correlation coefficients for random variables on a unit sphere or hypersphere summary a coefficient to measure association between two random variables each distributed on the surface of the unit sphere in p dimensional space is proposed analogous to an existing correlation coefficient for circular data the sample estimate is studied in detail for the case p 3 and comparisons are made with other statistics in current use some applications are given to studying remanent magnetization in rock samples

a decentred predictor for circular circular regression this paper considers the prediction of a angle given an x angle in a circular circular regression the predictor under study is a rotation of the decentred x angle defined as the direction of the sum of the unit vector corresponding to angle x plus a decentring vector the parameters are defined as those which maximise the average residual cosine this predictor is shown to perform well when the data come from a bivariate von mises distribution or from a wrapped bivariate normal model the large sample distributions of maximum cosine estimators are derived tests of fit for the rotational model which predicts by a rotation of x and for the independence model where is not related to x are presented the proposed methods are illustrated by the analysis of an earthquake dataset where the direction of ground movement is regressed on the direction of steepest descent

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

characterizing the manifest probabilities of latent trait models the problem of characterizing the manifest probabilities of a latent trait model is considered the item characteristic curve is transformed to the item passing odds curve and a corresponding transformation is made on the distribution of ability this results in a useful expression for the manifest probabilities of any latent trait model the result is then applied to give a characterization of the rasch model as a log linear model for a 2 j contingency table partial results are also obtained for other models the question of the identifiability of guessing parameters is also discussed

continuous and discrete latent structure models for item response data relations are examined between latent trait and latent class models for item response data conditions are given for the two latent class and two parameter normal ogive models to agree and relations between their item parameters are presented generalizationss are then made to continuous models with more than one latent trait and discrete models with more than two latent classes and methods are presented for relating latent class models to factor models for dichotomized variables results are illustrated using data from the law school admission test previously analyzed by several authors

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

some numerical results on the block spin transformation for the 2d ising model at the critical point we study the block spin transformation for the 2d ising model at the critical temperaturetc we consider the model with the constraint that the total spin in each block is zero an old argument by cassandro and gallavotti strongly supports the gibbsianness of the transformed measure provided that such model has a critical temperaturet c lower thantc after describing a possible rigorous approach to the problem we present numerical evidence that indeedt c tc and study the dobrushin shlosman uniqueness condition

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

application of log sobolov inequality to the stochastic dynamics of unbounded spin systems on the lattice abstract we consider a ferromagnetic lattice spin system with unbounded spins and investigate the relaxation property for the associated stochastic dynamics the glauber dynamics in the finite volume case we prove that the following two conditions are equivalent 1 the log sobolev inequality for the finite volume gibbs states holds uniformly in both the volume and the boundary condition 2 the finite volume glauber dynamics relaxes to equilibrium exponentially fast uniformly in the volume whenever it starts from a tempered configuration this can be considered as a complementary result to the ones previously obtained for infinite volume glauber dynamics by b zegarlinski 1996 comm math phys 175 401 432 our result can also be viewed as an extension of the equivalence theorem known for compact spin space settings

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

paradoxical results in multidimensional item response theory in multidimensional item response theory mirt it is possible for the estimate of a subject s ability in some dimension to decrease after they have answered a question correctly this paper investigates how and when this type of paradoxical result can occur we demonstrate that many response models and statistical estimates can produce paradoxical results and that in the popular class of linearly compensatory models maximum likelihood estimates are guaranteed to do so in light of these findings the appropriateness of multidimensional item response methods for assigning scores in high stakes testing is called into question

on the layering transition of an sos surface interacting with a wall ii the glauber dynamics we continue our study of the statistical mechanics of a 2d surface above a fixed wall and attracted towards it by means of a very weak positive magnetic fieldh in the solid on solid sos approximation when the inverse temperature is very large in particular we consider a glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions using the results proved in the first paper of this series we show that for allh h k 1 h k h k being the critical values of the magnetic field found in the previous paper the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions on the contrary forh h k and free boundary conditions we show that the gap in a cube of sidel is bounded from above and from below by a negative exponential ofl our results provide a strong indication that contrary to what happens in two dimensions for the three dimensional dynamical ising model in a finite cube at low temperature and very small positive external field with boundary conditions that are opposite to the field on one face of the cube and are absent free on the remaining faces the rate of exponential convergence to equilibrium which is positive in infinite volume may go to zero exponentially fast in the side of the cube

