jeffreys prior analysis of the simultaneous equations model in the case with n 1 endogenous variables this paper analyzes the behavior of posterior distributions under the jereys prior in a simul taneous equations model the case under study is that of a general limited information setup with n 1 endogenous variables the jereys prior is shown to give rise to a marginal pos terior density which has cauchy like tails similar to that exhibited by the exactnite sample distribution of the corresponding liml estimator a stronger correspondence is established in the special case of a just identied orthonormal canonical model where the posterior density under the jereys prior is shown to have the same functional form as the density of thenite sample distribution of the liml estimator the work here generalizes that of chao and phillips j econ 87 1998 49 which gave analogous results for the special case of an equation with two endogenous variables c 2002 elsevier science b v all rights reserved jel classication c11 c31

rounding effects of quenched randomness on first order phase transitions frozen in disorder in an otherwise homogeneous system is modeled by interaction terms with random coefficients given by independent random variables with a translation invariant distribution for such systems it is proven that ind 2 dimensions there can be no first order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter discontinuities which would amount to a continuous symmetry breaking in systems which are stochastically invariant under the action of a continuous subgroup ofo n are suppressed by the randomness in dimensionsd 4 specific implications are found in the random field ising model for which we conclude that ind 2 dimensions at all h the gibbs state is unique for almost all field configurations and in the random bond potts model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point the results are explained by the argument of imry and ma 1 the proofs involve the analysis of fluctuations of free energy differences which are shown using martingale techniques to be gaussian on the suitable scale

the order parameter in a spin glass various possible precise definitions of an edwards anderson type of order parameter for an ising model spin glass are considered using boundary conditions for a finite system states of an infinite system and a duplicate system approach several of these definitions are shown to yield identical results

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

magnetic critical behavior of two dimensional random bond potts ferromagnets in confined geometries we present a numerical study of two dimensional random bond potts ferromagnets the model is studied both below and above the critical value qc 4 which discriminates between second and first order transitions in the pure system two geometries are considered namely cylinders and square shaped systems and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid even in the randomness induced second order phase transition regime q 4 in the cylinder geometry connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point the scaling dimensions then follow from the exponential decay of correlations along the strip monte carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice but the data are then perturbed by strong surface effects the conformal mapping of a semi infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions the techniques are applied to different values of q in the range 3 64

an intermediate phase with slow decay of correlations in one dimensional 1 x y 2 percolation ising and potts models we rigorously establish the existence of an intermediate ordered phase in one dimensional 1 x y 2 percolation ising and potts models the ising model truncated two point function has a power law decay exponent which ranges from its low and high temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary similar results are obtained for percolation and potts models

weak randomness for large q state potts models in two dimensions we have studied the effect of weak randomness on q state potts models for q 4 by measuring the central charges of these models using transfer matrix methods we obtain a set of new values for the central charges and then show that some of these values are related to one another by a factorization law

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

floquet spectrum for two level systems in quasiperiodic time dependent fields we study the time evolution ofn level quantum systems under quasiperiodic time dependent perturbations the problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged hilbert space or equivalently of a generalized floquet operator we discuss criteria for the appearance of pure point as well as continuous spectrum corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior we discuss two types of mechanisms that lead to instability the first one is due to near resonances while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation we treat explicitly an example of this type the stability of the pure point spectrum under small perturbations is proven using kam techniques

