high precision estimate of g4 in the 2d ising model we compute the renormalized four point coupling in the 2d ising model using transfer matrix techniques we greatly reduce the systematic uncertainties which usually affect this type of calculation by using the exact knowledge of several terms in the scaling function of the free energy our final result is g4 14 697 35 3

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

generators for the hyperelliptic torelli group and the kernel of the burau representation at t 1 we prove that the hyperelliptic torelli group is generated by dehn twists about separating curves that are preserved by the hyperelliptic involution this verifies a conjecture of hain the hyperelliptic torelli group can be identified with the kernel of the burau representation evaluated at t 1 and also the fundamental group of the branch locus of the period mapping and so we obtain analogous generating sets for those one application is that each component in torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added

approximating the exponential from a lie algebra to a lie group consider a differential equation y a t y y y 0 y0 with y 0 g and a r g g where g is a lie algebra of the matricial lie group g every b g can be mapped to g by the matrix exponential map exp tb with t r most numerical methods for solving ordinary differential equations odes on lie groups are based on the idea of representing the approximation y n of the exact solution y t n t n r by means of exact exponentials of suitable elements of the lie algebra applied to the initial value y 0 this ensures that y n g when the exponential is difficult to compute exactly as is the case when the dimension is large an approximation of exp tb plays an important role in the numerical solution of odes on lie groups in some cases rational or polynomial approximants are unsuitable and we consider alternative techniques whereby exp tb is approximated by a product of simpler exponentials in this paper we present some ideas based on the use of the strang splitting for the approximation of matrix exponentials several cases of g and g are considered in tandem with general theory order conditions are discussed and a number of numerical experiments conclude the paper

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

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

on the singularity structure of the 2d ising model susceptibility some simplifications of the integrals 2n 1 derived by wu et al 1976 13 316 that contribute to the zero field susceptibility of the 2d square lattice ising model are reported in particular several alternate expressions for the integrands in 2n 1 are determined which greatly facilitate both the generation of high temperature series and analytical analysis one can show that as series 2n 1 22n s 2 4n n 1 1 o s where s is the high temperature variable sin 2k with k the conventional normalized inverse temperature analysis of the integrals near symmetry points of the integrands shows that 2n 1 s is singular on the unit circle at sk exp ik where 2cos k cos 2 k 2n 1 cos 2 2n 1 n k n the singularities k 0 excepted are logarithmic branch points of order 2n n 1 1ln with 1 s sk there is numerical evidence from series that these van hove points in addition to the known points at s 1 and i exhaust the singularities on the unit circle barring cancellation from extra unobserved singularities one can conclude that s 1 is a natural boundary for the susceptibility

ising model on nonorientable surfaces exact solution for the mobius strip and the klein bottle closed form expressions are obtained for the partition function of the ising model on an mxn simple quartic lattice embedded on a mobius strip and a klein bottle the solutions all lead to the same bulk free energy but for finite m and n the expressions are different depending on whether the strip width m is odd or even finite size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions our results are consistent with the conformal field prediction of a central charge c 1 2 provided that the twisted mobius boundary condition is regarded as a free or fixed boundary

ising field theory in a magnetic field analytic properties of the free energy we study the analytic properties of the scaling function associated with the 2d ising model free energy in the critical domain t tc h 0 the analysis is based on numerical data obtained through the truncated free fermion space approach we determine the discontinuities across the yang lee and langer branch cuts we confirm the standard analyticity assumptions and propose extended analyticity roughly speaking the latter states that the yang lee branching point is the nearest singularity under langer s branch cut we support the extended analyticity by evaluating numerically the associated extended dispersion relation

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

efficient algorithms for the two dimensional ising model with a surface field the bond propagation and site propagation algorithms are extended to the two dimensional 2d ising model with a surface field with these algorithms we can calculate the free energy internal energy specific heat magnetization correlation functions surface magnetization surface susceptibility and surface correlations the method can handle continuous and discrete bond and surface field disorder and is especially efficient in the case of bond or site dilution to test these algorithms we study the wetting transition of the 2d ising model which was solved exactly by abraham we can locate the transition point accurately with a relative error of 10 8 we carry out the calculation of the specific heat and surface susceptibility on lattices with sizes up to 200 2 times 200 the results show that a finite jump develops in the specific heat and surface susceptibility at the transition point as the lattice size increases for lattice size 320 2 times 320 the parallel correlation length exponent is 1 86 while abraham s exact result is 2 0 the perpendicular correlation length exponent for lattice size 160 2 times 160 is 1 05 whereas its exact value is 1 0

