quenched invariance principle for random walks in balanced random environment we consider random walks in a balanced random environment in mathbb z d d 2 we first prove an invariance principle for d 2 and the transience of the random walks when d 3 recurrence when d 2 in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition then using percolation arguments we show that under mere ellipticity the above results hold for random walks in i i d balanced environments

operator norm localization property of metric spaces under finite decomposition complexity abstract the notions of operator norm localization property and finite decomposition complexity were recently introduced in metric geometry to study the coarse novikov conjecture and the stable borel conjecture in this paper we show that a metric space x has weak finite decomposition complexity with respect to the operator norm localization property if and only if x itself has the operator norm localization property it follows that any metric space with finite decomposition complexity has the operator norm localization property in particular we obtain an alternative way to prove a very recent result by e guentner r tessera and g yu that all countable linear groups have the operator norm localization property

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

the speed of a biased random walk on a percolation cluster at high density we study the speed of a biased random walk on a percolation cluster on z d in function of the percolation parameter p we obtain a first order expansion of the speed at p 1 which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice

invariance principle for the random conductance model with unbounded conductances we study a continuous time random walk x in an environment of i i d random conductances e 1 we obtain heat kernel bounds and prove a quenched invariance principle for x this holds even when e e

quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights we consider a stationary and ergodic random field omega e e in e d that is parameterized by the edge set of the euclidean lattice mathbb z d d ge 2 the random variable omega e taking values in 0 infty and satisfying certain moment bounds is thought of as the conductance of the edge e assuming that the set of edges with positive conductances give rise to a unique infinite cluster mathcal c infty omega we prove a quenched invariance principle for the continuous time random walk among random conductances under certain moment conditions an essential ingredient of our proof is a new anchored relative isoperimetric inequality

a finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment we consider a nearest neighbor symmetric random walk on a homogeneous ergodic random lattice z d the jump rates of the walk are independent identically bernoulli distributed random variables indexed by the bonds of the lattice a standard result from the homogenization theory see a de masi p a ferrari s goldstein w d wick an invariance principle for reversible markov processes applications to random walks in random environments j statist phys 55 3 4 1989 787 855 asserts that the scaled trajectory of the particle satisfies the functional central limit theorem the covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk we use the duality structure corresponding to the product bernoulli measure to construct a numerical scheme that approximates this parameter when d 3 the estimates of the convergence rates are also provided

quenched invariance principles for random walks on percolation clusters we consider a supercritical bernoulli percolation model in d2 and study the simple symmetric random walk on the infinite percolation cluster the aim of this paper is to prove the almost sure q

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

a quenched invariance principle for non elliptic random walk in i i d balanced random environment we consider a random walk on mathbb z d d ge 2 in an i i d balanced random environment that is a random walk for which the probability to jump from x in mathbb z d to nearest neighbor x e is the same as to nearest neighbor x e assuming that the environment is genuinely d dimensional and balanced we show a quenched invariance principle for p almost every environment the diffusively rescaled random walk converges to a brownian motion with deterministic non degenerate diffusion matrix within the i i d setting our result extend both lawler s uniformly elliptic result comm math phys 87 1 pp 81 87 1982 1983 and guo and zeitouni s elliptic result to appear in ptrf 2010 to the general non elliptic case our proof is based on analytic methods and percolation arguments

quenched large deviations for simple random walks on percolation clusters including long range correlations we prove a quenched large deviation principle ldp for a simple random walk on a supercritical percolation cluster srwpc on mathbb z d d geq 2 the models under interest include classical bernoulli bond and site percolation as well as models that exhibit long range correlations like the random cluster model the random interlacement and the vacant set of random interlacements for d geq 3 and the level sets of the gaussian free field d geq 3 inspired by the methods developed by kosygina et al commun pure appl math 59 1489 1521 2006 for proving quenched ldp for elliptic diffusions with a random drift and by yilmaz commun pure appl math 62 8 1033 1075 2009 and rosenbluth quenched large deviations for multidimensional random walks in a random environment a variational formula ph d thesis nyu arxiv 0804 1444v1 for similar results regarding elliptic random walks in random environment we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures of the environment markov chain in the non elliptic case of srwpc via a contraction principle this reduces easily to a quenched ldp for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas the main difficulty in our set up lies in the inherent non ellipticity as well as the lack of translation invariance stemming from conditioning on the fact that the origin belongs to the infinite cluster we develop a unifying approach for proving quenched large deviations for srwpc based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

random walk on the infinite cluster of the percolation model we consider random walk on the infinite cluster of bond percolation on d we show that in the supercritical regime whend 3 this random walk is a s transient this conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance it is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

