the circulant operator in the banach algebra of matrices abstract we study an operator c which maps every n by n matrix a n to a circulant matrix c a n that minimizes the frobenius norm a n c n f over all n by n circulant matrices c n the circulant matrix c a n called the optimal circulant preconditioner has proved to be a good preconditioner for a general class of toeplitz systems in this paper we give different formulations of the operator discuss its algebraic and geometric properties and compute its operator norms in different banach algebras of matrices using these results we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of i c n 1 a n f over all nonsingular circulant matrices c n

block preconditioners with circulant blocks for general linear systems block preconditioner with circulant blocks bpcb has been used for solving linear systems with block toeplitz structure since 1992 r chan x jin a family of block preconditioners for block systems siam j sci statist comput 13 1992 1218 1235 in this new paper we use bpcbs to general linear systems with no block structure usually the bpcbs are constructed by partitioning a general matrix into a block matrix with blocks of the same size and then applying t chan s optimal circulant preconditioner t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 to each block these bpcbs can be viewed as a generalization of t chan s preconditioner it is well known that the optimal circulant preconditioner works well for solving some structured systems such as toeplitz systems by using the preconditioned conjugate gradient pcg method but it is usually not efficient for solving general linear systems unlike t chan s preconditioner bpcbs used here are efficient for solving some general linear systems by the pcg method several basic properties of bpcbs are studied the relations of the block partition with the cost per iteration and the convergence rate of the pcg method are discussed numerical tests are given to compare the cost of the pcg method with different bpcbs

enlargement of obstacles for the simple random walk we consider a continuous time simple random walk moving among obstacles which are sites resp bonds of the lattice z d we derive in this context a version of the technique of enlargement of obstacles developed by sznitman in the brownian case this method gives controls on exponential moments of certain death times as well as good lower bounds for certain principal eigenvalues we give an application to recover an asymptotic result of donsker and varadhan on the number of sites visited by the random walk and another application to the number of bonds visited by the random walk

covering of a finite lattice by a random walk abstract we study the covering process by a simple random walk of a d dimensional periodic hypercubic lattice of n sites in d 1 the probability l n x for site x to be the last site visited in this covering process does not depend on x as long as x is not the starting point of the walk we argue that in dimensions d 2 the probability l n x approaches a constant value according to a coulomb law l n x 1 2 1 const x d 2 valid for x small on the scale n 1 d whereas it behaves logarithmically in d 2 also there is a dimension dependent characteristic time scale on which the last site is visited the structure of the set of sites not yet visited on this characteristic time scale is fractal like in d 2 in d 3 on the other hand this set is essentially distributed randomly through the lattice

random walks on discrete cylinders with large bases and random interlacements following the recent work of sznitman probab theory related fields 145 2009 143 174 we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form g n x z where g n is a large finite connected weighted graph and relate it to the model of random interlacements on infinite transient weighted graphs under suitable assumptions the set of points not visited by the random walk until a time of order g n 2 in a neighborhood of a point with z component of order g n converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the structure of the graph in the neighborhood of the point the level of the random interlacement depends on the local time of a brownian motion the result also describes the limit behavior of the joint distribution of the local pictures in the neighborhood of several distant points with possibly different limit models as examples of g n we treat the d dimensional box of side length n the sierpinski graph of depth n and the d ary tree of depth n where d 2

the best circulant preconditioners for hermitian toeplitz systems ii the multiple zero case in 10 14 circulant type preconditioners have been proposed for ill conditioned hermitian toeplitz systems that are generated by nonnegative continuous functions with a zero of even order the proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions it was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems in 10 14 it was claimed that this result can be the case where the generating functions have multiple zeros the main aim of this paper is to give a complete convergence proof of the method in 10 14 for this class of generating functions

a note on preconditioned block toeplitz matrices in siam j sci statist comput 13 1992 pp 948 966 ku and kuo proposed and analysed a block circulant preconditioner r mn for solving a family of block toeplitz systems t mn v b for a special class of block matrices called the quadrantally symmetric toeplitz matrices they proved that the eigenvalues of r mn 1 t mn are clustered around one except at most o m n outliers with t mn generated by a two dimensional rational function the superior convergence rate of the preconditioned conjugate gradient pcg method is explained by the clustering property of the spectrum of r mn 1 t mn however in their analysis there is no discussion on the positive definiteness of the matrix t mn and the preconditioner r mn is assumed to be invertible in this paper we give some results on these two aspects under the assumptions in the above referenced paper we prove that if the generating function f x y of t mn is positive then t mn is posit