cluster expansion for d dimensional lattice systems and finite volume factorization properties we consider classical lattice systems with finite range interactions ind dimensions by means of a block decimation procedure we transform our original system into a polymer system whose activity is small provided a suitable factorization property of finite volume partition functions holds in this way we extend a result of olivieri

mixing times of critical two dimensional potts models we study dynamical aspects of the q state potts model on an n n box at its critical c q heat bath glauber dynamics and cluster dynamics such as swendsen wang that circumvent low temperature bottlenecks are all expected to undergo critical slowdowns in the presence of periodic boundary conditions the inverse spectral gap which in the subcritical regime is o 1 should at criticality be polynomial in n for 1 4 in accordance with the predicted discontinuous phase transition this was confirmed for q 2 the ising model by the second author and sly and for sufficiently large q by borgs et al here we show that the following holds for the critical potts model on the torus for q 3 the inverse gap of glauber dynamics is no 1 for q 4 it is at most no log n and for every q 4 in the phase coexistence regime the inverse gaps of both glauber dynamics and swendsen wang dynamics are exponential in n for free or monochromatic boundary conditions and large q we show that the dynamics at criticality is faster than on the torus unlike the ising model where free periodic boundary conditions induce similar dynamical behavior at all temperatures the inverse gap of swendsen wang dynamics is exp no 1 2017 wiley periodicals inc

logarithmic sobolev inequality for zero range dynamics we prove that the logarithmic sobolev constant for zero range processes in a box of diameter l grows as l 2

a correlation coefficient for circular data summary a coefficient to measure association between angular variables is discussed its asymptotic distribution found and its properties developed comparisons with other statistics in current use are made and some examples given

on the link between cognitive diagnostic models and knowledge space theory the present work explores the connections between cognitive diagnostic models cdm and knowledge space theory kst and shows that these two quite distinct approaches overlap it is proved that in fact the multiple strategy dina deterministic input noisy and gate model and the cblim a competence based extension of the basic local independence model blim are equivalent to demonstrate the benefits that arise from integrating the two theoretical perspectives it is shown that a fairly complete picture on the identifiability of these models emerges by combining results from both camps the impact of the results is illustrated by an empirical example and topics for further research are pointed out

marginal maximum likelihood estimation of item parameters application of an em algorithm maximum likelihood estimation of item parameters in the marginal distribution integrating over the distribution of ability becomes practical when computing procedures based on an em algorithm are used by characterizing the ability distribution empirically arbitrary assumptions about its form are avoided the em procedure is shown to apply to general item response models lacking simple sufficient statistics for ability this includes models with more than one latent dimension

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

complete analyticity of the 2d potts model above the critical temperature we investigate the complete analyticity ca of the two dimensional q state potts model for large values of q we are able to prove it for every temperature provided we restrict ourselves to nice subsets their niceness depending on the temperature t contrary to this restricted complete analyticity rca the full ca is known to fail for some values of the temperature above

the ising partition function zeros and deterministic approximation we study the problem of approximating the partition function of the ferromagnetic ising model with both pairwise as well as higher order interactions equivalently in graphs as well as hypergraphs our approach is based on the classical lee yang theory of phase transitions along with a new lee yang theorem for the ising model with higher order interactions and on an extension of ideas developed recently by barvinok and patel and regts that can be seen as an algorithmic realization of the lee yang theory our first result is a deterministic polynomial time approximation scheme an fptas for the partition function in bounded degree graphs that is valid over the entire range of parameters beta the interaction and lambda the external field except for the case left lambda right 1 the zero field case a polynomial time randomized approximation scheme fpras for all graphs and all beta lambda based on markov chain monte carlo simulation has long been known unlike most other deterministic approximation algorithms for problems in statistical physics and counting our algorithm does not rely on the decay of correlations property but as pointed out above on lee yang theory this approach extends to the more general setting of the ising model on hypergraphs of bounded degree and edge size where no previous algorithms even randomized were known for a wide range of parameters in order to achieve this latter extension we establish a tight version of the lee yang theorem for the ising model on hypergraphs improving a classical result of suzuki and fisher