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

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

measurement error modelling with an approximate instrumental variable summary consider using regression modelling to relate an exposure predictor variable to a disease outcome response variable if the exposure variable is measured with error but this error is ignored in the analysis then misleading inferences can result this problem is well known and has spawned a large literature on methods which adjust for measurement error in predictor variables one theme is that the requisite assumptions about the nature of the measurement error can be stronger than what is actually known in many practical situations in particular the assumptions that are required to yield a model which is formally identified from the observable data can be quite strong the paper deals with one particular strategy for measurement error modelling namely that of seeking an instrumental variable i e a covariate s which is associated with exposure and conditionally independent of the outcome given exposure if these two conditions hold exactly then we call s an exact instrumental variable and an identified model results however the second is not checkable empirically since the actual exposure is unobserved in practice then investigators typically seek a covariate which is plausibly thought to satisfy it we study inferences which acknowledge the approximate nature of this assumption in particular we consider bayesian inference with a prior distribution that posits that s is probably close to conditionally independent of outcome given exposure we refer to this as an approximate instrumental variable assumption although the approximate instrumental variable assumption is more realistic for most applications concern arises that a non identified model may result thus the paper contrasts inferences arising from the approximate instrumental variable assumption with their exact instrumental variable counterparts with particular emphasis on the benefit of basing inferences on a more realistic model versus the cost of basing inferences on a non identified model

natural conjugate priors for the instrumental variables regression model applied to the angrist krueger data we propose a natural conjugate prior for the instrumental variables regression model the prior is a natural conjugate one since the marginal prior and posterior of the structural parameter have the same functional expressions which directly reveal the update from prior to posterior the jeffreys prior results from a specific setting of the prior parameters and results in a marginal posterior of the structural parameter that has an identical functional form as the sampling density of the limited information maximum likelihood estimator we construct informative priors for the angrist krueger 1991 does compulsory school attendance affect schooling and earnings quarterly journal of economics 106 979 1014 data and show that the marginal posterior of the return on education in the us coincides with the marginal posterior from the southern region when we use the jeffreys prior this result occurs since the instruments are the strongest in the southern region and the posterior using the jeffreys prior identical to maximum likelihood focusses on the strongest available instruments we construct informative priors for the other regions that make their posteriors of the return on education similar to that of the us and the southern region these priors show the amount of prior information needed to obtain comparable results for all regions

dynamical localization for two level systems periodically driven here we consider a two level system driven by an external periodic field we show that the coherent destruction of tunnelling as proved by grossmann and co workers 1991 phys rev lett 67 516 1992 europhys lett 18 571 in the case of a monochromatic field also appears for any periodic driving field given by an even regular function with zero mean value and satisfying a technical condition on the zeros of this function

some exact distribution theory for maximum likelihood estimators of cointegrating coefficients in error correction models the author derives some exact finite sample disbibutions and characterizes the tail behavior of maximum likelihood estimators of the cointegrating coefficients in error correction models the reduced rank regression estimator has a distribution with cauchy like tails and no finite moments of integer order the maximum likelihood estimator of the coefficients in a particular triangular system representation has matrix t distribution tails with finite integer moments to order t n r where t is the sample size n is the total number of variables and r is the dimension of cointegration space this helps explain some recent simulation studies where extreme outliers occur more frequently for the reduced rank regression estimator than for alternative asymptotically efficient procedures based on triangular representation copyright 1994 by the econometric society

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

monte carlo study of the triangular blume capel model under bond randomness the effects of bond randomness on the universality aspects of a two dimensional d 2 blume capel model embedded in the triangular lattice are discussed the system is studied numerically in both its first and second order phase transition regimes by a comprehensive finite size scaling analysis for a particularly suitable value of the disorder strength we find that our data for the second order phase transition emerging under random bonds from the second order regime of the pure model are compatible with the universality class of the two dimensional 2d random ising model furthermore we find evidence that the second order transition emerging under bond randomness from the first order regime of the pure model belongs again to the same universality class although the first finding reinforces the scenario of strong universality in the 2d ising model with quenched disorder the second is in difference from the critical behavior emerging under randomness in the cases of the ex first order transitions of the potts model finally our results verify previous renormalization group calculations on the blume capel model with disorder in the crystal field coupling

invaded cluster algorithm for potts models the invaded cluster algorithm a method for simulating phase transitions is described in detail theoretical albeit nonrigorous justification of the method is presented and the algorithm is applied to potts models in two and three dimensions the algorithm is shown to be useful for both first order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities the dynamic properties of the invaded cluster algorithm are studied numerical evidence suggests that the algorithm has no critical slowing for ising models s1063 651x 96 09208 2