m preconditioner for m matrices linear systems with m matrices often appear in a wide variety of areas of biological physical and social sciences see a berman r plemmons nonnegative matrices in mathematical sciences siam press philadelphia 1994 in this paper we propose a preconditioner for solving the system with symmetric nonsingular m matrix by the preconditioned conjugate gradient pcg method we show that our proconditioner increases the convergence rate of the pcg method and reduces the operation cost numerical results are also given

the generalized superoptimal preconditioner abstract for any given n by n matrix a a specific circulant preconditioner t f a introduced by tyrtyshnikov e tyrtyshnikov optimal and super optimal circulant preconditioners siam j matrix anal appl 13 1992 459 473 is defined to be the solution of min c i c 1 a f over all n by n nonsingular circulant matrices c the preconditioner t f a called the superoptimal circulant preconditioner has been proved to be a good preconditioner for a large class of structured systems in this paper we study this preconditioner in the general case by using the moore penrose inverse we give a formula for the superoptimal preconditioner and discuss the stability properties of this preconditioner a spectral relation between the optimal and superoptimal preconditioned matrices in the general case is also given

strong invariance for local times let y 1 y 2 be a sequence of i i d random variables with distribution p y 1 k p k k 1 2 e y 1 0 e y 1 2 2 put t n y 1 y n and n x n k 0 k n t k x extending the result of revesz 1981 it is shown that for appropriate skorohod construction we have 1 mathop text sup limits x in mathbb z text l x n sigma 2 text sigma text 2 n x n o n 1 mathord left vphantom 1 4 varepsilon right kern nulldelimiterspace 4 varepsilon text a text s text provided all moments e y 1 m m 0 exists where l is the local time of a wiener process certain rate of convergence is given also under weaker conditions and for l x n 2 2 n x n too when x is fixed

creating kappa like distributions from a galton board abstract the galton board is a classical device demonstrating geometrically the formation of normal distributions in view of long range interactions governed by nonextensive statistics the concept of the galton board is extended providing numerically the corresponding power law distributions within nonextensive statistics kappa distributions which are linked to q gaussians are a consequence from entropy generalization in this way the transition from normal distributions to kappa like distributions where kappa is the entropic index is available within a galton board concept where both the generalized distributions for positive and negative kappa like values are reproduced based on two similar numerical approaches it is shown how positive kappa like exponents are related to memory effects and negative kappa like exponents to enhanced interactions of increased order in the system as compared to the normal distribution case

cover times for brownian motion and random walks in two dimensions lett x denote the rst hitting time of the disc of radius centered at x for brownian motion on the two dimensional torus t 2 we prove that sup x2t2t x j log j 2 2 as 0 the same applies to brownian motion on any smooth compact connected two dimensional riemannian manifold with unit area and no boundary as a consequence we prove a conjecture due to aldous 1989 that the number of steps it takes a simple random walk to cover all points of the lattice torus z 2 is asymptotic to 4n 2 logn 2 determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete so far this structure was only studied non rigorously in the physics literature we also establish a conjecture due to kesten and r ev esz that describes the asymptotics for the number of steps needed by simple random walk in z 2 to cover the disc of radius n

stochastic one dimensional lorentz gas on a lattice we stud ya one dimensional stochastic lorentz gas wher ea light particle moves i na fixed array of nonidentical random scatterers arranged i na lattice each scatterer is characterized b ya random transmissionreflection coefficient we consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables a symbolic program is presented which generates the exact velocity autocorrelation functio n vacf in terms of the moments of the transmission coefficients the vacf is found for different types of disorder for times up to 20 collision times we then conside ra specific type of disorder a two state lorentz gas in which two types of scatterers are arranged randomly i na lattice the na lattice point is occupied b ya scatterer whose transmission coefficient is with probability p or with probability 1 p a perturbation expansion with respect to is derived the 2 term in this expansion shows that the vacf oscillates with time the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor the coarse grained vacf decays for long times like t 32 which is similar to the decay of the vacf of the random lorentz gas wit ha single type of scatterer the perturbation results and the exact one s found up to 20 collision times show good agreement

