asymptotics for euclidean minimal spanning trees on random points asymptotic results for the euclidean minimal spanning tree onn random vertices inrd can be obtained from consideration of a limiting infinite forest whose vertices form a poisson process in allrd in particular we prove a conjecture of robert bland the sum of thed th powers of the edge lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofrd tends to a constant asn

universality of high dimensional spanning forests and sandpiles we prove that the wired uniform spanning forest exhibits mean field behaviour on a very large class of graphs including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph several of our results are new even in the case of mathbb z d d ge 5 in particular we prove that every tree in the forest has spectral dimension 4 3 and walk dimension 3 almost surely and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1 2 respectively the past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest we obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth and is 1 on any bounded degree nonamenable graph we deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the abelian sandpile model are 2 and 1 2 respectively for any transitive graph with polynomial growth of dimension at least five and are 1 and 1 2 respectively for any bounded degree nonamenable graph in the case of mathbb z d d ge 5 some of our results regarding critical exponents recover earlier results of bhupatiraju et al electron j probab 22 85 51 2017 in this case we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant order multiplicative errors rather than the polylogarithmic order multiplicative errors present in that work

noise induced drift in stochastic differential equations with arbitrary friction and diffusion in the smoluchowski kramers limit we consider the dynamics of systems with arbitrary friction and diffusion these include as a special case systems for which friction and diffusion are connected by einstein fluctuation dissipation relation e g brownian motion we study the limit where friction effects dominate the inertia i e where the mass goes to zero smoluchowski kramers limit using the ito stochastic integral convention we show that the limiting effective langevin equations has different drift fields depending on the relation between friction and diffusion alternatively our results can be cast as different interpretations of stochastic integration in the limiting equation which can be parametrized by interestingly in addition to the classical ito 0 stratonovich 0 5 and anti ito 1 integrals we show that position dependent x and even stochastic integrals with 0 1 arise our findings are supported by numerical simulations

field theory conjecture for loop erased random walks we give evidence that the functional renormalization group frg developed to study disordered systems may provide a field theoretic description for the loop erased random walk lerw allowing to compute its fractal dimension in a systematic expansion in e 4 d up to two loop the frg agrees with rigorous bounds correctly reproduces the leading logarithmic corrections at the upper critical dimension d 4 and compares well with numerical studies we obtain the universal subleading logarithmic correction in d 4 which can be used as a further test of the conjecture

critical percolation and the minimal spanning tree in slabs the minimal spanning forest on d is known to consist of a single tree for d 2 and is conjectured to consist of infinitely many trees for large d in this paper we prove that there is a single tree for quasi planar graphs such as 2 0 k d 2 our method relies on generalizations of the gluing lemma of duminil copin sidoravicius and tassion a related result is that critical bernoulli percolation on a slab satisfies the box crossing property its proof is based on a new russo seymour welsh type theorem for quasi planar graphs thus at criticality the probability of an open path from 0 of diameter n decays polynomially in n this strengthens the result of duminil copin et al where the absence of an infinite cluster at criticality was first established 2017 wiley periodicals inc

infinite volume limit of the abelian sandpile model in dimensions d 3 we study the abelian sandpile model on mathbb z d in d 3 we prove existence of the infinite volume addition operator almost surely with respect to the infinite volume limit of the uniform measures on recurrent configurations we prove the existence of a markov process with stationary measure and study ergodic properties of this process the main techniques we use are a connection between the statistics of waves and uniform two component spanning trees and results on the uniform spanning forest measure on mathbb z d