limited information bayesian analysis of a simultaneous equation with an autocorrelated error term and its application to the u s gasoline market using markov chain monte carlo algorithms within the limited information bayesian framework we estimate the parameters of the structural equation of interest and test weak exogeneity in a simultaneous equation model with white noise as well as autocorrelated error terms a numerical example and an estimation of the supply and demand equations of the u s gasoline market show that if we ignore autocorrelation we obtain unreasonable posterior distributions of the parameters of interest also we find that the hypothesis of the asymmetric effect of the changes in oil price on the changes in gasoline price is rejected oil inventory has a significant negative effect on the gasoline price

high order harmonic generation and dynamic localization in a driven two level system a non perturbative solution using the floquet green formalism non perturbative exact analytical results are obtained for the harmonically driven two level system with the use of a floquet green operator formalism from this operator we find the quasi energies and the fourier components of the floquet eigenstates which we use to construct the time evolution operator of the system as an application of our formalism we study dynamic localization and high order harmonic generation in this system our results show the existence of an initial condition that leads to an emission spectrum with only hyper raman lines for any frequency and strength of the driving field we show the important role of the initial condition in determining the emission spectrum of this system in the non perturbative regime the emission spectrum is found to be in general very sensitive to the amplitude of the driving field we also derive an expression for the coherent emission spectrum which shows its dependence on the degree of localization of the system

a universal amplitude ratio for the q 4 potts model from a solvable lattice model the universal amplitude ratio r for the q 4 state potts model in two dimensions is determined by using results for the dilute a model in regime 1 the nature of the relationship between the potts model and the dilute a model is discussed

models of two level atoms in quasiperiodic external fields we prove that the spectrum of the generalized quasienergy operator of a plane polarized two level atom in a strong external quasiperiodic electromagnetic field with nonzero constant fourier component is pure point under diophantine conditions on the frequency ratio and excluding a small subset of resonant values the widespread belief that there may be only pure point spectrum in such models is briefly discussed in section 2 and the circularly polarized case a well known soluble model is revisited from the point of view of the quasienergy operator

bayesian and classical approaches to instrumental variable regression we establish relationships between certain bayesian and classical approaches to instrumental variable regression we determine the form of priors that lead to posteriors for structural parameters that have similar properties as classical 2sls and liml and in doing so provide some new insight especially in the context of weak instruments to the small sample behavior of bayesian and classical procedures using a reduced rank restriction on a multivariate linear model we determine the priors that give rise to posteriors that are identical in functional form to the sampling densities of the classical 2sls and liml estimators

posterior distributions in limited information analysis of the simultaneous equations model using the jeffreys prior abstract this paper studies the use of the jeffreys prior in bayesian analysis of the simultaneous equations model sem exact representations are obtained for the posterior density of the structural coefficient in canonical sems with two endogenous variables for the general case with m endogenous variables and an unknown covariance matrix the laplace approximation is used to derive an analytic formula for the same posterior density both the exact and the approximate formulas we derive are found to exhibit cauchy like tails analogous to comparable results in the classical literature on liml estimation moreover in the special case of a two equation just identified sem in canonical form the posterior density of is shown to have the same infinite series representation as the density of the finite sample distribution of the corresponding liml estimator this paper also examines the occurrence first documented in kleibergen and van dijk 1994a of a nonintegrable asymptotic cusp in the posterior distribution of the coefficient matrix of the reduced form equations for the included endogenous regressors an explanation for this phenomenon is provided in terms of the jacobian of the mapping from the structural model to the reduced form this interpretation assists in understanding the success of the jeffreys prior in resolving this problem

