a very long time it remains such and the tip of the unit vector parallel to the angular momentum precesses up to a small noise along the level curves of a regular function on the unit sphere if the proper rotations are resonant chaotic motions with positive lyapunov exponents are possible but chaos unlike the case of ordinary motions that is motions not close to proper rotations is always localized and does not affect in an essential way the motion of the angular momentum in space preliminary numerical results indicate that the theory is in many aspects optimal although in some points it can still be improved

energy localization on q tori long term stability and the interpretation of fermi pasta ulam recurrences we focus on two approaches that have been proposed in recent years for the explanation of the so called fermi pasta ulam fpu paradox i e the persistence of energy localization in the low q fourier modes of fermi pasta ulam nonlinear lattices preventing equipartition among all modes at low energies in the first approach a low frequency fraction of the spectrum is initially excited leading to the formation of natural packets exhibiting exponential stability while in the second emphasis is placed on the existence of q breathers i e periodic continuations of the linear modes of the lattice which are exponentially localized in fourier space following ideas of the latter we introduce in this paper the concept of q tori representing exponentially localized solutions on low dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the fpu paradox

the statistics of the points where nodal lines intersect a reference curve we study the intersection points of a fixed planar curve with the nodal set of a translationally invariant and isotropic gaussian random field r and the zeros of its normal derivative across the curve the intersection points form a discrete random process which is the object of this study the field probability distribution function is completely specified by the correlation g r r r r given an arbitrary g r r we compute the two point correlation function of the point process on the line and derive other statistical measures repulsion rigidity which characterize the short and long range correlations of the intersection points we use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks we apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards of special interest is the observation that for monochromatic random waves the number variance of the intersections with long straight segments grows like lln l as opposed to the linear growth predicted by the percolation model which was successfully used to predict other long range nodal properties of that field

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

geometric properties of two dimensional o n loop configurations we study the fractal geometry of o n loop configurations in two dimensions by means of scaling and a monte carlo method and compare the results with predictions based on the coulomb gas technique the monte carlo algorithm is applicable to models with noninteger n and uses local updates although these updates typically lead to nonlocal modifications of loop connectivities the number of operations required per update is only of order 1 the monte carlo algorithm is applied to the honeycomb o n model for several values of n including noninteger ones we thus determine scaling exponents that describe the fractal nature of o n loops at criticality the results of the numerical analysis agree with the theoretical predictions

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

efficient simulation of the random cluster model the simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down a number of cluster algorithms usually based on the fortuin kasteleyn representation of the potts model and suitable generalizations for continuous spin models have been used to increase simulation efficiency the first algorithm making use of this representation suggested by sweeny in 1983 has not found widespread adoption due to problems in its implementation however it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well known algorithm due to swendsen and wang here we present an efficient implementation of sweeny s approach for the random cluster model using recent algorithmic advances in dynamic connectivity algorithms

on the formulation of new explicit conditions for steepness from a former result of n n nekhoroshev the application of the nekhoroshev theorem to many problems arising in different fields of physics and astronomy depends on a non degeneracy property called steepness that a suitable hamiltonian approximation must satisfy since steepness is implicitly defined we have the problem of recognizing whether a given function is steep or not for this purpose we here consider some sufficient conditions for steepness provided by nekhoroshev in 1979 based on the solvability of a collection of systems depending on the number n of degrees of freedom the derivatives of the function up to a certain order r and some auxiliary parameters these conditions are really explicit only for r 2 corresponding to quasi convexity and for r 3 instead for r 4 the conditions are implicit since they require an elaborate computation of the closure of a certain set in this paper we first revisit nekhoroshev s result and we show that the number of parameters in the collections of systems can be suitably reduced th

averaging slaving and balance dynamics in a simple atmospheric model we report numerical results from a study of balance dynamics using a simple model of atmospheric motion that is designed to help address the question of why balance dynamics is so stable the non autonomous hamiltonian model has a chaotic slow degree of freedom representing vortical modes coupled to one or two linear fast oscillators representing inertia gravity waves the system is said to be balanced when the fast and slow degrees of freedom are separated we find adiabatic invariants that drift slowly in time this drift is consistent with a random walk behaviour at a speed which qualitatively scales even for modest time scale separations as the upper bound given by neishtadt s and nekhoroshev s theorems moreover a similar type of scaling is observed for solutions obtained using a singular perturbation slaving technique in resonant cases where nekhoroshev s theorem does not apply we present evidence that the smaller lyapunov exponents of the system scale exponentially as well the results suggest that the observed stability of nearly slow motion is a consequence of the approximate adiabatic invariance of the fast motion