iterative tomographic image reconstruction using fourier based forward and back projectors iterative image reconstruction algorithms play an increasingly important role in modern tomographic systems especially in emission tomography with the fast increase of the sizes of the tomographic data reduction of the computation demands of the reconstruction algorithms is of great importance fourier based forward and back projection methods have the potential to considerably reduce the computation time in iterative reconstruction additional substantial speed up of those approaches can be obtained utilizing powerful and cheap off the shelf fast fourier transform fft processing hardware the fourier reconstruction approaches are based on the relationship between the fourier transform of the image and fourier transformation of the parallel ray projections the critical two steps are the estimations of the samples of the projection transform on the central section through the origin of fourier space from the samples of the transform of the image and vice versa for back projection interpolation errors are a limitation of fourier based reconstruction methods we have applied min max optimized kaiser bessel interpolation within the nonuniform fft nufft framework and devised ways of incorporation of resolution models into the fourier based iterative approaches numerical and computer simulation results show that the min max nufft approach provides substantially lower approximation errors in tomographic forward and back projection than conventional interpolation methods our studies have further confirmed that fourier based projectors using the nufft approach provide accurate approximations to their space based counterparts but with about ten times faster computation and that they are viable candidates for fast iterative image reconstruction

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

forward backward stochastic differential equations on infinite horizon and quasilinear elliptic pdes 2019 elsevier inc a class of infinite horizon forward backward stochastic differential equations fbsdes is investigated under some monotonicity conditions the existence and uniqueness of solutions in an arbitrarily large space for fbsdes on infinite horizon is obtained the probabilistic interpretations for a large class of quasilinear elliptic partial differential equations pdes in a global space is then given by virtue of the solutions of fbsdes on infinite horizon

convolution based interpolation for fast high quality rotation of images this paper focuses on the design of fast algorithms for rotating images and preserving high quality the basis for the approach is a decomposition of a rotation into a sequence of one dimensional translations as the accuracy of these operations is critical we introduce a general theoretical framework that addresses their design and performance we also investigate the issue of optimality and present an improved least square formulation of the problem this approach leads to a separable three pass implementation of a rotation using one dimensional convolutions only we provide explicit filter formulas for several continuous signal models including spline and bandlimited representations finally we present rotation experiments and compare the currently standard techniques with the various versions of our algorithm our results indicate that the present algorithm in its higher order versions outperforms all standard high accuracy methods of which we are aware both in terms of speed and quality its computational complexity increases linearly with the order of accuracy the best quality results are obtained with the sine based algorithm which can be implemented using simple one dimensional ffts

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

linear interpolation revitalized we present a simple original method to improve piecewise linear interpolation with uniform knots we shift the sampling knots by a fixed amount while enforcing the interpolation property we determine the theoretical optimal shift that maximizes the quality of our shifted linear interpolation surprisingly enough this optimal value is nonzero and close to 1 5 we confirm our theoretical findings by performing several experiments a cumulative rotation experiment and a zoom experiment both show a significant increase of the quality of the shifted method with respect to the standard one we also observe that in these results we get a quality that is similar to that of the computationally more costly high quality cubic convolution

high precision entropy values for spanning trees in lattices shrock and wu have given numerical values for the exponential growth rate of the number of spanning trees in euclidean lattices we give a new technique for numerical evaluation that gives much more precise values together with rigorous bounds on the accuracy in particular the new values resolve one of their questions

a multilevel domain decomposition algorithm for fast o n sup 2 logn reprojection of tomographic images a novel algorithm for fast computation of tomographic image projections is presented the method comprises a decomposition of an image into subimages followed by an aggregation of projections computed for the subimages the multilevel domain decomposition algorithm is formulated as a recursive procedure the computational cost of the proposed algorithm is comparable to that of fft based techniques while it appears to be more flexible than the latter numerical results demonstrate the effectiveness of the method

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

stochastic averaging in continuous time and its applications to extremum seeking we investigate stochastic averaging theory in continuous time for locally lipschitz systems and the applications of this theory to stability analysis of stochastic extremum seeking algorithms first we establish a general stochastic averaging principle and some related stability theorems for a class of continuous time nonlinear systems with stochastic perturbations and remove or weaken several significant restrictions present in existing results global lipschitzness of the nonlinear vector field equilibrium preservation under the stochastic perturbation global exponential stability of the average system and compactness of the state space of the perturbation process then we propose a continuous time extremum seeking algorithm with stochastic excitation signals instead of deterministic periodic signals we analyze the stability of stochastic extremum seeking for static maps and for general nonlinear dynamic systems

critical behaviour of self avoiding walks that cross a square consider the set of all self avoiding walks in the square lattice which start at 0 0 end at l l and are entirely contained in the square 0 l 0 l associate a fugacity x with each step of the walk whittington and guttmann 1990 showed that the dominant walks have o l steps when x is small and o l2 steps when x is large and they conjectured that there is a single transition point at x mu 1 where mu is the inverse of the connective constant for unconstrained self avoiding walks we present a rigorous proof of this conjecture and its analogue in higher dimensions we also discuss what can be said rigorously about two scaling exponents associated with this phase transition and compare this with analogous results that have been obtained exactly and rigorously on the discrete sierpinski gasket by hattori hattori and kusuoka 1990