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

large q asymptotics of the random bond potts model we numerically examine the large q asymptotics of the q state random bond potts model special attention is paid to the parametrisation of the critical line which is determined by combining the loop representation of the transfer matrix with zamolodchikov s c theorem asymptotically the central charge seems to behave like c q 1 2 log 2 q o 1 very accurate values of the bulk magnetic exponent x 1 are then extracted by performing monte carlo simulations directly at the critical point as q infinity these seem to tend to a non trivial limit x 1 0 192 0 002

high temperature series analysis of the free energy and susceptibility of the 2d random bond ising model abstract we derive high temperature series expansions for the free energy and susceptibility of the two dimensional random bond ising model with a symmetric bimodal distribution of two positive coupling strengths j1 and j2 and study the influence of the quenched random bond disorder on the critical behavior of the model by analysing the series expansions over a wide range of coupling ratios j2 j1 covering the crossover from weak to strong disorder we obtain for the susceptibility with two different methods compelling evidence for a singularity of the form t 7 4 ln t 7 8 as predicted theoretically by shalaev shankar and ludwig for the specific heat our results are less convincing but still compatible with the theoretically predicted log log singularity

quenched bond dilution in two dimensional potts models we report a numerical study of the bond diluted two dimensional potts model using transfer matrix calculations for different numbers of states per spin we show that the critical exponents at the random fixed point are the same as in self dual random bond cases in addition we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin spin correlation function in the cylinder geometry we show that the behaviour is fully compatible with the one observed in the random bond case confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder dilution self dual binary random bond self dual continuous random bond

noninformative priors for one parameter of many summary we consider the problem of constructing a prior that is noninformative for a single parameter in the presence of nuisance parameters our approach is to require that the resulting marginal posterior intervals have accurate frequentist coverage stein 1985 derived nonrigorously a sufficient condition for such a prior through the use of orthogonal parameters we give a general form for the class of priors satisfying stein s condition the priors are proportional to the square root of the information element for the parameter of interest times an arbitrary function of the nuisance parameters this is in contrast to jeffreys 1946 invariant prior for the overall parameter which is proportional to the square root of the determinant of the information matrix several examples are given and comparisons are made to the reference priors of bernardo 1979

on the unicity of discontinuous transitions in the two dimensional potts and ashkin teller models we investigate the uniqueness of the discontinuous phase transition of the q state potts model and the symmetric q s s cubic model generalized ashkin teller model on the integer square lattice with q 1 for the ferromagnetic cases when there is a discontinuous phase transition we show that the discontinuity can only occur at a point of self duality

finite size scaling study of two dimensional dilute potts models the author uses transfer matrices to calculate the free energy and various thermodynamic quantities of the two and three state potts models with random bond interactions at the critical temperature he verifies the harris criterion for the three state potts model and estimates the new random critical exponents for the ising case the results are consistent with randomness being irrelevant for the critical behaviour

self averaging distribution of pseudocritical temperatures and finite size scaling in critical disordered systems we evaluate by monte carlo simulations various singular thermodynamic quantities x for ensembles of quenched random ising and ashkin teller models the measurements are taken at t c and we study how the distributions p x and in particular their relative squared width r x over the ensemble depend on the system size l the ashkin teller model was studied in the regime where bond randomness is irrelevant and we found weak self averaging r x ensuremath sim l ensuremath alpha ensuremath nu ensuremath rightarrow 0 where ensuremath alpha l0 and ensuremath nu are the exponents of the pure model fixed point governing the transition for the site dilute ising model on a cubic lattice known to be governed by a random fixed point we find that r x tends to a constant as predicted by aharony and harris we tested whether this constant is universal however this constant is different for canonical and grand canonical disorder we identify the pseudocritical temperature of each sample i t c i l as the temperature at which the susceptibility reaches its maximal value ensuremath chi mathrm max the distribution of these sample dependent t c i l was investigated we found that its variance scales as ensuremath delta t c l 2 ensuremath sim l ensuremath 2 ensuremath nu our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold we did observe deviations from a single function which imply that sample dependent scaling functions are needed these deviations are however relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ensuremath chi mathrm max than the commonly used ensuremath chi t c ensuremath infty

