the flanders theorem over division rings let d be a division ring and f be a subfield of the center of d over which d has finite dimension d let n p r be positive integers and v be an affine subspace of thef vector space mn p d in which every matrix has rank less than or equal to r using a new method we prove that dimf v max n p rd and we characterize the spaces for which equality holds this extends a famous theorem of flanders which was known only for fields

density of states potts zeros and fisher zeros of the q state potts model for continuous q the q state potts model can be extended to noninteger and even complex q by expressing the partition function in the fortuin kasteleyn f k representation in the f k representation the partition function z q a is a polynomial in q and v a ensuremath 1 a e ensuremath beta j and the coefficients of this polynomial ensuremath phi b c are the number of graphs on the lattice consisting of b bonds and c connected clusters we introduce the random cluster transfer matrix to compute ensuremath phi b c exactly on finite square lattices with several types of boundary conditions given the f k representation of the partition function we begin by studying the critical potts model z mathrm cp z q a c q where a c q 1 sqrt q we find a set of zeros in the complex w sqrt q plane that map to or close to the beraha numbers for real positive q we also identify q c l the value of q for a lattice of width l above which the locus of zeros in the complex p v sqrt q plane lies on the unit circle by finite size scaling we find that 1 q c l ensuremath rightarrow 0 as stackrel ensuremath rightarrow l ensuremath infty we then study zeros of the antiferromagnetic af potts model in the complex q plane and determine q c a the largest value of q for a fixed value of a below which there is af order we find excellent agreement with baxter s conjecture q c mathrm af a 1 ensuremath a a 3 we also investigate the locus of zeros of the ferromagnetic potts model in the complex q plane and confirm that q c mathrm fm a a ensuremath 1 2 we show that the edge singularity in the complex q plane approaches q c as q c l ensuremath sim q c al ensuremath y q and determine the scaling exponent y q for several values of q finally by finite size scaling of the fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y t as a function of q in the range 2l ql 3 using data for lattices of size 3l ll 8 we find that y t is a smooth function of q and is well fitted by y t 1 au bu 2 c du where u ensuremath 2 ensuremath pi mathrm cos ensuremath 1 sqrt q 2 for q 3 we find y t ensuremath simeq 0 6 however if we include lattices up to l 12 we find y t ensuremath simeq 0 50 8 in rough agreement with a recent result of ferreira and sokal j stat phys 96 461 1999

a mean field renormalisation group approach to ising and q 3 state potts models with long range interactions in one dimension abstract we study the one dimensional potts model with long range interactions decaying with distance r as r 1 an extended mean field renormalisation group procedure is applied such that three finite size linear lattices are compared in order to evaluate critical temperatures and exponents for the q 2 ising model and q 3 such that the transition is a continuous one cases good results are obtained whenever comparison with exact results or with other procedures is possible moreover we evaluate the surface field exponent for these models we have been able to go to rather large lattices and then a suitable finite size scaling procedure is employed to obtain the results in the thermodynamic limit

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

partition function zeros of the one dimensional potts model the recursive method the yang lee fisher and potts zeros of the one dimensional q state potts model are studied using the theory of dynamical systems an exact recurrence relation for the partition function is derived it is shown that zeros of the partition function may be associated with neutral fixed points of the recurrence relation further a general equation for zeros of the partition function is found and a classification of the yang lee fisher and potts zeros is given it is shown that the fisher zeros in a nonzero magnetic field are located on several lines in the complex temperature plane and that the number of these lines depends on the value of the magnetic field analytical expressions for the densities of the yang lee fisher and potts zeros are derived it is shown that densities of all types of zeros of the partition function are singular at the edge singularity points with the same critical exponent

exact results for the zeros of the partition function of the potts model on finite lattices the yang lee zeros of the q state potts model are investigated in one two and three dimensions analytical results derived from the transfer matrix for the one dimensional model reveal a systematic behavior of the locus of zeros as a function of q for 1 2 they lie outside the unit circle for finite temperature in the special case q 2 the zeros lie exactly on the unit circle as proved by lee and yang in two and three dimensions the zeros are calculated numerically and behave in the same way results are also presented for the critical line of the potts model in an external field as determined from the zeros of the partition function in the complex temperature plane