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

random graph asymptotics on high dimensional tori we investigate the scaling of the largest critical percolation cluster on a large d dimensional torus for nearest neighbor percolation in sufficiently high dimensions or when d 6 for sufficiently spread out percolation we use a relatively simple coupling argument to show that this largest critical cluster is with high probability bounded above by a large constant times v 2 3 and below by a small constant times v 2 3 log v 4 3 where v is the volume of the torus we also give a simple criterion in terms of the subcritical percolation two point function on mathbb z d under which the lower bound can be improved to small constant times v 2 3 i e we prove random graph asymptotics for the largest critical cluster on the high dimensional torus this establishes a conjecture by 1 apart from logarithmic corrections we discuss implications of these results on the dependence on boundary conditions for high dimensional percolation

on brownian motion simple paths and loops we provide a decomposition of the trace of the brownian motion into a simple path and an independent brownian soup of loops that intersect the simple path more precisely we prove that any subsequential scaling limit of the loop erased random walk is a simple path a new result in three dimensions which can be taken as the simple path of the decomposition in three dimensions we also prove that the hausdorff dimension of any such subsequential scaling limit lies in 1 frac 5 3 we conjecture that our decomposition characterizes uniquely the law of the simple path if so our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance

loop erased random walk on a torus in dimensions 4 and above we show that the statistics of loop erased random walks above the upper critical dimension 4 are different between the torus and the full space the typical length of the path connecting a pair of sites at distance l which scales as l2 in the full space changes under the periodic boundary conditions to ld 2 the results are precise for dimensions 5 for the dimension d 4 we prove an upper bound conjecturally sharp up to subpolyonmial factors

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

curl flux induced drift in stochastic differential equations in the zero mass limit abstract we consider the nonlinear stochastic dynamics of dissipative hamiltonian systems with state dependent friction and diffusion connected by the fluctuation dissipation relation in high dimensions the system under study has a close connection to ao s framework in constructing a dynamical potential for non equilibrium processes without detailed balance we study the limiting case where the mass approaches zero and give a new and complete derivation of effective stochastic differential equations using the ito stochastic integral convention we show that the limiting effective langevin equations have a new drift term this extra term happens to be identical to the corresponding anti ito or isothermal integral requiring constant temperature in one dimension we however cannot obtain this additional drift term using conventional stochastic integrals in high dimension it is interesting to note that in a high dimensional system a curl flux induced drift may appear even if the diffusion matrix is constant our findings are supported by numerical simulations we further analyze and discuss the role of this new drift term in calculating the classic escape time for the first time to our knowledge the relation between the escape rate and the anti ito integral is presented we also demonstrate that the derived diffusion equations give a new sampling algorithm which can increase convergence speed in a simple two dimensional example

distances between transition probabilities of diffusions and applications to nonlinear fokker planck kolmogorov equations abstract we estimate the total variation and kantorovich distances between transition probabilities of two diffusions with different diffusion matrices and drifts via a natural quadratic distance between the drifts and diffusion matrices applications to nonlinear fokker planck kolmogorov equations optimal control and mean field games are given

a fast discrete approximation algorithm for the radon transform this paper addresses fast parallel methods for the computation of the radon or hough transform the radon transform of an image is a set of projections of the image taken at different angles its computation is important in image processing and computer vision for problems such as pattern recognition and reconstruction of medical images a unique new method for combining partial results is presented from which an algorithm is constructed that computes a provably good approximation to the discrete radon transform the approximate discrete radon transform adrt algorithm computes 4 n 4 projections through an n times n image in time o n 2 mathrm lg n the majority of previous algorithms are o n 3 the method is quite simple and easy to parallelize a parallel version of the adrt requires only o mathrm lg n parallel steps on o n 2 processors ignoring communication time an additional property of the algorithm is that it can be applied directly to compute the backprojection step of the inverse rt

