central limit theorem for a random walk with random obstacles in mathrm r d for some positive constants r and ro with r ro for a probability measure v on the space t we write p ffv d q p we call the process x t p a random walk with random obstacles let t be the set of all q ee tr such that p x l e has an unbounded support and let u be a gibbs state associated with a pair potential d which is translation invariant one of the main results of this paper is the central limit theorem for x t pa where 4w is a conditional probability measure of the gibbs state u given the event t that is 4t a i1 in a previous paper 10 we considered a system of infinitely many hard balls with the same diameter r moving discontinuously in rd we constructed the markov process 4t which describes the system this process has a gibbs state as a stationary measure we showed the ergodicity of the stationary markov

variable transformation to obtain geometric ergodicity in the random walk metropolis algorithm a random walk metropolis sampler is geometrically ergodic if its equilibrium density is super exponentially light and satisfies a curvature condition stochastic process appl 85 2000 341 361 many applications including bayesian analysis with conjugate priors of logistic and poisson regression and of log linear models for categorical data result in posterior distributions that are not super exponentially light we show how to apply the change of variable formula for diffeomorphisms to obtain new densities that do satisfy the conditions for geometric ergodicity sampling the new variable and mapping the results back to the old gives a geometrically ergodic sampler for the original variable this method of obtaining geometric ergodicity has very wide applicability

meaningful design and contrast estimability in fmri optimising the efficiency of an experimental design is known to be of great importance however existing methods for calculating design rank deficiency and contrast estimability an important aspect of experimental design relate to computational precision rather than image noise and are therefore not very meaningful for example a contrast between two experimental conditions may be mathematically estimable while requiring a huge differential bold response for statistical significance to be reached in this paper we formulate standard efficiency equations in terms of required bold effect and use this to generate measures of rank estimability which are meaningful this takes into account the strength and smoothness of the timeseries noise and is applicable to complex contrasts we show how to re express several regressors and an associated contrast vector as a single equivalent regressor so that we can calculate the contrast s effective peak peak height unambiguously we also present some example results on typical designs and characterise noise results from a range of typical fmri acquisitions in order to allow experimenters to apply efficiency estimation in advance of acquiring data

efficient implementation of markov chain monte carlo when using an unbiased likelihood estimator summary when an unbiased estimator of the likelihood is used within a metropolis hastings chain it is necessary to trade off the number of monte carlo samples used to construct this estimator against the asymptotic variances of the averages computed under this chain using many monte carlo samples will typically result in metropolis hastings averages with lower asymptotic variances than the corresponding averages that use fewer samples however the computing time required to construct the likelihood estimator increases with the number of samples under the assumption that the distribution of the additive noise introduced by the loglikelihood estimator is gaussian with variance inversely proportional to the number of samples and independent of the parameter value at which it is evaluated we provide guidelines on the number of samples to select we illustrate our results by considering a stochastic volatility model applied to stock index returns

diffusive limit of a tagged particle in asymmetric simple exclusion processes invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d 3 the method of proof is by martingale techniques which rely on the fact that symmetric random walks are transient in high dimensions c 2000 john wiley sons inc

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

einstein relation for reversible diffusions in a random environment we consider reversible diffusions in a random environment and prove the einstein relation for this model it says that the derivative at 0 of the effective velocity under an additional local drift equals the diffusivity of the model without drift the einstein relation is conjectured to hold for a variety of models but so far it has only been proved in particular cases our proof makes use of homogenization arguments the girsanov transform and a refinement of the regeneration times introduced by shen 2011 wiley periodicals inc

distinguishability of particles in glass forming systems abstract the distinguishability of particles has important implications for calculating the partition function in statistical mechanics while there are standard formulations for systems of identical particles that are either fully distinguishable or fully indistinguishable many realistic systems do not fall into either of these limiting cases in particular the glass transition involves a continuous transition from an ergodic liquid system of indistinguishable particles to a nonergodic glassy system where the particles become distinguishable while the question of partial distinguishability of microstates has been treated previously in quantum information theory this issue has not yet been addressed for a system of classical particles in this paper we present a general theoretical formalism for quantifying particle distinguishability in classical systems this formalism is based on a classical definition of relative entropy such as applied in quantum information theory example calculations for a simple glass forming system demonstrate the continuous onset of distinguishability as temperature is lowered we also examine the loss of distinguishability in the limit of long observation time coinciding with the restoration of ergodicity we discuss some of the general implications of our work including the direct connection to topological constraint theory of glass we also discuss qualitative features of distinguishability as they relate to the second and third laws of thermodynamics