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

on a conjecture of alley and alder for fluids and lorentz models we discuss a conjecture of alley and alder predicting a relation between the four point and the two point velocity autocorrelation functions for fluids and lorentz models at sufficiently long times if the conjecture is correct a modified burnett coefficient can be defined which has a finite value contrary to the ordinary burnett coefficient which is divergent the conjecture is tested for four classes of models with different methods for three dimensional fluids mode coupling theory yields a negative result the conjecture is confirmed for thed dimensional deterministic lorentz gas d 2 and for a class ofd dimensional stochastic lorentz models d 1 by low density kinetic theory as well as by rigorous results available for one dimension for yet another class of one dimensional stochastic lorentz models which are exactly solvable in one dimension the result is negative again all four classes of models show long time tails in the velocity autocorrelation function and have a finite diffusion coefficient

families of graphs with chromatic zeros lying on circles we define an infinite set of families of graphs which we call p wheels and denote wh p n that generalize the wheel p 1 and biwheel p 2 graphs the chromatic polynomial for wh p n is calculated and remarkably simple properties of the chromatic zeros are found i the real zeros occur at q 0 1 p 1 for n p even and q 0 1 p 2 for n p odd and ii the complex zeros all lie equally spaced on the unit circle q p 1 1 in the complex q plane in the n to infty limit the zeros on this circle merge to form a boundary curve separating two regions where the limiting function w wh p q is analytic viz the exterior and interior of the above circle connections with statistical mechanics are noted

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

multicanonical monte carlo simulations canonical monte carlo simulations of disordered systems like spin glasses and systems undergoing first order phase transitions are severely hampered by rare event states which lead to exponentially diverging autocorrelation times with increasing system size and hence to exponentially large statistical errors one possibility to overcome this problem is the multicanonical reweighting method using standard local update algorithms it could be demonstrated that the dependence of autocorrelation times on the system size v is well described by a less divergent power law v with 1 3 depending on the system after a brief review of the basic ideas combinations of multicanonical reweighting with non local update algorithms will be discussed with the multibondic algorithm which combines multicanonical reweighting with cluster updates the dynamical exponent can be reduced to unity the optimal value one would expect from a random walk argument asymptotically for large system sizes the multibondic algorithm therefore always performs better than the standard multicanonical method finally it is shown that a combination with multigrid update techniques improves the performance of multicanonical simulations by roughly one order of magnitude uniformly for all system sizes

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

monte carlo study of the random field ising model using a cluster flipping monte carlo algorithm combined with a generalization of the histogram reweighting scheme of ferrenberg and swendsen we have studied the equilibrium properties of the thermal random field ising model on a cubic lattice in three dimensions we have equilibrated systems of l x l x l spins with values of l up to 32 and for these systems the cluster flipping method appears to a large extent to overcome the slow equilibration seen in single spin flip methods from the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling for the exponents we find v 1 02 0 06 b 0 06 0 07 y 1 9 0 2 and mean y 2 9 0 2

understanding and improving the wang landau algorithm we present a mathematical analysis of the wang landau algorithm prove its convergence and identify sources of errors and strategies for optimization in particular we found the histogram increases uniformly with small fluctuations after a stage of initial accumulation and the statistical error is found to scale as sqrt mathrm ln phantom rule 0 2em 0ex f with the modification factor f this has implications for strategies for obtaining fast convergence

polynomial identities for the jordan algebra of a symmetric bilinear form 1 the main purpose of this paper is to obtain a description of the relatively free algebras fj b and fj of the varieties of unitary jordan algebras 2b var g and 2 var gk the g module structure of these algebras is established and their hilbert or poincare series are computed a formula for the codimensions of 22 is given and the codimensions of i and 2bk are estimated asymptotically the varieties 2bk are described as subvarieties of 93 and lbk the proofs of these results use the representation theory of the symmetric and general linear groups the essential results are contained in sections 5 7 while the first four sections give the necessary preliminaries section 8 lists some open problems on varieties of jordan algebras motivated by results and problems from the associative theory 1

one dimensional models with 1 r2 interactions the critical behaviour of a general discrete one dimensional model with inverse square interactions is discussed using renormalisation group methods

a localization technique for linear preserver problems abstract we introduce a new general technique for solving linear preserver problems the idea is to localize a given linear preserver at each non zero vector in such a way we get vector valued linear maps on the space of matrices which inherit certain properties from if we can prove that such induced maps have a standard form then itself has either a standard form or a very special degenerate form we apply this technique to characterize linear preservers of full rank using this technique we further reprove two classical results describing the general form of linear preservers of rank one and linear preservers of the unitary or orthogonal group