a circulant preconditioner for the systems of lmf based ode codes in this paper a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations odes is investigated most ode codes based on implicit formulas at each integration step need the solution of one or more unsymmetric linear systems that are often large and sparse here the boundary value methods a class of implicit methods for the numerical integration of odes based on linear multistep formulas are considered more in detail for initial value problems theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems moreover the p circulants a recently introduced circulant approximation for unsymmetric toeplitz matrices are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner

random walks on discrete cylinders and random interlacements we explore some of the connections between the local picture left by the trace of simple random walk on a cylinder mathbb z n mathbb z d times mathbb z d 2 running for times of order n2d and the model of random interlacements recently introduced in sznitman http www math ethz ch u sznitman preprints in particular we show that for large n in the neighborhood of a point of the cylinder with vertical component of order nd the complement of the set of points visited by the walk up to times of order n2d is close in distribution to the law of the vacant set of random interlacements with a level which is determined by an independent brownian local time the limit behavior of the joint distribution of the local pictures in the neighborhood of finitely many points is also derived

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

on new examples of ballistic random walks in random environment in this article we show that random walks in random environment on zd d 3 with transition probabilities which are e perturbations of the simple random walk and such that the expectation of the local drift has size bigger than e with 52 when d 3 3 when d 4 fulfill the condition t introduced by sznitman prob theory related fields 2002 122 509 544 when e is small as a result these walks satisfy a law of large numbers with nondegenerate limiting velocity a central limit theorem and several large deviation controls in particular this provides examples of ballistic random walks in random environment which do not satisfy kalikow s condition in the terminology of sznitman and zerner ann probab 1999 27 1851 1869 an important tool in this work is the effective criterion of sznitman

on infinite volume mixing in the context of the long standing issue of mixing in infinite ergodic theory we introduce the idea of mixing for observables possessing an infinite volume average the idea is borrowed from statistical mechanics and appears to be relevant at least for extended systems with a direct physical interpretation we discuss the pros and cons of a few mathematical definitions that can be devised testing them on a prototypical class of infinite measure preserving dynamical systems namely the random walks

a generalization of t chan s preconditioner abstract in this paper we propose a new preconditioner for solving linear systems by the preconditioned conjugate gradient pcg method the preconditioner can be thought of as a generalization of the well known t chan s preconditioner for hermitian positive definite matrix we observe that our preconditioner is also hermitian positive definite the operation cost and convergence of the pcg method are discussed numerical experiments have been performed on structured problems to show the competitiveness of this preconditioner

a survey and some extensions of t chan s preconditioner abstract for any given n by n matrix an t chan s circulant preconditioner cf an proposed by t chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 is defined to be the solution of min c n a n c n f over all n by n circulant matrices cn the cf an called the optimal circulant preconditioner by t chan has been proved to be a good preconditioner for a large class of structured systems a generalization of t chan s circulant preconditioner was given by huckle t huckle circulant and skew circulant matrices for solving toeplitz matrix problems siam j matrix anal appl 13 1992 767 777 which can be used for solving some general linear systems in this paper we review some old and develop some new properties of t chan s circulant preconditioner

on the size of a finite vacant cluster of random interlacements with small intensity in this paper we establish some properties of percolation for the vacant set of random interlacements for d 5 and small intensity u the model of random interlacements was first introduced by sznitman in ann math arxiv 0704 2560 2010 it is known that for small u almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u see sidoravicius and sznitman commun pure appl math 62 6 831 858 2009 and teixeira ann appl probab 19 1 454 466 2009 we estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin given that it is finite this comes as a by product of our main theorem which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube as another application we show that with high probability the unique infinite connected component of the vacant set is ubiquitous in large neighborhoods of the origin

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

a note on t chan s preconditioner abstract a matrix is said to be stable if the real part of all the eigenvalues are negative in this note for some class of matrices we show that t chan s preconditioner is stable by using this result we prove the invertibility of some preconditioners proposed in numerical odes recently