comparison of physiological noise at 1 5 t 3 t and 7 t and optimization of fmri acquisition parameters previous studies have shown that under some conditions noise fluctuations in an fmri time course are dominated by physiological modulations of the image intensity with secondary contributions from thermal image noise and that these two sources scale differently with signal intensity susceptibility weighting te and field strength the snr of the fmri time course was found to be near its asymptotic limit for moderate spatial resolution measurements at 3 t with only marginal gains expected from acquisition at higher field strengths in this study we investigate the amplitude of image intensity fluctuations in the fmri time course at magnetic field strengths of 1 5 t 3 t and 7 t as a function of image resolution flip angle and te the time course snr was a similar function of the image snr regardless of whether the image snr was modulated by flip angle image resolution or field strength for spatial resolutions typical of those currently used in fmri e g 3 3 3 mm3 increases in image snr obtained from 7 t acquisition produced only modest increases in time course snr at this spatial resolution the ratio of physiological noise to thermal image noise was 0 61 0 89 and 2 23 for 1 5 t 3 t and 7 t at a resolution of 1 1 3 mm3 however the physiological to thermal noise ratio was 0 34 0 57 and 0 91 for 1 5 t 3 t and 7 t for te near t2 thus by reducing the signal strength using higher image resolution the ratio of physiologic to image noise could be reduced to a regime where increased sensitivity afforded by higher field strength still translated to improved snr in the fmri time series

random walks in asymmetric random environments we consider random walks on zd with transition ratesp x y given by a random matrix ifp is a small random perturbation of the simple random walk we show that the walk remains diffusive for almost all environmentsp ifd 2 the result also holds for a continuous time markov process with a random drift the corresponding path space measures converge weakly in the scaling limit to the wiener process for almost everyp

design of a class of luenberger observers for descriptor systems a new approach to the design of a class of luenberger observers for descriptor systems is presented the design is performed in a straightforward manner the conditions required in the design are given in terms of the original system matrices the relationship among these and other well known conditions is highlighted

comprehensive admissibility for descriptor systems this paper presents a complete solution to the problem of comprehensive admissibility for descriptor systems with unstable and nonproper weights in such non standard circumstances it is hard to render the closed loop system admissible due to the uncontrollable and unobservable weights the necessary and sufficient condition is given and a specifically structured output feedback controller is conducted in terms of the dynamics of the weighs it is shown that determining suitable controllers for a given descriptor system with the presence of unstable and nonproper weights requires solving an admissibility problem for an augmented system explicitly constructed in this paper a numerical example is included to illustrate the current result

on the functional observers for linear descriptor systems this paper is concerned with the design of functional observers for linear time invariant descriptor systems contrary to the functional observers considered for the standard systems in darouach 8 where the order of these observers is equal to the dimension r of the functional to be estimated in this paper the order can be of dimension different of that of this functional when this order is equal to the dimension of the functional and the system is in a standard form the presented design method becomes that of darouach 8 it also generalizes the existing results for descriptor systems the approach is based on the new definition of partial impulse observability sufficient conditions for the existence and stability of these observers are given continuous time and discrete time systems are considered two numerical examples are given to illustrate our approach

tracer diffusion in lattice gases it has been proved that a tracer particle in a reversible lattice gas converges to brownian motion however only in a few particular cases has a strictly positive self diffusion coefficientd been established here we supply the missing piece and show thatd 0 in general the exceptions are one dimensional lattice gases with nearest neighbor jumps only for whichd 0 the proof establishes a variational formula ford which could be used to obtain realistic bounds

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

observers and observer based control for descriptor systems revisited this technical note is devoted to the observers and observer based control for linear descriptor systems first the complete proof of necessary and sufficient conditions for the existence of luenberger observers for an arbitrary order is given then the existence conditions for the controller based on these observers are presented in the lmi formulation the design approach is simple and based on the new study of the constrained generalized sylvester equations it unifies all the observers and observer based controllers of different orders a numerical example is presented to illustrate our approach

aging and relaxation dynamics in free energy landscapes with multiple minima abstract we consider the stochastic dynamics of a system thermally relaxing in a free energy landscape with multiple attractors and show that lack of translational homogeneity in this landscape leads to aging effects e g to the dependence of the susceptibilities on the time elapsed from a thermal quench to the imposition of the probing field we then prove an inequality between response and correlation which generalizes the fluctuation dissipation theorem to a situation far from thermodynamical equilibrium as an application and a check we specialize our formalism in a way which we suggest is appropriate for spin glass systems we assume a hierarchical organization of the landscape and find aging behavior in the response curves in good agreement with relevant experimental data we finally conclude with a summary and a brief discussion of different approaches to slow relaxation in complex systems

