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

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

convergence rates of the gibbs sampler the metropolis algorithm and other single site updating dynamics sampling from a markov random field ii can be performed efficiently via monta carlo methods by simulating a markov chain that converges weakly to ii we consider a class of local updating dynamics that are reversible with respect to ii it includes the metropolis algorithm mii and the gibbs sampler gs we investigate the speed of weak convergence of these markov chains in terms of their second largest eigenvalues in absolute value we study the general algebraic structure and then the stochastic ising model in detail

bounding the convergence time of the gibbs sampler in bayesian image restoration agibbs mathstat yorku ca summary this paper shows how coupling methodology can be used to give precise a priori bounds on the convergence time of markov chain monte carlo algorithms for which a partial order exists on the state space which is preserved by the markov chain transitions this methodology is applied to give a bound on the convergence time of the random scan gibbs sampler used in the bayesian restoration of an image of n pixels for our algorithm in which only one pixel is updated at each iteration the bound is a constant times n2 the proportionality constant is given and is easily calculated these bounds also give an indication of the running time of coupling from the past algorithms

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

improved bounds for sampling colorings we consider the problem of sampling uniformly at random from the set of proper k colorings of a graph with maximum degree our main result is the design of a simple markov chain that converges in o nk log n time to the desired distribution when k 116

random sampling for the monomer dimer model on a lattice in the monomer dimer model on a graph each matching collection of non overlapping edges m has a probability proportional to lambda m where lambda 0 is the model parameter and m denotes the number of edges in m an approximate random sample from the monomer dimer distribution can be obtained by running an appropriate markov chain each step of which involves an elementary local change in the configuration sufficiently long jerrum and sinclair have shown roughly speaking that for an arbitrary graph and fixed lambda and varepsilon the maximal allowed variational distance from the desired distribution o lambda 2 e steps suffice where e is the number of edges and lambda the number of vertices of the graph for sufficiently nice subgraphs e g cubes of the d dimensional cubic lattice we give an explicit recipe to generate approximate random samples in asymptotically siginificantly fewer steps namely for fixed lambda and varepsilon o lambda ln lambda 2

bound on the mass gap for finite volume stochastic ising models at low temperature we consider a sequence of finite volume z d d 2 reversible stochastic ising models in the low temperature regime and having invariant measures satisfying free boundary conditions we show that associated with the models are random hitting times whose expectations regarded as a function of grow exponentially in d 1 d moreover the mass gaps for the models shrink exponentially fast in d 1 d a geometrical lemma is employed in the analysis which states that if a peierls contour is sufficiently small relative to the faces of then the fraction of the contour tangent to the faces is less than a constant smaller than one

geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms we develop results on geometric ergodicity of markov chains and apply these and other recent results in markov chain theory to multidimensional hastings and metropolis algorithms for those based on random walk candidate distributions we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution by phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly used statistical forms from these results we develop central limit theorems for the metropolis algorithm converse results showing non geometric convergence rates for chains where the rejection rate is not bounded away from unity are also given these show that the negative definiteness property is not redundant

minorization conditions and convergence rates for markov chain monte carlo abstract general methods are provided for analyzing the convergence of discrete time general state space markov chains such as those used in stochastic simulation algorithms including the gibbs sampler the methods provide rigorous a priori bounds on how long these simulations should be run to give satisfactory results results are applied to two models of the gibbs sampler a bivariate normal model and a hierarchical poisson model with gamma conditionals the methods use the notion of minorization conditions for markov chains

computational complexity of markov chain monte carlo methods for finite markov random fields summary this paper studies the computational complexity of markov chain monte carlo algorithms with finite valued markov random fields on a finite regular lattice as target distributions we state conditions under which the complexity for approximate convergence is polynomial in n the number of variables approximate convergence takes time o n log n as n oo if the target field satisfies certain spatial mixing conditions otherwise if the field has a potential with finite interaction range independent of n the complexity is exponential in n with y 1 which is still more favourable than enumerating all the states when the interaction range grows with n the algorithms can converge exponentially in n analogous results are provided for an expectation approximated by an average along the chain

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

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

global nonexistence for a semilinear petrovsky equation abstract in this paper we consider a semilinear petrovsky equation with damping and source terms it is proved that the solution blows up in finite time if the positive initial energy satisfies a suitable condition moreover for the linear damping case we show that the solution blows up in finite time even for vanishing initial energy this is an important breakthrough since it is only well known that the solution blows up in finite time if the initial energy is negative from all the previous literature

fast deconvolution by a two step method a method for solving a discretized fredholm integral equation of the first kind with convolution kernel on the d dimensional torus d geq1 is presented a two step regularization method is used based on the minimization of two objective functions made up of a convex coherence to data term and a term which enforces a roughness penalty this is convex in the first initialization step and nonconvex in the second one the computational aspect is the main point addressed in the paper a deterministic as opposed to stochastic approach is followed auxiliary variables are used the separability of the original and the auxiliary variables is proven and exploited to reduce the computational burden the first step provides an initial estimate by computing the unique minimum of the convex functional the second step refines this initial estimate by computing a stationary point of the nonconvex functional experiments are reported on simulated data and in magnetic resonance mr imaging a significant reduction in computer time and similar restoration quality is observed when the method is compared with dynamic monte carlo methods applied to global optimization of the nonconvex functional

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

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

regeneration in markov chain samplers abstract markov chain sampling has recently received considerable attention in particular in the context of bayesian computation and maximum likelihood estimation this article discusses the use of markov chain splitting originally developed for the theoretical analysis of general state space markov chains to introduce regeneration into markov chain samplers this allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance the approach is applied to several samplers including certain metropolis samplers that can be used on their own or in hybrid samplers and is illustrated in several examples