first order transition in the one dimensional three state potts model with long range interactions the first order phase transition in the three state potts model with long range interactions decaying as 1 r 1 ensuremath sigma has been examined by numerical simulations using the recently proposed luijten bl ote algorithm by applying scaling arguments to the interface free energy the binder s fourth order cumulant and the specific heat maximum the change in the character of the transition through variation of parameter ensuremath sigma was studied

the critical behaviour of the long range potts chain from the largest cluster probability distribution abstract we present the numerical study of the one dimensional potts model with power law decaying ferromagnetic interactions the largest cluster probability distribution is obtained by monte carlo simulations using the swendsen wang cluster algorithm with cumulative probabilities the finite size scaling analysis of the largest cluster is used to derive the critical behaviour in the non classical regime of this model for various values of q

first order transitions for n vector models in two and more dimensions rigorous proof we prove that various so n invariant n vector models with interactions which have a deep and narrow enough minimum have a first order transition in the temperature the result holds in dimensions two or more and is independent of the nature of the low temperature phase

on invertibility preserving linear mappings simultaneous triangularization and property l abstract let a be a unital matrix algebra a m n c a unital linear mapping and b the algebra generated by a the mapping is a homomorphism modulo the jacobson radical in b if and only if for k dim b dim a 3 the mapping id a m k c b m k c preserves invertibility this result is closely related to the question on the relationship between property l and simultaneous triangularization we introduce a property kl and give criteria in terms of conditions on matrix pencils with coefficients in m k c rather than in c which ensure simultaneous triangularization

long range one dimensional potts models a cluster mean field and extrapolation approach we obtain approximations for the critical temperature tc of q state ferromagnetic potts models on one dimensional lattices with algebraically decaying ferromagnetic pair interactions i e decaying as 1 r with 1 2 initially we use a cluster mean field method to get approximations to the critical temperature and we look at increasingly large clusters this gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms we see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate tc estimates our best estimates we believe to be correct to four figure accuracy

2 graded polynomial identities for the jordan algebra of the symmetric matrices of order two abstract the jordan algebra of the symmetric matrices of order two over a field k has two natural gradings by z 2 the cyclic group of order 2 we describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2 we exhibit bases for these identities in each of the two cases in one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from invariant theory moreover we extend the latter grading to a z 2 grading on b n the jordan algebra of a symmetric bilinear form in a vector space of dimension n n 1 2 we call this grading the scalar one since its even part consists only of the scalars as a by product we obtain finite bases of the z 2 graded identities for b n in fact the last result describes the weak jordan polynomial identities for the pair b n v n

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

the one dimensional potts model with long range interactions a renormalization group approach the one dimensional q state potts model with ferromagnetic pair interactions which decay with the distance r as is considered we calculate through a real space renormalization group technique using kadanoff blocks of length b the critical temperature and the correlation length critical exponent as a function of for different values of q some of the very few known rigorous results for general q are reproduced by our approach several asymptotic behaviours are derived analytically for q 2 3 in the limit we also obtain extrapolated critical temperatures for arbitrary values of and for q 2 3 4 which we believe approximate the exact ones well except in the region near furthermore the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q we also verify that which is consistent with a recent conjecture of tsallis

a numerical method to compute exactly the partition function with application to z n theories in two dimensions i present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions the time for the computation for ann state model on anl d lattice scales like n l d 1 nl d i show examples of the use of this method by computing the partition function of the 2d ising and 3 state potts models for maximum lattice sizes 10 10 and 8 8 respectively the critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values the distribution of zeros of the partition function of the potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity

fast flat histogram method for generalized spin models we present a monte carlo method that efficiently computes the density of states for spin models having any number of interaction per spin by combining a random walk in the energy space with collective updates controlled by the microcanonical temperature our method yields dynamic exponents close to their ideal random walk values reduced equilibrium times and very low statistical error in the density of states the method can host any density of states estimation scheme including the wang landau algorithm and the transition matrix method our approach proves remarkably powerful in the numerical study of models governed by long range interactions where it is shown to reduce the algorithm complexity to that of a short range model with the same number of spins we apply the method to the q state potts chains 3 ensuremath leqslant q ensuremath leqslant 12 with power law decaying interactions in their first order regime we find that conventional local update algorithms are outperformed already for sizes above a few hundred spins by considering chains containing up to 2 16 spins which we simulated in fairly reasonable time we obtain estimates of transition temperatures correct to five figure accuracy finally we propose several efficient schemes aimed at estimating the microcanonical temperature