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

finite time stabilizability detectability and dynamic output feedback finite time stabilization of linear systems in this note we deal with linear systems and we extend to the finite time setting the concepts of stabilizability and detectability it will be shown that similarly to what happens in the classical lyapunov framework even in the finite time context stabilizability and detectability play a role into the existence of stabilizing dynamical controllers we prove that a dynamic output feedback controller which finite time stabilizes the overall closed loop system exists if and only if the open loop system is finite time detectable and stabilizable plus a further linear matrix inequality coupling condition we also show that in the finite time context the equivalence between stabilizability via output feedback and stabilizability via observer based controllers is no longer true

joint estimation and correction of geometric distortions for epi functional mri using harmonic retrieval magnetic resonance imaging mri uses applied spatial variations in the magnetic field to encode spatial position therefore nonuniformities in the main magnetic field can cause image distortions in order to correct the image distortions it is desirable to simultaneously acquire data with a field map in registration we propose a joint estimation je framework with a fast noniterative approach using harmonic retrieval hr methods combined with a multi echo echo planar imaging epi acquisition the connection with hr establishes an elegant framework to solve the je problem through a sequence of 1 d hr problems in which efficient solutions are available we also derive the condition on the smoothness of the field map in order for hr techniques to recover the image with high signal to noise ratio compared to other dynamic field mapping methods this method is not constrained by the absolute level of the field inhomogeneity over the slice but relies on a generous pixel to pixel smoothness moreover this method can recover image field map and t2 map simultaneously

green kubo formula for weakly coupled systems with noise we study the green kubo formula kappa varepsilon varsigma for the heat conductivity of an infinite chain of d dimensional finite systems cells coupled by a smooth nearest neighbor potential varepsilon v the uncoupled systems evolve according to hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength varsigma noting that kappa varepsilon varsigma exists and is finite whenever varsigma 0 we are interested in what happens when the strength of the noise varsigma to 0 for this we start in this work by formally expanding kappa varepsilon varsigma in a power series in varepsilon kappa varepsilon varsigma varepsilon 2 sum n geq 2 varepsilon n 2 kappa n varsigma and investigating the formal equations satisfied by kappa n varsigma we show in particular that kappa 2 varsigma is well defined when no pinning potential is present and coincides formally with the heat conductivity obtained in the weak coupling van hove limit where time is rescaled as varepsilon 2 t for the cases where the latter has been established liverani and olla in jams 25 555 583 2012 dolgopyat and liverani in commun math phys 308 201 225 2011 for one dimensional systems we investigate kappa 2 varsigma as varsigma to 0 in three cases the disordered harmonic chain the rotor chain and a chain of strongly an harmonic oscillators moreover we formally identify kappa 2 varsigma with the conductivity obtained by having the chain between two reservoirs at temperature t and t delta t in the limit delta t to 0 n to infty varepsilon to 0

impulse controllability and observability of rectangular descriptor systems the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observability is considered it is shown that the equivalence between impulse controllability observability and the existence of an impulse eliminating semistate feedback output injection gain although true for square descriptor systems does not hold for rectangular descriptor systems in general hence necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback output injection gain are presented

homogenization of parabolic equations with a continuum of space and time scales this paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available namely we consider divergence form linear parabolic operators in omega subset mathbb r n with l infty omega times 0 t coefficients it appears that the inverse operator maps the unit ball of l 2 omega times 0 t into a space of functions which at small time and space scales are close in h 1 norm to a functional space of dimension n it follows that once one has solved these equations at least n times it is possible to homogenize them both in space and in time reducing the number of operation counts necessary to obtain further solutions in practice we show under a cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in l 2 instead of h 1 with euclidean coordinates if the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the parabolic equation

limit theory for random walks in degenerate time dependent random environments abstract we study continuous time variable speed random walks in random environments on z d d 2 where at time t the walk at x jumps across edge x y at time dependent rate a t x y the rates which we assume stationary and ergodic with respect to space time shifts are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time we formulate conditions on the environment under which the law of diffusively scaled random walk path tends to brownian motion for almost every sample of the rates the proofs invoke moser iteration to prove sublinearity of the corrector in pointwise sense a key additional input is a conversion of certain weighted energy norms to ordinary ones our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the helffer sjostrand representation of gradient models with certain non strictly convex potentials