irrelevant operators in the two dimensional ising model by using conformal field theory we classify the possible irrelevant operators for the ising model with nearest neighbour interactions on the square and triangular lattices we analyse the existing results for the free energy and its derivatives and for the correlation length showing that they are in agreement with the conformal field theory predictions moreover these results imply that the nonlinear scaling field of the t operator where t is the energy momentum tensor vanishes at the critical point several other peculiar cancellations are explained in terms of a number of general conjectures we show that all existing results on the square and triangular lattices are consistent with the assumption that only nonzero spin operators are present

the level four braid group by evaluating the burau representation at t 1 we obtain a symplectic representation of the braid group we study the resulting congruence subgroups of the braid group namely the preimages of the principal congruence subgroups of the symplectic group our main result is that the level 4 congruence subgroup is equal to the group generated by squares of dehn twists we also show that the image of the brunnian subgroup of the braid group under the symplectic representation is the level four congruence subgroup

scaling and universality in the two dimensional ising model with a magnetic field the scaling function of the two dimensional ising model on the square and triangular lattices is obtained numerically via baxter s variational corner transfer matrix approach the use of aharony fisher nonlinear scaling variables allowed us to perform calculations sufficiently away from the critical point and to confirm all predictions of the scaling and universality hypotheses our results are in excellent agreement with quantum field theory calculations of fonseca and zamolodchikov as well as with many previously known exact and numerical calculations including susceptibility results by barouch mccoy tracy and wu

geometry thermodynamics and finite size corrections in the critical potts model we establish an intriguing connection between geometry and thermodynamics in the critical q state potts model on two dimensional lattices using the q state bond correlated percolation model qbcpm representation we find that the number of clusters nc of the qbcpm has an energylike singularity for qfi1 which is reached and supported by exact results numerical simulation and scaling arguments we also establish that the finite size correction to the number of bonds nb has no constant term and explains the divergence of related quantities as q 4 the multicritical point similar analyses are applicable to a variety of other systems s1063 651x 99 09612 9

addendum to on the singularity structure of the 2d ising model susceptibility a remarkable product formula first derived by palmer and tracy 1981 adv appl math 2 329 for the integrand of the two dimensional ising model susceptibility expansion coefficients 2n for temperatures t less than the critical t c is shown to apply equally for 2n 1 for t t c and agrees with formulae derived by yamada 1984 prog theor phys 71 1416 this new representation simplifies the derivation of the results in the original paper of this title 1999 j phys a math gen 32 3889 to the extent that the leading series behaviour and the singularity structure can be deduced almost by inspection the derivation of series is also simplified and i show using extended series and knowledge of the singularity structure that there is now unambiguous evidence for correction to scaling terms in the susceptibility beyond those inferred from a nonlinear scaling field analysis

on the sections of universal hyperelliptic curves abstract in this paper we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves c h g n k h g n k when g 3 and the image of the l adic cyclotomic character g k z is infinite furthermore we will study the nonabelian phenomena associated to the universal hyperelliptic curves for example we will show that the section conjecture holds for c h g k h g k and the unipotent analogue of the conjecture holds for the pointed cases this work is an extension of hain s original work on the rational points of universal curves to the hyperelliptic case

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

image of the burau representation at d th roots of unity we show that the image of the braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of branch points is suciently large compared to the degree d this is deduced by proving the arithmeticity of the image of the braid group on n 1 letters under the burau representation evaluated at d th roots of unity when n 2d

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

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

asymptotic expansion for the resistance between two maximally separated nodes on an m by n resistor network we analyze the exact formulae for the resistance between two arbitrary notes in a rectangular network of resistors under free periodic and cylindrical boundary conditions obtained by wu j phys a 37 6653 2004 based on such expression we then apply the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive the exact asymptotic expansions of the

critical behavior of the pure and random bond two dimensional triangular ising ferromagnet we investigate the effects of quenched bond randomness on the critical properties of the two dimensional ferromagnetic ising model embedded in a triangular lattice the system is studied in both the pure and disordered versions by the same efficient two stage wang landau method in the first part of our study we present the finite size scaling behavior of the pure model for which we calculate the critical amplitude of the specific heat s logarithmic expansion for the disordered system the numerical data and the relevant detailed finite size scaling analysis along the lines of the two well known scenarios logarithmic corrections versus weak universality strongly support the field theoretically predicted scenario of logarithmic corrections a particular interest is paid to the sample to sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality

finite size effects for anisotropic 2d ising model with various boundary conditions we analyze the exact partition function of the anisotropic ising model on finite m n rectangular lattices under four different boundary conditions periodic periodic pp periodic antiperiodic pa antiperiodic periodic ap and antiperiodic antiperiodic aa obtained by kaufman 1949 phys rev 76 1232 wu and hu 2002 j phys a math gen 35 5189 and kastening 2002 phys rev e 66 057103 we express the partition functions in terms of the partition functions z j k with 0 0 1 2 0 0 1 2 and 1 2 1 2 j is an interaction coupling and k is an anisotropy parameter based on such expressions we then extend the algorithm of ivashkevich et al 2002 j phys a math gen 35 5543 to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above our result is f fbulk p 0fp k s p 1 where f is the free energy of the system fbulk is the free energy of the bulk s mn is the area of the lattice and m n is the aspect ratio all coefficients in this expansion are expressed through analytical functions we have introduced the effective aspect ratio eff sinh 2jc and show that for pp and aa boundary conditions all finite size correction terms are invariant under the transformation eff 1 eff this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

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

accurate expansions of internal energy and specific heat of critical two dimensional ising model with free boundaries the bond propagation algorithm for the specific heat of the two dimensional ising model is developed and that for the internal energy is completed using these algorithms we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries comparing with previous works phys rev e 86 041149 2012 phys rev e 87 022124 2013 we reach much higher accuracy 10 28 of the internal energy and specific heat compared to the accuracy 10 11 of the internal energy and 10 9 of the specific heat reached in the previous works this leads to much more accurate estimations of the edge and corner terms the exact values of all edge and corner terms are therefore conjectured the accurate forms of finite size scaling for the internal energy and specific heat are determined for the rectangle shaped square lattice with various aspect ratios and various shaped triangular lattice for the rectangle shaped square and triangular lattices and the triangle shaped triangular lattice there is no logarithmic correction terms of order higher than 1 s with s the area of the system for the triangular lattice in rhombus trapezoid and hexagonal shapes there exist logarithmic correction terms of order higher than 1 s for the internal energy and logarithmic correction terms of all orders for the specific heat

non analyticity of the callan symanzik β function of two dimensional o n models we discuss the analytic properties of the callan symanzik function g associated with the zero momentum four point coupling g in the two dimensional 4 model with o n symmetry using renormalization group arguments we derive the asymptotic behaviour of g at the fixed point g we argue that g g o g g 1 7 for n 1 and g g o 1 log g g for n 3 our claim is supported by an explicit calculation in the ising lattice model and by a 1 n calculation for the two dimensional 4 theory we discuss how these non analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

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

universal ratio of magnetization moments in two dimensional ising models the authors calculate the universal critical point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic ising models on the square and on the triangular lattices periodic boundary conditions are used in accordance with the four fold and six fold rotational symmetries of the respective lattices these results which are obtained by means of an analysis of finite size data computed with a transfer matrix technique have an accuracy of the order of one millionth this analysis is also applied to rectangular systems with arbitrary aspect ratios

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

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

fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition abstract in recent years a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials where the complexity of solving numerically all microscale details is not feasible in this paper we present a multiscale framework based on the decomposition of the displacement field into coarse macro and fine micro scales earlier proposed in the variational multiscale approach or the hp d method the novelty of this work lies in solving the microscale step where a multidimensional parametrized model of a generic rve is solved by means of the multidimensional model reduction technique named as proper generalized decomposition pgd as result of this previous and off line step the displacement field over the rve is obtained by simple algebraic operations for any combination of parameters boundary condition material properties loads etc allowing a significant reduction in computational cost when solving the macro scale problem for problems where the detailed structure of the localized displacement strain and stress fields are of interest recovery of the fine scale components can be performed immediately the basic concepts and several 1 d and 2 d linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving

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

finite size effects for the ising model on helical tori we analyze the exact partition function of the ising model on a square lattice under helical boundary conditions obtained by liaw et al phys rev e 73 055101 r 2006 based on such an expression we then extend the algorithm of ivashkevich izmailian and hu j phys a 35 5543 2002 to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point from such results we find that the shift exponent for the specific heat is ensuremath lambda 1 for all values of the helicity factor d we also find that finite size corrections for the free energy the internal energy and the specific heat of the model in a crucial way depend on the helicity factor of the lattice

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

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

higher order alternate directions methods abstract alternate direction methods are based on the lie formula ea neb n n ea b o 1 n for square matrices a and b a more precise formula is known ea 2neb nea 2n n ea b o 1 n2 the search for more accurate product formulae is negative in the following sense there is no integer k and no choice of nonnegative real numbers j and j for 1 j k such that e 1a ne 1b n e ka ne kb n n ea b is of order 3 unless the commutators a a b or b b a are linearly dependent replacing exponentials by approximation of exponentials does not improve the situation however linear combinations of product formula can be constructed they lead to stable numerical schemes of order 3 and more applications to domain decomposition methods are given as well as some numerical experiments

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

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

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

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