ergodicity of probabilistic cellular automata a constructive criterion we give a sequence of criteria of increasing complexity for the exponential ergodicity of discrete time interacting particle systems each criterion involves estimating the dependence on initial conditions of the process on finite space time volumes it generalizes and improves the existing single site condition and is the analog of the dobrushin shlosmanc v condition in equilibrium statistical mechanics our dynamic criterion may also be used to prove the uniqueness of gibbs state in situations where thec v condition fails as a converse we prove that if there is a certain form of convergence to the stationary measure faster thann d wheren is the time andd is the dimension of the lattice then our condition holds for some space time volumes and hence the convergence must be exponentially fast

statistical mechanics of the knapsack problem the knapsack problem is an np complete combinatorial optimization problem with inequality constraints using the replica method of statistical physics we study the space of its solutions for a large problem size it turns out that this problem is closely related to the theory of the binary perceptron

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

neural networks for optimization problems with inequality constraints the knapsack problem a strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented the constraints x d 0 are encoded by x x terms in the energy function a careful treatment of the mean field approximation for the self coupling parts of the energy is crucial and results in an essentially parameter free algorithm this methodology is extensively tested on the knapsack problem of size up to 103 items the algorithm scales like nm for problems with n items and m constraints comparisons are made with an exact branch and bound algorithm when this is computationally possible n d 30 the quality of the neural network solutions consistently lies above 95 of the optimal ones at a significantly lower cpu expense for the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach for nonhomogeneous problems these produce good solutions whereas for the more difficult homogeneous problems the neural approach is a winner with respect to solution quality and or cpu time consumption the approach is of course also applicable to other problems of similar structure like set covering

critical behaviour of a surface reaction model with infinitely many absorbing states in a recent letter yaldram et al 1993 studied the critical behaviour of a simple lattice gas model of the co no catalytic reaction the model exhibits a second order non equilibrium phase transition from an active state into one out of infinitely many absorbing states estimates for the critical exponent beta suggested that the model belongs to a new universality class the results reported in this article contradict this notion as estimates for various critical exponents show that the model belongs to the universality class of directed percolation

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

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

dirichlet operators on loop spaces essential self adjointness and log sobolev inequality for each 0 1 and potential function v rd r we consider the dirichlet form e and the associated dirichlet operator h for the gibbs measure on the loop space e c 0 1 rd 0 1 the gibbs measure is related to the gibbs state of the quantum anharmonic oscillator with the potential v via the feynman kac formula we formulate dirichlet forms in the framework of rigged hilbert spaces which are related to the loop space e we then give an approximate criterion for the essential self adjointness of dirichlet operators associated with dirichlet forms given by probability measures on hilbert spaces under appropriate conditions on the potential we apply the approximate criterion to show that the dirichlet operator h is essentially self adjoint on the domain of smooth cylinder functions in addition if the potential satisfies a uniform convexity condition we prove that the dirichlet operator h has a gap at the lower end of spectrum we also show that the gibbs measure satisfie

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

topologies on the smirnov class the smirnov class n of analytic functions on the disk has traditionally been studied with the complete metric topology given by f g sup0 r 112 tlog 1 f rei g rei d however the space can be represented as the union of certain weighted hardy spaces h2 w the closure of the polynomials in l2 w d and hence given the inductive limit topology h we give a new proof of yanagihara s characterisation of the dual of n we prove that n h is an incomplete metrizable topological algebra in which fourier series converge we then study the individual spaces h2 w prove asymptotic versions of szego s theorem and the helson szego theorem on these spaces and characterise the universal multipliers of their duals

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

optimal pruning in neural networks we study pruning strategies in simple perceptrons subjected to supervised learning our analytical results obtained through the statistical mechanics approach to learning theory are independent of the learning algorithm used in the training process we calculate the post training distribution mathcal p j of synaptic weights which depends only on the overlap ensuremath rho 0 achieved by the learning algorithm before pruning and the fraction ensuremath kappa of relevant weights in the teacher network from this distribution we calculate the optimal pruning strategy for deleting small weights the optimal pruning threshold grows from zero as ensuremath theta mathrm opt ensuremath rho 0 ensuremath kappa ensuremath propto ensuremath rho 0 ensuremath ensuremath rho c ensuremath kappa 1 2 above some critical value ensuremath rho c ensuremath kappa thus the elimination of weak synapses enhances the network performance only after a critical learning period possible implications for biological pruning phenomena are discussed

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

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

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

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

learning in neural networks by eliminating frustrated bonds the authors study neural network models in which the initial interaction matrix elements are drawn from an arbitrary probability distribution and in which patterns are subsequently stored by eliminating the frustrated bonds generalizing a proposal by kinzel 1985 they show that the optimal choice for the a priori distribution corresponds to choosing uniform ferromagnetic initial interactions for the optimal model they study analytically the dynamical behaviour the equilibrium properties the sizes of the domains of attraction and a number of information theoretical performance measures