a role for a brachistochrone in the broken ergodicity scenario abstract we show that the time evolution of ergodic decomposition is outlined by a variational principle for a rather universal class of complex systems with a quasicontinuous spectrum of intrinsic relaxation timescales in turn this variational approach enables us to single out a logarithmic scaling law for the barriers on the hierarchical level such a logarithmic dependence has already been shown to hold valid for certain complex systems moreover it represents a key factor to explain the experimentally ubiquitous debye kohlrausch relaxation law and plays a central role on different theoretical model descriptions within this field

generalized ensemble and tempering simulations a unified view from the underlying master equations we derive one dimensional stochastic processes that describe generalized ensemble simulations as well as tempering simulated and parallel simulations the representations obtained are either in the form of a one dimensional fokker planck equation or a hopping process on a one dimensional chain in particular we discuss the conditions under which these representations are valid approximate markovian descriptions of the random walk in order parameter or control parameter space they allow a unified discussion of the stationary distribution on as well as of the stationary flow across each space we demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and discuss the consequences of our results for optimizing simulations finally we point out the limitations of these representations under conditions of broken ergodicity

optimization of functional mri for detection decoding and high resolution imaging of the response patterns of cortical columns the capacity of functional mri fmri to resolve cortical columns depends on several factors these include the spatial scale of the columnar pattern the point spread of the fmri response the voxel size and the signal to noise ratio snr considering thermal and physiological noise however it remains unknown how these factors combine and what is the voxel size that optimizes fmri of cortical columns here we combine current knowledge into a quantitative model of fmri of realistic patterns of cortical columns with different spatial scales and degrees of irregularity we compare different approaches for identifying patterns of cortical columns including univariate and multivariate based detection multi voxel pattern analysis mvpa based decoding and high resolution imaging and reconstruction of the pattern of cortical columns we present the dependence of the performance of each approach on the parameters of the imaged pattern as well as those of the data acquisition in addition we predict voxel sizes that optimize fmri of cortical columns under various scenarios we found that all measures associated with multivariate detection and decoding could be approximately calculated from a measure we termed multivariate contrast to noise ratio mv cnr which is a function of the contrast to noise ratio cnr and number of voxels furthermore mv cnr implied that the optimal voxel width for detection and decoding is independent of changes in response amplitude snr and imaged volume that are not caused by changes in voxel size for regular patterns optimal voxel widths for detection decoding and imaging reconstructing the pattern of cortical columns were approximately half the main cycle length of the organization optimal voxel widths for irregular patterns were less dependent on the main cycle length and differed between univariate detection multivariate detection and decoding and reconstruction we compared the effects of different factors of gradient echo fmri at 3 tesla t gradient echo fmri at 7t and spin echo fmri at 7t on the detection decoding and reconstruction measures considered and found that in all cases the width of the fmri point spread had the most significant effect in contrast different response amplitudes and noise characteristics played a relatively minor role we recommend specific voxel widths for optimal univariate detection for multivariate detection and decoding and for high resolution imaging of cortical columns under these three data acquisition scenarios our study supports the planning optimization and interpretation of high resolution fmri of cortical columns and the decoding of information conveyed by these columns

rank based inference in the proportional hazards model for interval censored data summary a marginal likelihood approach to fitting the proportional hazards model to interval censored or grouped data is proposed this approach maximises a likelihood that is the sum over all rankings of the data that are consistent with the observed censoring intervals as in the usual proportional hazards model the method does not require specification of the baseline hazard function the score equations determining the maximum marginal likelihood estimator can be written as the expected value of the score of the usual proportional hazards model with respect to a certain distribution of rankings a gibbs sampling scheme is given to generate rankings from this distribution and stochastic approximation is used to solve the score equations simulation results under various censoring schemes give point estimates that are close to estimates obtained using actual failure times

positivity of the self diffusion matrix of interacting brownian particles with hard core we prove the positivity of the self diffusion matrix of interacting brownian particles with hard core when the dimension of the space is greater than or equal to 2 here the self diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle we will do this for all activities z 0 of gibbs measures in particular for large z the case of high density particles a typical example of such a particle system is an infinite amount of hard core brownian balls

quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction we consider a ballistic random walk in an i i d random environment that does not allow retreating in a certain fixed direction we prove an invariance principle functional central limit theorem under almost every fixed environment the assumptions are nonnestling at least two spatial dimensions and a 2 e moment for the step of the walk uniformly in the environment the main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively

on homogenization of time dependent random flows we study a diffusion with a random time dependent drift we prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition this result generalizes the results of 13 7

asymptotic efficiency of the sample mean in markov chain monte carlo schemes in many cases a univariate time series derived from a time reversible stationary markov chain has the property that there is a distribution function on 1 1 0 1 such that for all non negative integers l its lth moment is the ith lag autocorrelation of the time series we discuss some consequences of this property effects of the asymptotic efficiency of the sample mean on the efficiency of subsampling a stationary markov chain are discussed the stationarity assumption is then relaxed we show that under certain regularity conditions the sample mean of a strongly asymptotically stationary process is asymptotically efficient in that it has the smallest mean square error asymptotically in a broad class of linear estimators

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

relations between invasion percolation and critical percolation in two dimensions we study invasion percolation in two dimensions we compare connectivity properties of the origin s invaded region to those of a the critical percolation cluster of the origin and b the incipient infinite cluster to exhibit similarities we show that for any k 1 the k point function of the first socalled pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin more prominent though are the differences we show that there are infinitely many ponds that contain many large disjoint pc open clusters further for k 1 we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin we finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

brief paper a controller design based on a functional h filter for descriptor systems the time and frequency domain cases in this paper we investigate the time and frequency domain designs of a controller based on a functional filter for both continuous time and discrete time multivariable descriptor systems the order of this controller is equal to the dimension of the state to be estimated and the control gain is designed using h techniques the time procedure is based on the unbiasedness of the estimation error using a sylvester equation then a new method for avoiding the derivative of the disturbance vector in the observation error dynamics is proposed this problem is solved by means of linear matrix inequalities lmis the frequency domain procedure design is derived from time domain results where we propose some suitable matrix fraction descriptions mfds and most importantly the factorization approach permits us to give a polynomial parameterization of the functional h filter based controller

observer design for discrete time descriptor systems an lmi approach abstract in this paper the observer design problem for discrete time descriptor systems is considered for the discrete time linear descriptor systems the necessary and sufficient conditions for the existence and convergence of the proposed observer are given and proved and a systemic design approach is presented via the linear matrix inequalities formulation furthermore an extension to a class of nonlinear descriptor systems with lipschitz constraints is investigated simulation examples are given to illustrate the estimation performance of the proposed method

design of observers for descriptor systems in this note simple and straightforward methods to design full and reduced order observers for linear time invariant descriptor systems are presented the approach for the reduced order observer design is based on the generalized sylvester equation sufficient conditions for the existence of the observers are given an illustrative example is included

brief paper h observers design for a class of nonlinear singular systems the present paper is devoted to the h observer or filter design problem for a class of lipschitz nonlinear singular systems the approach is based on the parameterization of the obtained algebraic constraints from the estimation errors sufficient conditions for the existence of the observers which guarantee stability and the worst case observers error energy over all bounded energy disturbances is minimized are given the method also unifies the design for the full order reduced order minimal order observers for continuous time systems it can be seen that the obtained conditions when applied to linear systems are also necessary a numerical example is given to show the applicability of our results

scaling limit for a class of gradient fields with nonconvex potentials we consider gradient fields x x d whose law takes the gibbs boltzmann form z 1 exp x y v v x where the sum runs over nearest neighbors we assume that the potential v admits the representation v log d exp 1 2 2 where is a positive measure with compact support in 0 hence the potential v is symmetric but nonconvex in general while for strictly convex v s the translation invariant ergodic gradient gibbs measures are completely characterized by their tilt a nonconvex potential as above may lead to several ergodic gradient gibbs measures with zero tilt still every ergodic zero tilt gradient gibbs measure for the potential v above scales to a gaussian free field

reduced order observer design for descriptor systems with unknown inputs a new method for the design of reduced order observers for descriptor systems with unknown inputs is presented the approach is based on the generalized constrained sylvester equation sufficient conditions for the existence of the observer are given