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

comparison of quenched and annealed invariance principles for random conductance model we show that there exists an ergodic conductance environment such that the weak annealed invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

existence of the harmonic measure for random walks on graphs and in random environments we give a sufficient condition for the existence of the harmonic measure from infinity of transient random walks on weighted graphs in particular this condition is verified by the random conductance model on d d 3 when the conductances are i i d and the bonds with positive conductance percolate the harmonic measure from infinity also exists for random walks on supercritical clusters of 2 this is proved using results of barlow ann probab 32 3024 3084 2004 and barlow and hambly electron j probab 14 1 1 27 2009

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

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

quenched invariance principle for random walks with time dependent ergodic degenerate weights we study a continuous time random walk xx on zdzd in an environment of dynamic random conductances taking values in 0 0 we assume that the law of the conductances is ergodic with respect to space time shifts we prove a quenched invariance principle for the markov process xx under some moment conditions on the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

optimal space varying regularization in iterative image restoration it has been shown that space variant regularization in image restoration provides better results than space invariant regularization however the optimal choice of the regularization parameter is usually unknown a priori in previous work the generalized cross validation gcv criterion was shown to provide accurate estimates of the optimal regularization parameter the author introduces a modified form of the gcv criterion that incorporates space variant regularization and data error terms furthermore he presents an efficient method for estimating the gcv criterion for the space variant case using iterative image restoration techniques this method performs nearly as well as the exact criterion for the image restoration problem in addition he proposes a wiener filter interpretation for choosing the local weighting of the regularization this interpretation suggests the use of a multistage estimation procedure to estimate the optimal choice of the local regularization weights experiments confirm the value of the modified gcv estimation criterion as well as the multistage procedure for estimating the local regularization weights

isoperimetry and heat kernel decay on percolation clusters we prove that the heat kernel on the infinite bernoulli percolation cluster in zd almost surely decays faster than t d 2 we also derive estimates on the mixing time for the random walk confined to a finite box our approach is based on local isoperimetric inequalities some of the results of this paper were previously announced in the note of mathieu and remy c r acad sci paris ser i math 332 2001 927 931

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

disorder entropy and harmonic functions we study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on z d we prove that the vector space of harmonic functions growing at most linearly is d 1 dimensional almost surely in particular there are no non constant sublinear harmonic functions thus implying the uniqueness of the corrector the main ingredient of the proof is a quantitative annealed version of the kaimanovich vershik entropy argument this also provides bounds on the derivative of the heat kernel simplifying and generalizing existing results the argument applies to many different environments even reversibility is not necessary we also mention several open problems and conjectures on the behavior of harmonic functions on stationary random graphs

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

diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice we construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal a dirichlet form can be constructed on the limit set and we consider the properties of the associated laplace operator and diffusion process in particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on diagonal heat kernel for the percolation clusters and for the underlying lattice in this setting a number of features of the lattice are inherited by the critical cluster

anchored nash inequalities and heat kernel bounds for static and dynamic degenerate environments abstract we introduce anchored versions of the nash inequality they allow to control the l 2 norm of a function by dirichlet forms that are not uniformly elliptic we then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments as an example we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

trapping in the random conductance model we consider random walks on d among nearest neighbor random conductances which are i i d positive bounded uniformly from above but whose support extends all the way to zero our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time 2n we show that in the situations when the heat kernel exhibits subdiffusive decay which is known to occur in dimensions d 4 the walk gets trapped for a time of order n in a small spatial region this shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant in addition we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions

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

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models