random cluster models on the triangular lattice we study percolation and the random cluster model on the triangular lattice with 3 body interactions starting with percolation we generalize the star triangle transformation we introduce a new parameter the 3 body term and identify configurations on the triangles solely by their connectivity in this new setup necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non percolation next we apply this set of ideas to the q 1 random cluster model we derive duality relations for the suitable random cluster measures prove necessary and sufficient conditions for them to have positive correlations and finally prove some rigorous theorems concerning phase transitions

nekhoroshev stability of elliptic equilibria of hamiltonian systems we prove a conjecture by n n nekhoroshev about the long time stability of elliptic equilibria of hamiltonian systems without any diophantine condition on the frequencies higher order terms of the hamiltonian are used to provide convexity the singularity of the action angle coordinates at the origin is overcome by working in cartesian coordinates

effective stability for a hamiltonian system near an elliptic equilibrium point with an application to the restricted three body problem abstract we consider an n degrees of freedom hamiltonian system near an elliptic equilibrium point the system is put in normal form up to an arbitrary order and with respect to some resonance module and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form a bound to the rate of diffusion is thus found and by optimizing the order of normalization exponential estimates of nekhoroshev s type are obtained this provides explicit estimates for the stability properties of the elliptic point and leads in some cases to effective stability i e stability up to finite but long times an application to the stability of the triangular libration points in the spatial restricted three body is also given

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

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

monte carlo study of the site percolation model in two and three dimensions we investigate the site percolation problem on the square and simple cubic lattices by means of a monte carlo algorithm that in fact simulates systems with size l d 1 x infinity where l specifies the linear system size this algorithm can be regarded either as an extension of the hoshen kopelman method or as a special case of the transfer matrix monte carlo technique various quantities such as the magnetic correlation function are sampled in the finite directions of the above geometry simulations are arranged such that both bulk and surface quantities can be sampled on the square lattice we locate the percolation threshold at p c 0 592 746 5 4 and determine two universal quantities as q gbc 0 930 34 1 and q gsc 0 793 72 3 which are associated with bulk and surface correlations respectively these values agree well with the exact values 2 5 48 and 2 1 3 respectively which follow from conformal invariance on the simple cubic lattice we locate the percolation threshold at p c 0 311 607 7 4 we further determine the bulk thermal and magnetic exponents as y t 1 1437 6 and y h 2 5219 2 respectively and the surface magnetic exponent at the ordinary phase transition as y o hs 1 0248 3

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

asymptotic stability estimates near an equilibrium point abstract we use the error bounds for adiabatic invariants found in the work of chartier murua and sanz serna 3 to bound the solutions of a hamiltonian system near an equilibrium over exponentially long times our estimates depend only on the linearized system and not on the higher order terms as in kam theory nor do we require any steepness or convexity conditions as in nekhoroshev theory we require that the equilibrium point where our estimate applies satisfy a type of formal stability called lie stability

stability of some solutions for elliptic equations on a cylindrical domain we study analytically a class of solutions for the elliptic equation formula 1 where 0 and e is a small parameter this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every 0 it contains solutions that are defined for large values of time and they are very close of order o e to a linear torus for long times of order o e 1 the proof uses the fact that the equation leaves invariant a smooth centre manifold and for the restriction of the system to the centre manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates 1 embed graphic 1 gif

crossover from isotropic to directed percolation we generalize the directed percolation dp model by relaxing the strict directionality of dp such that propagation can occur in either direction but with anisotropic probabilities we denote the probabilities as p p pd and p p 1 pd with p representing the average occupation probability and pd controlling the anisotropy the leath alexandrowicz method is used to grow a cluster from an active seed site we call this model with two main growth directions biased directed percolation bdp standard isotropic percolation ip and dp are the two limiting cases of the bdp model corresponding to pd 1 2 and pd 0 1 respectively in this work besides ip and dp we also consider the 1 2 pd 1 region extensive monte carlo simulations are carried out on the square and the simple cubic lattices and the numerical data are analyzed by finite size scaling we locate the percolation thresholds of the bdp model for pd 0 6 and 0 8 and determine various critical exponents these exponents are found to be consistent with those for standard dp we also determine the renormalization exponent associated with the asymmetric perturbation due to pd 1 2 6 0 near ip and confirm that such an asymmetric scaling field is relevant at ip