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

phase space geometry and slow dynamics we describe a non arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space we show that such a mechanism is at work both in a family of mean field spin glass models without any domain structure and in the case of ferromagnetic domain growth the marginality of spin glass dynamics as well as the existence of a quasi equilibrium regime can be understood within this scenario we discuss the question of ergodicity in an out of equilibrium situation

exponential growth of ponds in invasion percolation on regular trees in invasion percolation the edges of successively maximal weight the outlets divide the invasion cluster into a chain of ponds separated by outlets on the regular tree the ponds are shown to grow exponentially with law of large numbers central limit theorem and large deviation results the tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster with a logarithmic correction

large deviations for the symmetric simple exclusion process in dimensions d 3 we consider symmetric simple exclusion processes with l macr nd particles in a periodic d dimensional lattice of width n we perform the diffusive hydrodynamic scaling of space and time the initial condition is arbitrary and is typically far away form equilibrium it specifies in the scaling limit a density profile on the d dimensional torus we are interested in the large deviations of the empirical process n d l1 xi as random variables taking values in the space of measures on d 0 1 we prove a large deviation principle with a rate function that is more or less universal involving explicity besides the initial profile only such canonical objects as bulk and self diffusion coefficients

slowdown and neutral pockets for a random walk in random environment we consider a d dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift pointing to the right we obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion both under the annealed and quenched measures these estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk

noise can speed markov chain monte carlo estimation and quantum annealing carefully injected noise can speed the average convergence of markov chain monte carlo mcmc estimates and simulated annealing optimization this includes quantum annealing and the mcmc special case of the metropolis hastings algorithm mcmc seeks the solution to a computational problem as the equilibrium probability density of a reversible markov chain the algorithm must cycle through a long burn in phase until it reaches equilibrium because the markov samples are statistically correlated the special injected noise reduces this burn in period in mcmc a related theorem shows that it reduces the cooling time in simulated annealing simulations showed that optimal noise gave a 76 speed up in finding the global minimum in the schwefel optimization benchmark the noise boosted simulations found the global minimum in 99 8 of trials compared with only 95 4 of trials in noiseless simulated annealing simulations also showed that the noise boost is robust to accelerated cooling schedules and that noise decreased convergence times by more than 32 under aggressive geometric cooling molecular dynamics simulations showed that optimal noise gave a 42 speed up in finding the minimum potential energy configuration of an eight argon atom gas system with a lennard jones 12 6 potential the annealing speed up also extends to quantum monte carlo implementations of quantum annealing noise improved ground state energy estimates in a 1024 spin simulated quantum annealing simulation by 25 6 the quantum noise flips spins along a trotter ring the noisy mcmc algorithm brings each markov step closer on average to equilibrium if an inequality holds between two expectations gaussian or cauchy jump probabilities reduce the noise benefit inequality to a simple quadratic inequality simulations show that noise boosted simulated annealing is more likely than noiseless annealing to sample high probability regions of the search space and to accept solutions that increase the search breadth

finite dimensional approximation of the self diffusion coefficient for the exclusion process we show that for the symmetric simple exclusion process on z d the self diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices the proof depends on a similar stability property of the asymptotic variance of additive functionals of mean 0 this requires establishing a property for the dirichlet space known as the liouville d property

equilibrium fluctuations for nabla_ varphi interface model we study the large scale space time fluctuations of an interface which is modeled by a massless scalar field with reversible langevin dynamics for a strictly convex interaction potential we prove that on a large space time scale these fluctuations are governed by an infinite dimensional ornstein uhlenbeck process its effective diffusion type covariance matrix is characterized through a variational formula

the quest for the best the impact of different epi sequences on the sensitivity of random effect fmri group analyses we compared the sensitivity of standard single shot 2d echo planar imaging epi to three advanced epi sequences i e 2d multi echo epi 3d high resolution epi and 3d dual echo fast epi in fixed effect and random effects group level fmri analyses at 3t the study focused on how well the variance reduction in fixed effect analyses achieved by advanced epi sequences translates into increased sensitivity in the random effects group level analysis the sensitivity was estimated in a functional mri experiment of an emotional learning and a reward based learning tasks in a group of 24 volunteers each experiment was acquired with the four different sequences the task related response amplitude contrast level and respective t value were proxies for the functional sensitivity across the brain all three advanced epi methods increased the sensitivity in the fixed effects analyses but standard single shot 2d epi provided a comparable performance in random effects group analysis when whole brain coverage and moderate resolution are required in this experiment inter subject variability determined the sensitivity of the random effects analysis for most brain regions making the impact of epi pulse sequence improvements less relevant or even negligible for random effects analyses an exception concerns the optimization of epi reducing susceptibility related signal loss that translates into an enhanced sensitivity e g in the orbitofrontal cortex for multi echo epi thus future optimization strategies may best aim at reducing inter subject variability for higher sensitivity in standard fmri group studies at moderate spatial resolution