the ising model with long range ferromagnetic interactions a variational method based on the nearest neighbour ising model is used to derive an expression for the low temperature correlation function in the ising model with long range ferromagnetic interactions also by expanding the correlation function directly the authors demonstrate that the characteristic power law behaviour at large distances is irrelevant in the critical region when above the marginal range of interaction

critical behaviour of the 1d q state potts model with long range interactions the critical behaviour of the one dimensional q state potts model with long range interactions decaying with distance r as r 1 sigma has been studied in the wide range of parameters 0 sigma or 1 and 1 16 or q or 64 a transfer matrix has been constructed for a truncated range of interactions for integer and continuous q and finite range scaling has been applied results for the phase diagram and the correlation length critical exponent are presented

finite size scaling of the density of zeros of the partition function in first and second order phase transitions the finite size scaling form for the density of zeros of the partition function in first and second order phase transitions is derived using the finite size scaling of the density of zeros the order of a phase transition can be easily determined and the order parameter calculated from finite size data we illustrate the scaling theory using exact values for the zeros of the partition function of the two dimensional ising model in the complex magnetic field plane

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

reexamination of the long range potts model a multicanonical approach we investigate the critical behavior of the one dimensional q state potts model with long range lr interaction 1 r d sigma using a multicanonical algorithm the recursion scheme initially proposed by berg is improved so as to make it suitable for a large class of lr models with unequally spaced energy levels the choice of an efficient predictor and a reliable convergence criterion is discussed we obtain transition temperatures in the first order regime which are in far better agreement with mean field predictions than in previous monte carlo studies by relying on the location of spinodal points and resorting to scaling arguments we determine the threshold value sigma c q separating the first and second order regimes to two digit precision within the range 3 leq q leq 9 we offer convincing numerical evidence supporting sigma c q 1 0 for all q by virtue of an unusual finite size effect namely finite size scaling predicts a continuous transition in the thermodynamic limit despite the first order nature of the transition at finite size a qualitative account in terms of correlation lengths is provided finally we find the crossover between the lr and short range regimes to occur inside a narrow window 1 0 sigma 1 2 thus lending strong support to sak s scenario

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

chromatic polynomials potts models and all that the q state potts model can be defined on an arbitrary finite graph and its partition function encodes much important information about that graph including its chromatic polynomial flow polynomial and reliability polynomial the complex zeros of the potts partition function are of interest both to statistical mechanicians and to combinatorists i give a pedagogical introduction to all these problems and then sketch two recent results a construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q plane except possibly for the disc q 1 1 b proof of a universal upper bound on the q plane zeros of the chromatic polynomial or antiferromagnetic potts model partition function in terms of the graph s maximum degree

dynamical behavior of the multibondic and multicanonic algorithm in the 3d q state potts model we investigate the dynamical behavior of the recently proposed multibondic cluster monte carlo algorithm in applications to the three dimensional q state potts models with q 3 4 and 5 in the vicinity of their first order phase transition points for comparison we also report simulations with the standard multicanonical algorithm similar to the findings in two dimensions we show that for the multibondic cluster algorithm the dependence of the autocorrelation time on the system size vis well described by the power law v and that the dynamical exponent is consistent with the optimal random walk estimate 1 for the multicanonical simulations we obtain as expected a larger value of 1 2

overcoming the slowing down of flat histogram monte carlo simulations cluster updates and optimized broad histogram ensembles we study the performance of monte carlo simulations that sample a broad histogram in energy by determining the mean first passage time to span the entire energy space of d dimensional ferromagnetic ising potts models we first show that flat histogram monte carlo methods with single spin flip updates such as the wang landau algorithm or the multicanonical method perform suboptimally in comparison to an unbiased markovian random walk in energy space for the d 1 2 3 ising model the mean first passage time ensuremath tau scales with the number of spins n l d as ensuremath tau ensuremath propto n 2 l z the exponent z is found to decrease as the dimensionality d is increased in the mean field limit of infinite dimensions we find that z vanishes up to logarithmic corrections we then demonstrate how the slowdown characterized by zg0 for finite d can be overcome by two complementary approaches char22 cluster dynamics in connection with wang landau sampling and the recently developed ensemble optimization technique both approaches are found to improve the random walk in energy space so that ensuremath tau ensuremath propto n 2 up to logarithmic corrections for the d 1 2 ising model