finite volume glauber dynamics in a small magnetic field we consider glauber dynamics on a finite cube in d dimensional lattice d 2 which is associated with basic ising model at temperature t 1 1 under a magnetic field h 0 we prove that if the effective magnetic field is positive then the relaxation of the glauber dynamics in the uniform norm is exponentially fast uniformly over the size of underlying cube the result covers the case of the free boundary condition with arbitrarily small positive magnetic field this paper is a continuation of an attempt initiated earlier by schonmann and yoshida to shed more light on the relaxation of the finite volume glauber dynamics when the thermodynamic parameter h is so near the phase transition line h c h 0 that the dobrushin shlosman mixing condition is no longer available

a general structural equation model with dichotomous ordered categorical and continuous latent variable indicators a structural equation model is proposed with a generalized measurement part allowing for dichotomous and ordered categorical variables indicators in addition to continuous ones a computationally feasible three stage estimator is proposed for any combination of observed variable types this approach provides large sample chi square tests of fit and standard errors of estimates for situations not previously covered two multiple indicator modeling examples are given one is a simultaneous analysis of two groups with a structural equation model underlying skewed likert variables the second is a longitudinal model with a structural model for multivariate probit regressions

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

ill defined block spin transformations at arbitrarily high temperatures examples are presented of block spin transformations which map the gibbs measures of the ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non gibbsian measures in this way we provide the first example of this kind of pathology for very high temperatures and as a corollary also the first example of such a pathology happening at a critical point

isotropic probabilistic models for directions planes and referential systems although simple probabilistic models for directions in two dimensional 2d and three dimensional 3d spaces are well known respectively the von mises and fisher distributions probabilistic models for directions planes and referential systems in an n dimensional n d space are either not popular or not available the results for directions in 2d and 3d can be generalized for i arbitrary directions ii planes iii referential systems and iv symmetric random tensors and in turn all these results can be generalized to n d spaces in this paper we will focus on and build up isotropic probabilistic models for directions planes and referential systems in n d space

on the ergodic properties of glauber dynamics we show that if there is an infinite volume gibbs measure which satisfies a logarithmic sobolev inequality with local coefficients of moderate growth then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm

relaxation of disordered magnets in the griffiths regime we study the relaxation to equilibrium of discrete spin systems with random finite range not necessarily ferromagnetic interactions in the griffiths regime we prove that the speed of convergence to the unique reversible gibbs measure is almost surely faster than any stretched exponential at least if the probability distribution of the interaction decays faster than exponential e g gaussian furthermore if the interac tion is uniformly bounded the average over the disorder of the time autocorrelation function goes to equilibrium as exp k log t d d 1 in d 1 in agreement with previous results obtained for the dilute ising model

some statistical methods for bivariate circular data summary a function of the smallest singular value of the cross product matrix between two circular unit vectors is suggested as a measure of correlation it possesses most of the desirable properties for a correlation coefficient given by jupp and mardia 1980 the concept of cluster dependence is introduced a good predictor of one unit vector given the other is shown to depend on the type of dependence observed the wrapped bivariate normal is characterized

relaxed criteria of the dobrushin shlosman mixing condition an interacting particle system glauber dynamics which evolves on a finite subset in thed dimensional integer lattice is considered it is known that a mixing property of the gibbs state in the sense of dobrushin and shlosman is equivalent to several very strong estimates in terms of the glauber dynamics we show that similar but seemingly much milder estimates are again equivalent to the dobrushin shlosman mixing condition hence to the original ones found by stroock and zegarlinski this may be understood as the absence of intermediate speed of convergence to equilibrium

the logarithmic sobolev inequality for continuous spin systems on a lattice abstract in this article we give cf theorem 1 6 a sufficient condition on a local specification for the associated gibbs state to satisfy a logarithmic sobolev inequality in the final section cf theorem 3 10 we relate our condition to the type of mixing condition introduced by dobrushin and shlosman in connection with their work on complete analyticity

absence of renormalization group pathologies near the critical temperature two examples we consider real space renormalization group transformations for ising type systems which are formally defined by exp left h sigma right sum limits sigma t sigma sigma exp left h sigma right wheret is a probability kernel i e t 1 for every configuration for each choice of the block spin configuration let let be the measure on spin configurations which is formally given by taking the probability of to be proportional tot exp h we give a condition which is sufficient to imply that the renormalized hamiltonianh is defined roughly speaking the condition is that the collection of measures is in the high temperature phase uniformly in the block spin configuration the proof of this result uses methods of olivieri and picco we use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples decimation with spacingb 2 on the square lattice with 1 36 c and the kadanoff transformation with parameterp on the trian gular lattice in a subset of the p plane that includes values of greater than c

