antiferromagnetic potts model in a magnetic field a finite size scaling study abstract the finite size scaling method is used to investigate the phase diagrams of the q state antiferromagnetic potts model in an external magnetic field on a square lattice critical lines where the system undergoes second order phase transitions between ordered and disordered phases are found which are in complete qualitative agreement with known results for q 3 also new unexpected features are found for q 4 where the transition vanishes in the negative field region while in the positive field region as q increases the critical line flattens out with increasing slopes both at the t 0 h h c point where h c is the critical field at zero temperature and at the t 0 h 0 point thermal exponents y t are also calculated for arbitrary q along the critical lines

phase diagrams of the q state potts model on bethe lattices abstract we use a rigorous recently developed method of solution of spin models on bethe lattices cayley trees it allows us to detect all the phases of the ferromagnetic and antiferromagnetic q state potts models in a field fpm and apm respectively the phase diagrams of the fpm are characterized by three critical lines of first order transitions in the apm the ordered phases which appear always for h 0 and under certain conditions for h 0 and or h

existence of twisted constant scalar curvature kahler metrics with a large twist suppose that there exist two kahler metrics omega and alpha such that the metric contraction of alpha with respect to omega is constant i e varlambda omega alpha text const we prove that for all large enough r 0 there exists a twisted constant scalar curvature kahler metric omega in the cohomology class omega satisfying s omega r varlambda omega alpha text const we discuss its implication to k stability and the continuity method recently proposed by chen

on the convergence and singularities of the j flow with applications to the mabuchi energy the j flow of s k donaldson and x x chen is a parabolic flow on kahler manifolds with two kahler metrics it is the gradient flow of the j functional that appears in chen s formula for the mabuchi energy we find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the j flow to a critical metric we use this result to show that on manifolds with ample canonical bundle the mabuchi energy is proper on all kahler classes in an open neighborhood of the canonical class defined by a positivity condition this improves previous results of chen and of the second author we discuss the implications of this for the problem of the existence of constant scalar curvature kahler metrics we also study the singularities of the j flow and under certain conditions which always hold for dimension 2 derive some estimates away from a subvariety we discuss the conjectural remark of donaldson that if the j flow does not converge on a kahler surface then it should blow up over some curves of negative self intersection 2007 wiley periodicals inc

critical temperature of an anisotropic two dimensional potts model the phase transition temperature tc of the q state potts model with ferromagnetic interactions jx and antiferromagnetic ones jy in the two respective directions of a 2d lattice is calculated by the high temperature series expansion htse of the appropriate staggered susceptibility as well a by the cluster variate approximation cva for q 3 the htse result in the range of mod jy mod jx or approximately 1 5 shows a good coincidence with the kosterlitz thouless type phase transition temperature calculated on the prescription by ostlund which is given by 1 16 pi exp 2 beta cjx coth beta cjy 2 beta 1 ktc the cva result shows similar behaviour of tc to that of the migdal kadanoff renormalisation group method obtained by kinzel et al but it is considered to be less reliable than the htse result

a potts model with infinitely degenerate ground state a q component potts model with ferromagnetic and antiferromagnetic interactions in the two respective directions of a square lattice is considered an argument is given showing that an order phase can exist in this model even though the ground state is disordered and infinitely degenerate the authors use the migdal kadanoff transformation to obtain a closed form expression for its critical point they also carry out a monte carlo simulation of the model for q 3 the specific heat exhibits a broad maximum which does not sharpen appreciably as the lattice size is increased this suggests that the phase transition if it exists is of an unconventional type

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

convexity of the k energy on the space of kahler metrics and uniqueness of extremal metrics we establish the convexity of mabuchi s k energy functional along weak geodesics in the space of kahler potentials on a compact kahler manifold thus confirming a conjecture of chen and give some applications in kahler geometry including a proof of the uniqueness of constant scalar curvature metrics or more generally extremal metrics modulo automorphisms the key ingredient is a new local positivity property of weak solutions to the homogeneous monge ampere equation on a product domain whose proof uses plurisubharmonic variation of bergman kernels

twisted constant scalar curvature kahler metrics and kahler slope stability on a compact kahler manifold we introduce a cohomological obstruction to the solvability of the constant scalar curvature csck equation twisted by a semipositive form appearing in works of fine and song tian as a special case we find an obstruction for a manifold to be the base of a holomorphic submersion carrying a csck metric in certain adiabatic classes we apply this to find new examples of general type threefolds with classes which do not admit a csck representative when the twist vanishes our obstruction extends the slope stability of ross thomas to effective divisors on a kahler manifold thus we find examples of non projective slope unstable manifolds

filtrations and test configurations we introduce a strengthening of k stability based on filtrations of the homogeneous coordinate ring this allows for considering certain limits of families of test configurations which arise naturally in several settings we prove that if a manifold with no automorphisms admits a csck metric then it satisfies this stronger stability notion we also discuss the relation with the birational transformations in the definition of b stability

phase diagrams of the blume emery griffiths model real space renormalization group investigation and finite size scaling analysis abstract an approximate real space renormalization group scheme based on the migdal kadanoff recursion relations and the finite size scaling method are used to study the phase diagrams of the two dimensional spin 1 ising model with nearest neighbor interactions both bilinear and biquadratic and with a crystal field interaction contrary to previous studies we clarify the behavior of the model at low temperature including the region j k

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

convergence of the j flow on toric manifolds we show that on a kahler manifold whether the j flow converges or not is independent of the chosen background metric in its kahler class on toric manifolds we give a numerical characterization of when the j flow converges verifying a conjecture in 19 m lejmi and g szekelyhidi the j flow and stability in this case we also strengthen existing results on more general inverse sigma k equations on kahler manifolds

construction of the l fuzzy concept lattice we propose two processes to obtain l fuzzy concepts based on finite l fuzzy contexts and the theory of cousot and cousot pacific j math 82 1979 43 the first algorithm calculates the l fuzzy concepts derived from an l fuzzy set and the second one constructs the whole l fuzzy concept lattice we also represent graphically the l fuzzy concept lattice

the kahler ricci flow on surfaces of positive kodaira dimension 4 estimates 9 4 1 the zeroth order and volume estimates 9 4 2 a partial second order estimate 13 4 3 gradient estimates 14 4 4 the second order estimates 18

critical behaviour in three state potts antiferromagnets on a bethe lattice abstract a rigorous method recently introduced for the solution of spin models on bethe lattices is used to solve the three state antiferromagnetic potts model in an external field and to find its phase diagram critical lines where the system undergoes first order or second order phase transitions are found which are in complete qualitative agreement with recent results by racz and vicsek also new unexpected ordered phases are found

the j flow and stability we study the j flow from the point of view of an algebro geometric stability condition in terms of this we give a lower bound for the natural associated energy functional and we show that the blowup behavior found by fang and lai 11 is reflected by the optimal destabilizer finally we prove a general existence result on complex tori