partition function zeros of the q state potts model for non integer q abstract the distribution of the zeros of the partition function in the complex temperature plane fisher zeros of the two dimensional q state potts model is studied for non integer q on l l self dual lattices studied l 8 no fisher zero lies on the unit circle p0 ei in the complex p e j 1 q plane for q 1 and the number of such zeros increases with increasing q the ferromagnetic and antiferromagnetic properties of the potts model are investigated using the distribution of the fisher zeros for the potts ferromagnet we verify the den nijs formula for the thermal exponent yt for the potts antiferromagnet we also verify the baxter conjecture for the critical temperature and present new results for the thermal exponents in the range 0

complex q zeros of the partition function of the potts model with long range interactions abstract the zeros of the partition function of the ferromagnetic q state potts model with long range interactions in the complex q plane are studied in the mean field case while preliminary numerical results are reported for the finite 1 d chains with power law decaying interactions in both cases at any fixed temperature the zeros lie on the arc shaped contours which cross the positive real axis at the value for which the given temperature is transition temperature for finite number of spins the positive real axis is free of zeros which approach to it in the thermodynamic limit the convergence exponent of the zero closest to the positive real q axis is found to have the same value as the temperature critical exponent 1

fractal structure of spin clusters and domain walls in the two dimensional ising model the fractal structure of spin clusters and their boundaries in the critical two dimensional ising model is investigated numerically the fractal dimensions of these geometrical objects are estimated by means of monte carlo simulations on relatively small lattices through standard finite size scaling the obtained results are in excellent agreement with theoretical predictions and partly provide significant improvements in precision over existing numerical estimates

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

the correlation length of the potts model at the first order transition point the authors consider the q state two dimensional potts model for q g i e in the first order phase transition regime following a scheme given by p martin 1991 the authors prove an identity between the spectra of the transfer matrices of the potts model and the transfer matrices connecting the diagonals of a 6 vertex model by using a bethe ansatz for the latter they obtain an exact expression for the correlation length of the potts model at the transition point

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

the singular linear preservers of non singular matrices given an arbitrary field k we reduce the determination of the singular endomorphisms f of mn k such that f gln k gln k to the classification of n dimensional division algebras over k our method which is based upon dieudonne s theorem on singular subspaces of mn k also yields a proof for the classical non singular case

invertibility preservers on central simple algebras abstract we solve kaplansky s problem concerning the structure of linear preservers of invertibility in the special case of maps on central simple algebras

optimized energy calculation in lattice systems with long range interactions we discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long range interactions although since the introduction of the luijten blote algorithm monte carlo simulations of these systems no longer pose a fundamental problem the energy calculation is still an o n 2 problem for systems of size n we show how this can be reduced to an o n log n problem with a break even point that is already reached for very small systems this allows the study of a variety of until now hardly accessible physical aspects of these systems in particular we combine the optimized energy calcula tion with histogram interpolation methods to investigate the specific heat of the ising model and the first order regime of the three state potts model with long range interactions

transfer matrices and partition function zeros for antiferromagnetic potts models i general theory and square lattice chromatic polynomial we study the chromatic polynomials zero temperature antiferromagnetic potts model partition functions p g q for m n rectangular subsets of the square lattice with m 8 free or periodic transverse boundary conditions and n arbitrary free longitudinal boundary conditions using a transfer matrix in the fortuin kasteleyn representation in particular we extract the limiting curves of partition function zeros when n which arise from the crossing in modulus of dominant eigenvalues beraha kahane weiss theorem we also provide evidence that the beraha numbers b2 b3 b4 b5 are limiting points of partition function zeros as n whenever the strip width m is 7 periodic transverse b c or 8 free transverse b c along the way we prove that a noninteger beraha number except perhaps b10 cannot be a chromatic root of any graph

first order phase transition in potts models with finite range interactions we consider the q state potts model on z d q 3 d 2 with kac ferromagnetic interactions and scaling parameter we prove the existence of a first order phase transition for large but finite potential ranges more precisely we prove that for small enough there is a value of the temperature at which coexist q 1 gibbs states the proof is obtained by a perturbation around mean field using pirogov sinai theory the result is valid in particular for d 2 q 3 in contrast with the case of nearest neighbor interactions for which available results indicate a second order phase transition putting both results together provides an example of a system which undergoes a transition from second to first order phase transition by changing only the finite range of the interaction

determination of the order of phase transitions in potts model by the graph weight approach we examine the order of the phase transition in the potts model by using the graph representation for the partition function which allows treating a non integer number of potts states the order of transition is determined by the analysis of the shape of the graph weight probability distribution the approach is illustrated on special cases of the one dimensional potts model with long range interactions and on its mean field limit