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

logarithmic sobolev inequality for lattice gases with mixing conditions let mu lambda l lambda gc denote the grand canonical gibbs measure of a lattice gas in a cube of sizel with the chemical potential and a fixed boundary condition let mu lambda l n c be the corresponding canonical measure defined by conditioning mu lambda l lambda gc on sigma x in lambda eta x n consider the lattice gas dynamics for which each particle performs random walk with rates depending on near by particles the rates are chosen such that for everyn andl fixed mu lambda l n c is a reversible measure suppose that the dobrushin shlosman mixing conditions holds for mu l lambda gc forall chemical potentials we prove that smallint f log fd mu lambda l n c leqq const l 2 d sqrt f for any probability densityf with respect to mu lambda l n c here the constant is independent ofn orl andd denotes the dirichlet form of the dynamics the dependence onl is optimal

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

log sobolev inequalities for infinite one dimensional lattice systems it is shown that a unique gibbs measure of infinite spin system with short range interaction on one dimensional lattice satisfies log sobolev inequality

universality of cutoff for the ising model on any locally finite geometry the stochastic ising model is known to be contractive when the inverse temperature beta is small enough via classical results of dobrushin and of holley in the 1970s by a general principle proposed by peres the dynamics is then expected to exhibit cutoff however so far cutoff for the ising model has been confirmed mainly for lattices heavily relying on amenability and log sobolev inequalities without these cutoff was unknown at any fixed beta 0 no matter how small even in basic examples such as the ising model on a binary tree or a random regular graph we use the new framework of information percolation to show that in any geometry there is cutoff for the ising model at high enough temperatures precisely on any sequence of graphs with maximum degree d the ising model has cutoff provided that beta kappa d for some absolute constant kappa a result which up to the value of kappa is best possible moreover the cutoff location is established as the time at which the sum of squared magnetizations drops to 1 and the cutoff window is o 1 just as when beta 0 finally the mixing time from almost every initial state is not more than a factor of 1 varepsilon beta faster then the worst one with varepsilon beta to0 as beta to0 whereas the uniform starting state is at least 2 varepsilon beta times faster

stretched exponential relaxation in the biased random voter model we study the relaxation properties of the voter model with i i d random bias we prove under mild conditions that the disorder averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent d d where 0 2 depends on the transition rates of the non biased voter model under an additional assumption we show that the above upper bound is optimal the main ingredient of our proof is a result of donsker and varadhan

dobrushin uniqueness theorem and logarithmic sobolev inequalities we formulate a condition on a local specification e on a countable product space m m being a riemannian manifold or a discrete set 1 1 assuring that the corresponding set of gibbs measures consists of a unique measure satisfying a logarithmic sobolev inequality

exact thresholds for ising gibbs samplers on general graphs we establish tight results for rapid mixing of gibbs samplers for the ferromagnetic ising model on general graphs we show that if d 1 tanh 1 then there exists a constant c such that the discrete time mixing time of gibbs samplers for the ferromagnetic ising model on any graph of n vertices and maximal degree d where all interactions are bounded by and arbitrary external fields are bounded by cnlogn moreover the spectral gap is uniformly bounded away from 0 for all such graphs as well as for infinite graphs of maximal degree d we further show that when dtanh 1 with high probability over the erd s renyi random graph g n d n it holds that the mixing time of gibbs samplers is n1 1 loglogn both results are tight as it is known that the mixing time for random regular and erd s renyi random graphs is with high probability exponential in n when d 1 tanh 1 and dtanh 1 respectively to our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs moreover our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees

a very simple proof of the lsi for high temperature spin systems abstract we present a very simple proof that the o n model satisfies a uniform logarithmic sobolev inequality lsi if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n this condition applies in particular to the sk spin glass model at inverse temperature 1 4 it is the first result of rapid relaxation for the sk model and requires significant cancellations between the ferromagnetic and anti ferromagnetic spin couplings that cannot be obtained by existing methods to prove log sobolev inequalities the proof also applies to more general bounded and unbounded spin systems it uses a single step of zero range renormalisation and bakry emery theory for the renormalised measure

quantum conditional relative entropy and quasi factorization of the relative entropy the existence of a positive log sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the markov approximation for classical spin systems such constant was proven to exist under the assumption of a mixing condition in the gibbs measure associated to their dynamics via a quasi factorization of the entropy in terms of the conditional entropy in some sub algebras in this work we analyze analogous quasi factorization results in the quantum case for that we define the quantum conditional relative entropy and prove several quasi factorization results for it as an illustration of their potential we use one of them to obtain a positive log sobolev constant for the heat bath dynamics with product fixed point