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

the galton board limit theorems and recurrence we study a particle moving in r2 under a constant external force and bouncing off a periodic array of convex domains scatterers the latter must satisfy a standard finite horizon condition to prevent ballistic collision free motion this model is known to physicists as galton board it is also identical to a periodic lorentz gas previous heuristic and experimental studies have suggested that the particle s speed v t should grow as t1 3 and its coordinate x t as t2 3 we prove these conjectures rigorously we also find limit distributions for the rescaled velocity t 1 3v t and position t 2 3x t in addition quite unexpectedly we discover that the particle s motion is recurrent that means that a ball dropped on an idealized galton board will roll down but from time to time it should bounce all the way back up with probability one

on the disconnection of a discrete cylinder by a random walk we investigate the large n behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder we show that when d 2 this time is roughly of order n 2 d and comparable to the cover time of the slice but substantially larger than the cover timer of the base by the projection of the walk further we show that by the time disconnection occurs a massive clogging typically takes place in the truncated cylinders of height these mechanisms are in contrast with what happens when d 1

some stability properties of t chan s preconditioner abstract a matrix is said to be stable if the real parts of all the eigenvalues are negative in this paper for any matrix a n we give some sufficient and necessary conditions for the stability of t chan s preconditioner c u a n

fluctuation of the entropy production for the lorentz gas under small external forces in this paper we study the physical and statistical properties of the periodic lorentz gas with finite horizon driven to a non equilibrium steady state by the combination of non conservative external forces and deterministic thermostats a version of this model was introduced by chernov eyink lebowitz and sinai and subsequently generalized by chernov and the third author non equilibrium steady states for these models are srb measures and they are characterized by the positivity of the steady state entropy production rate our main result is to establish that the entropy production in this context equal to the phase space contraction satisfies the gallavotti cohen fluctuation relation the main tool needed in the proof is the family of anisotropic banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques

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

painting a graph with competing random walks let x1 x2 be independent random walks on zdn d 3 each starting from the uniform distribution initially each site of zdn is unmarked and whenever xi visits such a site it is set irreversibly to i the mean of ai the cardinality of the set ai of sites painted by i once all of zdn has been visited is 12nd by symmetry we prove the following conjecture due to pemantle and peres for each d 3 there exists a constant d such that limn var ai hd n 14 d where h3 n n4 h4 n n4 logn and hd n nd for d 5 we will also identify d explicitly and show that d 1 as d this is a special case of a more general theorem which gives the asymptotics of var ai for a large class of transient vertex transitive graphs other examples include the hypercube and the caley graph of the symmetric group generated by transpositions

design and analysis of toeplitz preconditioners the solution of symmetric positive definite toeplitz systems ax b by the preconditioned conjugate gradient pcg method was recently proposed by strang 1986 and analyzed by chan 1989 and chan and strang 1989 the convergence rate of the pcg method depends on the choice of preconditioners for the given toeplitz matrices the authors present a general approach to the design of toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions all resulting preconditioners can therefore be inverted via various fast transform algorithms with o n log n operations for a wide class of problems the pcg method converges in a finite number of iterations independent of n so that the computational complexity for solving these toeplitz systems is o n log n

circulant preconditioned wr bvm methods for ode systems we consider the solution of a system of ordinary differential equations odes by waveform relaxation wr iterations in conjunction with boundary value methods bvms the wr method is a continuous in time analogue of the stationary method and it iterates with functions in each wr iteration we use bvms to discretize systems of odes bvms are relatively new ode solvers based on linear multistep formulae in this paper we discuss the use of the generalized minimal residual gmres method with block circulant circulant block preconditioners for solving the linear systems arising from the application of bvms in each wr iteration these preconditioners are effective in speeding up the convergence rate of the gmres method numerical experiments are presented to illustrate the effectiveness of our methods

the statistics of the trajectory of a certain billiard in a flat two torus we consider a billiard in the punctured torus obtained by removing a small disk of radius 0 from the flat torus 2 with trajectory starting from the center of the puncture in this case the phase space is given by the range of the velocity only let and respectively r denote the first exit time length of the trajectory and respectively the number of collisions with the side cushions when 2 is being identified with 0 1 2 we prove that the probability measures on 0 associated with the random variables and r are weakly convergent as varepsilon rightarrow 0 and explicitly compute the densities of the limits

