correction induced by irrelevant operators in the correlators of the two dimensional ising model in a magnetic field we investigate the presence of irrelevant operators in the two dimensional ising model perturbed by a magnetic field by studying the corrections induced by these operators in the spin spin correlator of the model to this end we perform a set of high precision simulations for the correlator both along the axes and along the diagonal of the lattice by comparing the numerical results with the predictions of a perturbative expansion around the critical point we find unambiguous evidence of the presence of such irrelevant operators it turns out that among the irrelevant operators the one which gives the largest correction is the spin 4 operator t 2 2 which accounts for the breaking of the rotational invariance due to the lattice this result agrees with what was already known for the correlator evaluated exactly at the critical point and also with recent results obtained in the case of the thermal perturbation of the model

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

inhomogeneous ising chain in a transverse field finite size scaling and asymptotic conformal spectrum the quantum version of the hilhorst van leeuwen model i e an inhomogeneous lsing model in a transverse field with a marginal perturbation of the bulk interaction varying like i with the distance i from the surface is studied numerically on a finite chain with symmetric interactions using fermion techniques and finite size scaling at the bulk critical point the surface exponents for the energy and the magnetisation are respectively x 2a 2 xm a 4when a a andx a f oandx a when as a where a is the critical value of the perturbation amplitude below which the surface becomes ordered at the bulk critical point although the magnetic exponents can be translated from the known classical values the surface thermal exponents seem to be ne the excitation spectrum obtained in the continuum limit exhibits an asymptotic conformal behaviour an analytic calculation of the leading finite size corrections when a ac and a test for the consistency of the numerical results when a c a are given in the appendix

displacement rank of the drazin inverse in this paper we study the displacement rank of the drazin inverse both sylvester displacement and the generalized displacement are discussed we present upper bounds for the ranks of the displacements of the drazin inverse the general results are applied to the group inverse of a structured matrix such as close to toeplitz generalized cauchy toeplitz plus hankel and bezoutians

the representations of the drazin inverse of differences of two matrices abstract in this paper we give explicit representations of p q d p pq d and pq of two matrices p and q as a function of p q pd and qd under the conditions p3q qp and q3p pq

the three dimensional ising model revisited numerically several basic universal amplitude ratios are studied afresh for three dimensional nearest neighbor ising models in revising earlier work modern estimates of the critical temperature and exponents are used in conjunction with biased inhomogeneous differential approximants to extrapolate the longest available series expansions to find the critical amplitudes c for the susceptibility 1 for the correlation length 1 a for the specific heat c t and b for the spontaneous magnetization m0 we find c c 4 95 0 15 1 1 1 960 0 01 a a 0 523 0 009 a c b 0 0581 0 0010 while a 1 3 verifies hyperuniversality to within 0 8 a method for calculating amplitude ratios which allows for corrections to scaling yields estimates for c c and 1 1 in excellent agreement with those derived from the individual amplitudes finally explicit formulae are given for the numerical evaluation of t 1 t c t and m0 t over the full temperature range from criticality to t 0 and corresponding plots and convenient near critical representations are also presented

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

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

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

criticality of o n symmetric models in the presence of discrete gauge symmetries we investigate the critical properties of the three dimensional antiferromagnetic rp n 1 model which is characterized by a global o n symmetry and a discrete z 2 gauge symmetry we perform a field theoretical analysis using the landau ginzburg wilson lgw approach and a numerical monte carlo study the lgw field theoretical results are obtained by high order perturbative analyses of the renormalization group flow of the most general 4 theory with the same global symmetry as the model assuming a gauge invariant order parameter field for n 4 no stable fixed point is found implying that any transition must necessarily be of first order this is contradicted by the numerical results that provide strong evidence for a continuous transition this suggests that gauge modes are not always irrelevant as assumed by the lgw approach but they may play an important role to determine the actual critical dynamics at the phase transition of o n symmetric models with a discrete z 2 gauge symmetry

some additive results on drazin inverses in this paper some additive results on drazin inverse of a sum of drazin invertible elements are derived some converse results are also presented

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

a note on the representations for the drazin inverse of 2 2 block matrices abstract in 1979 campbell and meyer proposed the problem of finding a formula for the drazin inverse of a 2 2 matrix m a b c d in terms of its various blocks where the blocks a and d are required to be square matrices special cases of the problems have been studied in particular a representation of the drazin inverse of m denoted by md has recently been obtained under the assumptions that c i aad b o and a i aad b o together with the condition that the generalized schur complement d cadb be either nonsingular or zero we derive an alternative representation for md under the same assumptions but with the condition on the schur complement in the hypothesis replaced by the condition that r caa d n b n d where r and n are the range and null space of a matrix