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

exponential relaxation of glauber dynamics with some special boundary conditions we consider attractive finite range glauber dynamics and show that if a certain mixing condition is satisfied then the system evolving on arbitrary subsets of the lattice with appropriate boundary conditions converges to equilibrium exponentially fast in the uniform sense uniformly over the subsets of the lattice this result applies for instance to the ferromagnetic nearest neighbor ising model in the so called basuev region where complete analyticity is expected to fail

on the relationship between item response theory and factor analysis of discretized variables equivalence of marginal likelihood of the two parameter normal ogive model in item response theory irt and factor analysis of dichotomized variables fa was formally proved the basic result on the dichotomous variables was extended to multicategory cases both ordered and unordered categorical data pair comparison data arising from multiple judgment sampling were discussed as a special case of the unordered categorical data a taxonomy of data for the irt and fa models was also attempted

a comparison of latent trait and latent class analyses of likert type data this paper brings together and compares two developments in the analysis of likert attitude scales the first is the generalization of latent class models to ordered response categories the second is the introduction of latent trait models with multiplicative parameter structures for the analysis of rating scales key similarities and differences between these two methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of life satisfaction data the way in which the latent trait model defines a unit of measurement takes into account the order of the response categories and scales the latent classes is discussed while the latent class model provides better fit to these data this is achieved at the cost of a logically inconsistent assignment of individuals to latent classes

renormalization group at criticality and complete analyticity of constrained models a numerical study we study the majority rule transformation applied to the gibbs measure for the 2d ising model at the critical point the aim is to show that the renormalized hamiltonian is well defined in the sense that the renormalized measure is gibbsian we analyze the validity of dobrushin shlosman uniqueness dsu finite size condition for the constrained models corresponding to different configurations of the image system it is known that dsu implies in our 2d case complete analyticity from which as recently shown by haller and kennedy gibbsianness follows we introduce a monte carlo algorithm to compute an upper bound to vasserstein distance appearing in dsu between finite volume gibbs measures with different boundary conditions we get strong numerical evidence that indeed the dsu condition is verified for a large enough volumev for all constrained models

renormalization group transformations under strong mixing conditions gibbsianness and convergence of renormalized interactions in this paper we study a renormalization group map the block averaging transformation applied to gibbs measures relative to a class of finite range lattice gases when suitable strong mixing conditions are satisfied using a block decimation procedure cluster expansion and detailed comparison between statistical ensembles we are able to prove gibbsianness and convergence to a trivial i e gaussian and product fixed point our results apply to the 2d standard ising model at any temperature above the critical one and arbitrary magnetic field

improved mixing condition on the grid for counting and sampling independent sets the hard core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics a multicast model of calls in communication networks and as a weighted independent set problem in combinatorics probability and theoretical computer science in this model each independent set i in a graph g is weighted proportionally to i for a positive real parameter for large computing the partition function namely the normalizing constant which makes the weighting a probability distribution on a finite graph on graphs of maximum degree 3 is a well known computationally challenging problem more concretely let lambda c mathbb t delta denote the critical value for the so called uniqueness threshold of the hard core model on the infinite regular tree recent breakthrough results of weitz proceedings of the 38th annual acm symposium on theory of computing stoc pp 140 149 2006 and sly proceedings of the 51st annual ieee symposium on foundations of computer science focs pp 287 296 2010 have identified lambda c mathbb t delta as a threshold where the hardness of estimating the above partition function undergoes a computational transition we focus on the well studied particular case of the square lattice mathbb z 2 and provide a new lower bound for the uniqueness threshold in particular taking it well above lambda c mathbb t 4 our technique refines and builds on the tree of self avoiding walks approach of weitz resulting in a new technical sufficient criterion of wider applicability for establishing strong spatial mixing and hence uniqueness for the hard core model our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure improving upon what can be achieved by just using the maximum degree applying our technique to mathbb z 2 we prove that strong spatial mixing holds for all 2 3882 improving upon the work of weitz that held for 27 16 1 6875 our results imply a fully polynomial deterministic approximation algorithm for estimating the partition function as well as rapid mixing of the associated glauber dynamics to sample from the hard core distribution