exponential convergence to equilibrium for a class of random walk models we prove exponential convergence to equilibrium l 2 geometric ergodicity for a random walk with inward drift on a sub cayley rooted tree this randomwalk model generalizes a monte carlo algorithm for the self avoiding walk proposed by berretti and sokal if the number of vertices of leveln in the tree grows asc n n n 1 we prove that the autocorrelation time satisfies n 2 n 1

random walk in random environment a counterexample we describe a family of random walks in random environments which have exponentially decaying correlations nearest neighbor transition probabilities which are bounded away from 0 and yet are subdiffusive in any dimensiond

convergence to fractional kinetics for random walks associated with unbounded conductances we consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail we prove that the scaling limit of this process is a fractional kinetics process that is the time change of a d dimensional brownian motion by the inverse of an independent stable subordinator we further show that the same process appears in the scaling limit of the non symmetric bouchaud s trap model

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

harmonic functions h transform and large deviations for random walks in random environments in dimensions four and higher we consider large deviations for nearest neighbor random walk in a uniformly elliptic i i d environment on d there exist variational formulae for the quenched and averaged rate functions i q and i a obtained by rosenbluth and varadhan respectively i q and i a are not identically equal however when d 4 and the walk satisfies the so called t condition of sznitman they have been previously shown to be equal on an open set a eq for every a eq we prove the existence of a positive solution to a laplace like equation involving and the original transition kernel of the walk we then use this solution to define a new transition kernel via the h transform technique of doob this new kernel corresponds to the unique minimizer of varadhan s variational formula at it also corresponds to the unique minimizer of rosenbluth s variational formula provided that the latter is slightly modified

on the sector condition and homogenization of diffusions with a gaussian drift abstract in this paper we present the functional central limit theorem for a class of markov processes whose l 2 generator satisfies the so called graded sector condition we apply the result to obtain homogenization theorems for certain classes of diffusions with a random gaussian drift additionally we present a result concerning the regularity of the effective diffusivity tensor with respect to the parameters related to the statistics of the drift the abstract central limit theorem see theorem 2 2 is obtained by applying the technique used in sethuraman et al comm pure appl math 53 2000 972 to the case of infinite particle systems

symmetric simple exclusion process regularity of the self diffusion coefficient we prove that the self diffusion coefficient of a tagged particle in the symmetric exclusion process in zd which is in equilibrium at density is of class c as a function of in the closed interval 0 1 the proof provides also a recursive method to compute the taylor expansion at the boundaries

einstein relation and steady states for the random conductance model we consider random walk among i i d uniformly elliptic conductances on zdzd and prove the einstein relation see theorem 1 it says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium for fixed bias we show that there is an invariant measure for the environment seen from the particle these invariant measures are often called steady states the einstein relation follows at least for d 3d 3 from an expansion of the steady states as a function of the bias see theorem 2 which can be considered our main result this expansion is proved for d 3d 3 in contrast to guo ann probab 44 2016 324 359 we need not only convergence of the steady states but an estimate on the rate of convergence see theorem 4

bold sensitivity and snr characteristics of parallel imaging accelerated single shot multi echo epi for fmri echo planar imaging epi is a standard procedure in functional magnetic resonance imaging fmri for measuring changes in the blood oxygen level dependent bold signal associated with neuronal activity the images obtained from fmri with epi however exhibit signal dropouts and geometric distortions parallel imaging pi due to its short readout accelerates image acquisition and might reduce dephasing in phase encoding direction the concomitant loss of signal to noise ratio snr might be compensated through single shot multi echo epi mepi we systematically compared the temporal snr and bold sensitivity of single echoes te 15 45 and 75ms and contrast optimized mepi with and without pi and mepi based denoising audio visual stimulation under natural viewing conditions activated distributed neural networks heterogeneous snr noise gain and sensitivity maps emerged in single echoes snr and bold sensitivity followed the predicted dependency on echo time te and were reduced under pi however the combination of echoes with mepi recovered the quality parameters and increased bold signal changes at circumscribed fronto polar and deep brain structures we suggest applying pi only in combination with mepi to reduce imaging artifacts and conserve bold sensitivity

broken ergodicity and the geometry of rugged landscapes we study the nature and mechanisms of broken ergodicity be in specific random walk models corresponding to diffusion on random potential surfaces in both one dimension and high dimension using both rigorous results and nonrigorous methods we confirm several aspects of the standard be picture and show that others apply in one dimension but need to be modified in higher dimensions these latter aspects include the notions that a fixed temperature confining barriers increase logarithmically with time that components are necessarily bounded regions of state space which depend on the observational time scale and that the system continually revisits previously traversed regions of state space we examine our results in the context of several experiments and discuss some implications of our results for the dynamics of disordered and or complex systems