the susceptibility of the square lattice ising model new developments we have made substantial advances in elucidating the properties of the susceptibility of the square lattice ising model we discuss its analyticity properties certain closed form expressions for subsets of the coefficients and give an algorithm of complexity o n6 to determine its first n coefficients as a result we have generated and analyzed series with more than 300 terms in both the high and low temperature regime we quantify the effect of irrelevant variables to the scaling amplitude functions in particular we find and quantify the breakdown of simple scaling in the absence of irrelevant scaling fields arising first at order t tc 9 4 though high low temperature symmetry is still preserved at terms of order t tc 17 4 and beyond this symmetry is no longer present the short distance terms are shown to have the form t tc p log t tc q with p q2 conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments

a monte carlo study of leading order scaling corrections of phi 4 theory on a three dimensional lattice we present a monte carlo study of the one component 4 model on a cubic lattice in three dimensions leading order scaling corrections are studied using the finite size scaling method we compute the corrections to scaling exponent with high precision we determine the value of the coupling at which leading order corrections to scaling vanish using this result we obtain estimates for critical exponents that are more precise than those obtained with field theoretic methods

critical behaviour at an edge the critical behaviour of a magnetic system with o n spin symmetry bounded by two d 1 dimensional hyperplanes meeting at an angle alpha is studied within mean field theory and in d 4 epsilon dimensions new exponents emerge for correlation functions and magnetisation and susceptibilities for spins close to the edge they can be expressed in terms of known bulk and surface exponents together with a single new edge exponent which depends however on the angle alpha this exponent is computed to first order in epsilon

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

improved high temperature expansion and critical equation of state of three dimensional ising like systems high temperature series are computed for a generalized three dimensional ising model with arbitrary potential three specific improved potentials suppressing leading scaling corrections are selected by monte carlo computation critical exponents are extracted from high temperature series specialized to improved potentials achieving high accuracy our best estimates are gamma 1 2371 4 nu 0 630 02 23 alpha 0 1099 7 eta 0 0364 4 beta 0 326 48 18 by the same technique the coefficients of the small field expansion for the effective potential helmholtz free energy are computed these results are applied to the construction of parametric representations of the critical equation of state a systematic approximation scheme based on a global stationarity condition is introduced the lowest order approximation reproduces the linear parametric model this scheme is used for an accurate determination of universal ratios of amplitudes a comparison with other theoretical and experimental determinations of universal quantities is presented

magnetization at corners in two dimensional ising models we investigate the corner spin magnetization of two dimensional ferromagnetic ising models in various wedge geometries results are obtained for triangular and square lattices by numerical studies on finite wedges using the star triangle transformation as well as analytic calculations using corner transfer matrices and a fermionic representation of the row to row transfer matrix the corner magnetizations vanish at the bulk critical temperature tc with an exponent c differing from the bulk exponent for isotropic systems with free edges we find that c is given simply by c 2 where is the angle at the corner for apex magnetizations of conical lattices we obtain the strikingly similar result a 4 these formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle eff determined by the relative strengths of the interactions

duality symmetry strong coupling expansion and universal critical amplitudes in two dimensional phi 4 field models we show that the exact beta function g in the continuous two dimensional g4 model possesses the kramers wannier duality symmetry the duality symmetry transformation d g such that d g d g g is constructed and the approximate values of g computed from the duality equation d g g are shown to agree with the available numerical results the calculation of the beta function g for the 2d scalar g4 field theory based on the strong coupling expansion is developed and the expansion of g in powers of g 1 is obtained up to order g 8 the numerical values calculated for the renormalized coupling constant g are in reasonably good agreement with the best modern estimates recently obtained from the high temperature series expansion and with those known from the perturbative four loop renormalization group calculations the application of cardy s theorem for calculating the renormalized isothermal coupling constant gc of the 2d ising model and the related universal critical amplitudes is also discussed

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

critical behaviour in parabolic geometries the authors study two dimensional systems with boundary curves described by power laws using conformal mappings they obtain the correlations at the bulk critical point three different classes of behaviour are found and explained by scaling arguments which also apply to higher dimensions for an ising system of parabolic shape the behaviour of the order at the tip is also found

gradient critical phenomena in the ising quantum chain we consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings in the direction of the gradient an interface region separates the ordered phase from the disordered one we develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region the scaling predictions are tested in the framework of the mean field ginzburg landau theory then we consider the ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two dimensional ising model the quantum hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator the energy density the magnetization profile and the two point correlation function are studied either analytically or by exact numerical calculations their scaling behaviour is in agreement with the predictions of the scaling theory

smoothly varying hopping rates in driven flow with exclusion we consider the one dimensional totally asymmetric simple exclusion process tasep with position dependent hopping rates the problem is solved in a mean field adiabatic approximation for a general smooth form of spatial rate variation numerical simulations of systems with hopping rates varying linearly against position constant rate gradient for both periodic and open boundary conditions provide detailed confirmation of theoretical predictions concerning steady state average density profiles and currents as well as open system phase boundaries to excellent numerical accuracy