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

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

ising models on the lattice sierpinski gasket ferromagnetic ising models on the lattice sierpinski gasket are considered we prove the dobrushin shlosmann mixing condition and discuss corresponding properties of the stochastic ising models

storage capacity of ultrametric committee machines the problem of computing the storage capacity of a feed forward network with l hidden layers n inputs and k units in the first hidden layer is analyzed using techniques from statistical mechanics we found that the storage capacity strongly depends on the network architecture c log k 1 1 2l and that the number of units k limits the number of possible hidden layers l through the relationship 2l 1 2log k

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

capacity of diluted multi state neural networks the optimal storage capacity is studied for diluted networks with multi state neurons and continuous respectively discrete couplings within the replica symmetric gardner approach the gardner derrida line the de almeida thouless line and the zero entropy line are compared and the validity of the replica symmetric approximation is discussed in detail the distribution of the synaptic couplings is determined the results are analysed in terms of the number of states of the neuron the distribution of the stored patterns the amount of dilution and the number of discrete values for the couplings

logarithmic sobolev inequalities on path spaces over riemannian manifolds let w o m be the space of paths of unit time length on a connected complete riemannian manifold m such that 0 o a fixed point on m and the wiener measure on w o m the law of brownian motion on m starting at o if the ricci curvature is bounded by c then the following logarithmic sobolev inequality holds

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

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

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

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

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

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

annealed estimates on the green function we consider a random uniformly elliptic coefficient field a x on the d dimensional integer lattice mathbb z d we are interested in the spatial decay of the quenched elliptic green function g a x y next to stationarity we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic sobolev inequality holds for the ensemble langle cdot rangle we prove that all stochastic moments of the first and second mixed derivatives of the green function that is langle nabla x g x y p rangle and langle nabla x nabla y g x y p rangle have the same decay rates in x y gg 1 as for the constant coefficient green function respectively this result relies on and substantially extends the one by delmotte and deuschel probab theory relat fields 133 358 390 2005 which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of g that is langle nabla x g x y 2 rangle and langle nabla x nabla y g x y rangle as an application we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast

strong decay to equilibrium in one dimensional random spin systems we consider a spin system on a lattice with finite range possibly unbounded random interactions we show that for such systems the glauber dynamics cannot decay to equilibrium exponentially fast inl2 even at high temperatures additionally for one dimensional systems with unbounded random couplings we prove that with probability one the corresponding glauber dynamics has a fast subexponential decay to equilibrium in the uniform norm provided that the distribution of random couplings satisfies some exponential bound

the log sobolev inequality for weakly coupled lattice fields we consider a ferromagnetic spin system with unbounded interactions on the d dimensional integer lattice d 1 under mild assumptions on the one body interactions so that arbitrarily deep double wells are allowed we prove that if the coupling constants are small enough then the finite volume gibbs states satisfy the log sobolev inequality uniformly in the volume and the boundary condition

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

learning by dilution in a neural network a perceptron with n random weights can store of the order of n patterns by removing a fraction of the weights without changing their strengths the critical storage capacity as a function of the concentration of the remaining bonds for random outputs and for outputs given by a teacher perceptron is calculated a simple hebb like dilution algorithm is presented which in the teacher case reaches the optimal generalization ability

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

a dilution algorithm for neural networks a dilution algorithm to enlarge the storage capacity per synapse alpha eff of neural networks is proposed the algorithm is a hybrid method where hebb s rule is used to select a fraction of couplings to be removed afterwards the perceptron of optimal stability for the remaining couplings is learned the authors present an analytical calculation and the results of the numerical simulations in comparison with the fully connected or the randomly diluted perceptron the effective storage capacity alpha eff is remarkably enlarged

convergence to equilibrium for classical and quantum spin systems the paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on euclidean spaces existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted hilbert and banach spaces this paper is an enlarged and rewritten version of the paper 7

partially connected models of neural networks a partially connected hopfield neural network model is studied under the restriction that w the ratio of connections per site to the size of the system remains finite as the size n to infinity with the connection structure at each site being the same the replica symmetric mean field theory equations for the order parameters are derived the zero temperature forms of these equations are then solved numerically for a few different local connectivity architectures showing phase transitions at different critical storage ratios alpha c where the states which the authors are trying to store in the network become discontinuously unstable they show that the information capacity per connection improves for partially connected systems

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

stochastic optimization algorithms for barrier dividend strategies this work focuses on finding optimal barrier policy for an insurance risk model when the dividends are paid to the share holders according to a barrier strategy a new approach based on stochastic optimization methods is developed compared with the existing results in the literature more general surplus processes are considered precise models of the surplus need not be known only noise corrupted observations of the dividends are used using barrier type strategies a class of stochastic optimization algorithms are developed convergence of the algorithm is analyzed rate of convergence is also provided numerical results are reported to demonstrate the performance of the algorithm

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