the central limit theorem for markov chains started at a point the aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic markov chain on a general state space with respect to the law of the chain started at a point no irreducibility assumption nor mixing conditions are imposed the only assumption bears on the growth of the l 2 norms of the ergodic sums for the function generating the additive functional which must be with the result holds almost surely with respect to the invariant probability of the chain

spectral measure and approximation of homogenized coefficients this article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations the originality of this work is the use of a well known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients in particular we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error and conversely combined with results by otto and the first author in low dimension and results by the second author in high dimension this allows us to prove that for any dimension d 2 there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem

transience of second class particles and diffusive bounds for additive functionals in one dimensional asymmetric exclusion processes consider a one dimensional exclusion process with finite range translation invariant jump rates with nonzero drift let the process be stationary with product bernoulli invariant distribution at density place a second class particle initially at the origin for the case 1 2 we show that the time spent by the second class particle at the origin has finite expectation this strong transience is then used to prove that variances of additive functionals of local mean zero functions are diffusive when 1 2 as a corollary to previous work we deduce the invariance principle for these functionals the main arguments are comparisons of h 1 norms a large deviation estimate for second class particles and a relation between occupation times of second class particles and additive functional variances

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

central limit theorems for additive functionals of the simple exclusion process definition 2 1 in the above paper is incorrectly stated in the proof of theo rem 2 1 which gives an invariance principle for certain processes satisfying defi nition 2 1 conditions in definition 2 1 are sufficient to deduce finite dimensional convergence but not enough to apply a maximal inequality for demimartingales to obtain tightness the problem is definition 2 1 as stated only considers pair increment associations and not more general associations needed for the demi martingale property we slightly strengthen the definition here in this correction so that the proof of tightness in theorem 2 1 holds details of how this is accom plished are given below by substituting the corrected definition 2 1 for the previous one all results in the article hold as written in particular proposition 2 1 which is the link between theorem 2 1 and the main results and which states certain additive processes sat isfy definition 2 1 holds with the same argument

limit theorems for 2d invasion percolation we prove limit theorems and variance estimates for quantities related to ponds and outlets for 2d invasion percolation we first exhibit several properties of a sequence o n of outlet variables the nth of which gives the number of outlets in the box centered at the origin of side length 2n the most important of these properties describes the sequence s renewal structure and exponentially fast mixing behavior we use these to prove a central limit theorem and strong law of large numbers for o n we then show consequences of these limit theorems for the pond radii and outlet weights

outlets of 2d invasion percolation and multiple armed incipient infinite clusters we study invasion percolation in two dimensions focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters iic s first we compute the exact decay rate of the distribution of both the weight of the k th outlet and the volume of the k th pond next we prove bounds for all moments of the distribution of the number of outlets in an annulus this result leads to almost sure bounds for the number of outlets in a box b 2 n and for the decay rate of the weight of the k th outlet to pc we then prove existence of multiplearmed iic measures for any number of arms and for any color sequence which is alternating or monochromatic we use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin

investigation of spatial resolution partial volume effects and smoothing in functional mri using artificial 3d time series this work addresses the balance between temporal signal to noise ratio tsnr and partial volume effects pve in functional magnetic resonance imaging fmri and investigates the impact of the choice of spatial resolution and smoothing in fmri since physiological time courses are monitored tsnr is of greater importance than image snr improving snr by an increase in voxel volume may be of negligible benefit when physiological fluctuations dominate the noise furthermore at large voxel volumes pve are more pronounced leading to an overall loss in performance artificial fmri time series based on high resolution anatomical data were used to simulate bold activation in a controlled manner the performance was subsequently quantified as a measure of how well the resulted activation matched the simulated activation the performance was highly dependent on the spatial resolution at high contrast to noise ratio cnr the optimal voxel volume was small i e in the region of 2 3 mm 3 it was also shown that using a substantially larger voxel volume in this case could potentially negate the cnr benefits the optimal smoothing kernel width was dependent on the cnr being larger at poor cnr at cnr 1 little or no smoothing proved advantageous the use of artificial time series gave an opportunity to quantitatively investigate the effects of partial volume and smoothing in single subject fmri it was shown that a proper choice of spatial resolution and smoothing kernel width is important for fmri performance