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

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

representation of the drazin inverse for special block matrix abstract for a c n n the matrix x c n n is said to be the drazin inverse of a if it holds that xax x ax xa a r xa a r where r is the smallest nonnegative integer such that rank a r rank a r 1 in this paper we give the representation for the drazin inverse of the block matrix a a b 0 a b c n n a 2 a

a note on the drazin inverses with banachiewicz schur forms let h be a hilbert space m the closed subspace of h with orthocomplement m according to the orthogonal decomposition email protected m every operator email protected b h can be written in a block form m abcd in this note we give necessary and sufficient conditions for a partitioned operator matrix m to have the drazin inverse with banachiewicz schur form in addition this paper investigates the relations among the drazin inverse the moore penrose inverse and the group inverse when they can be expressed in the banachiewicz schur forms

formulas for the drazin inverse of special block matrices properties of the drazin inverse of the matrix f iie0 with e square are investigated based on this approach it is obtained an explicit formula for the drazin inverse of matrices of the form a ip tquv t where u v p and q are nxk the representation for a d is given in terms of kxk matrices involving the individual blocks under some conditions some special cases are also analyzed

renormalized four point coupling constant in the three dimensional o n model with n 0 we simulate self avoiding walks on a cubic lattice and determine the second virial coefficient for walks of different lengths this allows us to determine the critical value of the renormalized four point coupling constant in the three dimensional n vector universality class for n 0 we obtain g 1 4005 5 where g is normalized so that the three dimensional field theoretical beta function behaves as beta g g g 2 for small g as a byproduct we also obtain precise estimates of the interpenetration ratio psi psi 0 24685 11 and of the exponent nu nu 0 5876 2

universality and double critical end points a double critical end point in the two dimensional spin 3 2 blume capel model is studied via extensive monte carlo simulations the resultant scaling character of the probability distribution of the mixing scaling operators allows us to locate the double critical end point precisely and also to convincingly show that it indeed belongs to the same universality class as the critical points

bulk surface and hull fractal dimension of critical ising clusters in d 2 the authors present accurate numerical calculations of the fractal dimension d and surface dimension ds of the critical ising cluster in d 2 the results clearly support the values d 187 96 ds 5 6 which are consistent with ising clusters being described by tricritical q 1 potts model exponents from this the hull dimension dh of critical ising clusters is found to be dh 11 8 consistent with numerical work of other authors

two point correlation function of three dimensional o n models the critical limit and anisotropy in three dimensional o n models we investigate the low momentum behavior of the two point green s function g x in the critical region of the symmetric phase we consider physical systems whose criticality is characterized by a rotationally invariant fixed point several approaches are exploited such as strong coupling expansion of lattice n vector model and 1 n expansion field theoretical methods within the ensuremath varphi 4 continuum formulation non gaussian corrections to the universal low momentum behavior of g x are evaluated and found to be very small in nonrotationally invariant physical systems with o n invariant interactions the vanishing of the spatial anisotropy approaching the rotationally invariant fixed point is described by a critical exponent ensuremath rho which is universal and is related to the leading irrelevant operator breaking rotational invariance at n ensuremath infty one finds ensuremath rho 2 we show that for all values of ng 0 ensuremath rho ensuremath simeq 2

high precision monte carlo study of the 3d xy universality class we present a monte carlo study of the two component 4 model on the simple cubic lattice in three dimensions by suitable tuning of the coupling constant we eliminate leading order corrections to scaling high statistics simulations using finite size scaling techniques yield 0 6723 3 8 and 0 0381 2 2 where the statistical and systematical errors are given in the first and second bracket respectively these results are more precise than any previous theoretical estimate of the critical exponents for the 3d xy universality class

corner exponents in the two dimensional potts model the critical behavior at a corner in two dimensional ising and three state potts models is studied numerically on the square lattice using transfer operator techniques the local critical exponents for the magnetization and the energy density for various opening angles are deduced from finite size scaling results at the critical point for isotropic or anisotropic couplings the scaling dimensions compare quite well with the values expected from conformal invariance provided the opening angle is replaced by an effective one in anisotropic systems

a note on the drazin inverse of an anti triangular matrix abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m under some conditions moreover some particular cases of these results are also considered

confluent singularities in 3d continuum φ4 theory resolving critical point discrepancies a new analysis of the three dimensional 4 continuum model series for critical exponents and amplitude ratios is presented by explicitly allowing for a second correction to scaling term with exponent 2 and 2 1 2 0 a revision is found of previous critical point estimates that are all in the direction of resolving the remaining discrepancies between the 3d and 4 analyses the new estimates are also consistent with those obtained from high temperature series analyses of 3d lattice ising models for example the exponents 1 238 0 630 0 0355 and correction to scaling amplitude ratio 0 77 are easily accommodated and possibly even preferred detailed verification will require further terms both in the low order perturbation theory and the high order asymptotic expansions