on the time taken by random walks on finite groups to visit every state consider a random walk on a finite group g suppose that the time taken for the transition probabilities to approach the uniform distribution is small compared with g then the time taken for the random walk to visit every state is approximately r g log g where r is the mean number of returns to the initial state in the short term

vacant set of random interlacements and percolation we introduce a model of random interlacements made of a countable collection of doubly infinite paths on d d 3 a nonnegative parameter u measures how many trajectories enter the picture this model describes in the large n limit the microscopic structure in the bulk which arises when considering the disconnection time of a discrete cylinder n d 1 x by simple random walk or the set of points visited by simple random walk on the discrete torus n d at times of order u n d in particular we study the percolative properties of the vacant set left by the interlacement at level u which is an infinite connected translation invariant random subset of d we introduce a critical value u such that the vacant set percolates for u u our main results show that u is finite when d 3 and strictly positive when d 7

recurrence properties of a special type of heavy tailed random walk in the proof of the invariance principle for locally perturbed periodic lorentz process with finite horizon a lot of delicate results were needed concerning the recurrence properties of its unperturbed version these were analogous to the similar properties of simple symmetric random walk however in the case of lorentz process with infinite horizon the analogous results for the corresponding random walk are not known either in this paper these properties are ascertained for the appropriate random walk this happens to be in the non normal domain of attraction of the normal law as a tool an estimation of the remainder term in the local limit theorem for the corresponding random walk is computed

percolation for the vacant set of random interlacements we investigate random interlacements on d d 3 this model recently introduced in 8 corresponds to a poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times a nonnegative parameter u measures how many trajectories enter the picture our main interest lies in the percolative properties of the vacant set left by random interlacements at level u we show that for all d 3 the vacant set at level u percolates when u is small this solves an open problem of 8 where this fact has only been established when d 7 it also completes the proof of the nondegeneracy in all dimensions d 3 of the critical parameter u of 8 2008 wiley periodicals inc

convergence analysis of superoptimal pcg algorithm for toeplitz systems with a fisher hartwig singularity recently lu and hurvich y lu c hurvich on the complexity of the preconditioned conjugate gradient algorithm for solving toeplitz systems with a fisher hartwig singularity siam j matrix anal appl 27 2005 638 653 used the preconditioned conjugate gradient method with the optimal circulant preconditioner proposed in chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 for solving the toeplitz system t n f x b where the generating function f is given by f 1 e i 2d h with d e 7 7 0 the function h is positive continuous on and differentiable on 0 in this paper we will use the superoptimal circulant preconditioner proposed by tyrtyshnikov e tyrtyshnikov optimal and superoptimal circulant preconditioners siam j matrix anal appl 13 1992 459 473 to solve the same problem when 0 d 1 2 our convergence analysis shows that the number of iterations is bounded by o log 3 n and therefore the complexity of our algorithm is o n log 4 n we notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner

optimal preconditioners for functions of matrices abstract the optimal preconditioner c u a of a given matrix a was proposed in 1988 by t chan 6 since then it has been proved to be efficient for solving a large class of structured systems in this paper we construct the optimal preconditioners for different functions of matrices more precisely let f be a function of matrices from c n n to c n n given a c n n there are two possible optimal preconditioners for f a c u f a and f c u a in the paper we study properties of both c u f a and f c u a for different functions of matrices numerical experiments are given to illustrate the efficiency of the optimal preconditioners when they are used to solve f a x b

decoupling inequalities and interlacement percolation on g z we study the percolative properties of random interlacements on g where g is a weighted graph satisfying certain sub gaussian estimates attached to the parameters 1 and 2 1 describing the respective polynomial growths of the volume on g and of the time needed by the walk on g to move to a distance we develop decoupling inequalities which are a key tool in showing that the critical level u for the percolation of the vacant set of random interlacements is always finite in our set up and that it is positive when 1 2 we also obtain several stretched exponential controls both in the percolative and non percolative phases of the model even in the case where g d d 2 several of these results are new