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

analytical and non analytical corrections to finite size scaling using conformal invariance the author calculates finite size corrections to the scaled spectra of one dimensional quantum chains at the critical point an explicit application to the ising and three state potts model gives good agreement with analytical and numerical results respectively

percolation phenomena in low and high density systems we consider the 2d quenched disordered q state potts ferromagnets and show that in the translation invariant measure averaged over the disorder at self dual points any amalgamation of q 1 species will fail to percolate despite an overall high density of 1 q 1 further in the dilute bond version of these systems if the system is just above threshold then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density finally we demonstrate both phenomena in a single model by considering a perturbation of the dilute model that has a self dual point we also demonstrate that these phenomena occur by a similar mechanism in a simple coloring model invented by o haggstrom

quenched randomness at first order transitions abstract a rigorous theorem due to aizenman and wehr asserts that there can be no latent heat in a two dimensional system with quenched random impurities we examine this result and its possible extensions to higher dimensions in the context of several models for systems whose pure versions undergo a strong first order transition we show that there is an asymptotically exact mapping to the random field ising model at the level of the interface between the ordered and disordered phases this provides a physical explanation for the above result and also implies a correspondence between the problems in higher dimensions including scaling relations between their exponents the particular example of the q state potts model in two dimensions has been considered in detail by various authors and we review the numerical results obtained for this case turning to weak fluctuation driven first order transitions we describe analytic renormalization group calculations which show how the continuous nature of the transition is restored by randomness in two dimensions

dynamical localization criterion for driven two level systems in this paper we consider a two level system under the effect of an external time dependent field we give a precise criterion for dynamical localization we then apply our result to the cases of external ac dc and bichromatic fields

perturbative analysis of dynamical localization in this paper we present a mathematical investigation of the phenomenon of dynamical localization in a class of quasi periodically and periodically time dependent two level systems our results are based on a sort of renormalization procedure which is developed here in a systematic way in order to adapt it to the case of dynamical localization in the quasi periodic case this procedure leads to a formal perturbative expansion free of secular terms in the periodic case a convergent perturbative solution is obtained and in particular a convergent perturbative expansion for the secular frequency is presented the case of ac dc fields is discussed in some detail leading to the conclusion that the phenomenon of dynamical localization is not exact in that situation

bayesian intervals with good frequentist behaviour in the presence of nuisance parameters given a random sample from a distribution with density function that depends on an unknown parameter 1 p we are concerned with the problem of setting confidence intervals for a particular component of it say 1 treating the remaining components as a nuisance parameter adopting an objective bayesian approach we show that the bayes intervals with a certain conditional prior density on the parameter of interest 1 are confidence intervals as well having nearly the correct frequency of coverage the frequentist performance of the proposed intervals is tested in a simulation study for gamma mean and shape parameters

two level systems driven by a stochastic perturbation here we consider a two level system driven by an external harmonic field whose amplitude is perturbed by a white noise term in the limit of small splitting dynamical localization i e coherent destruction of tunneling is proved for times of the order of 1 e where e is the two level splitting the same type of localization is proved if the driving field is simply the white noise

tests of conformal invariance in randomness induced second order phase transitions laboratoire de physique des mat eriaux universit e henri poincar e nancy 1 bo ite postale 239 f 54506 vand uvre les nancy cedex france received 6 july 1998 revised manuscript 29 september 1998 the conformal covariance of correlation functions is checked in the second order transition in duced by random bonds in the two dimensional eight state potts model the decay of correlationsis obtained via transfer matrix calculations in a cylinder geometry and large scale monte carlosimulations provide access to the correlations and the pro les inside a square with free or xedboundary conditions in both geometries conformal transformations constrain the form of the spa tial dependence leading to accurate determinations of the order parameter scaling index in goodagreement with previous independent determinations obtained through standard techniques theenergy density exponent is also computed pacs numbers 64 60 cn 05 50 q 05 70 jk 64 60 fr

jeffreys prior for logit models abstract this paper investigates jeffreys prior for logit models with covariates in conditional logit models it is not recommended and in multinomial logit models it has properties similar to a neutral natural conjugate prior but none of the computational advantages