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

averaging principle of sde with small diffusion moderate deviations consider the following stochastic differential equation in rd dxetxe0 b xet t e dt e a xet t e dwt x0 where the random environment t is an exponentially ergodic markov process independent of the wiener process wt with invariant probability measure and e is some small parameter in this paper we prove the moderate deviations for the averaging principle of xe that is deviations of xet around its limit averaging system x t given by d dx t b x t dt and x 0 x0 where b x e b x more precisely we obtain the large deviation estimation about et xet x te h e t 0 1 in the space of continuous trajectories as e decreases to 0 where h e is some deviation scale satisfying 1 h e e 1 2 our strategy will be first to show the exponential tightness and then the local moderate deviation principle which relies on some new method involving a conditional schilder s theorem and a moderate deviation principle for inhomogeneous integral functionals of markov processes previously established by the author

fast tomographic reconstruction via rotation based hierarchical backprojection we introduce a family of fast algorithms for tomographic backprojection in the parallel beam geometry the algorithms reduce the computational cost of backprojecting p projections onto an n times n pixel image from the conventional o n 2 p to o n 2 log p the new algorithms aggregate the projections in a hierarchical structure with images in the hierarchy formed by the rotation and addition of other images made up of fewer projections while these algorithms are related to existing fast algorithms this work places them within the signal processing framework providing a systematic means to optimize and adjust the trade off between computational cost and accuracy rotations are performed separably in order that higher order interpolators may be used with low computational cost the same ideas are applied to create a tomographic projection algorithm which computes projections of an n times n pixel image onto p view angles at a cost of o n 2 log p

geometry of the uniform spanning forest transitions in dimensions 4 8 12 the uniform spanning forest usf in d is the weak limit of random uniformly chosen spanning trees in n n d pemantle 11 proved that the usf consists a s of a single tree if and only if d 4 we prove that any two components of the usf in d are adjacent a s if 5 d 8 but not if d 9 more generally let n x y be the minimum number of edges outside the usf in a path joining x and y in d then max n x y x y in mathbb z d left lfloor d 1 4 right rfloor textrm a s

shear based fast hierarchical backprojection for parallel beam tomography we introduce a family of fast algorithms for 2 d parallel beam tomographic backprojection they aggregate the projections in a hierarchical structure involving the shearing and addition of sparsely sampled images the algorithms achieve a computational cost of o n2 logp when backprojecting an ntimesn pixel image from p projections the algorithms provide a systematic means guided by a fourier domain interpretation to adjust and optimize the tradeoff between computational cost and accuracy in an example with n 512 and p 1458 the algorithms provide high accuracy with more than an order of magnitude reduction in operation counts

o n sup 2 log sub 2 n filtered backprojection reconstruction algorithm for tomography we present a new fast reconstruction algorithm for parallel beam tomography the new algorithm is an accelerated version of the standard filtered backprojection fbp reconstruction and uses a hierarchical decomposition of the backprojection operation to reduce the computational cost from o n sup 3 to o n sup 2 log sub 2 n we discuss the choice of the various parameters that affect the behavior of the algorithm and present numerical studies that demonstrate the cost versus distortion tradeoff comparisons with fourier reconstruction algorithms and a multilevel inversion algorithm by brandt et al both of which also have o n sup 2 log sub 2 n cost suggest that the proposed hierarchical scheme has a superior cost versus distortion performance it offers rms reconstruction errors comparable to the fbp with considerable speedup for an example with a 512 spl times 512 pixel image and 1024 views the speedup achieved with a particular implementation is over 40 fold with reconstructions visually indistinguishable from the fbp

indistinguishability of trees in uniform spanning forests we prove that in both the free and the wired uniform spanning forest fusf and wusf of any unimodular random rooted network in particular of any cayley graph it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely this confirms a conjecture of benjamini et al ann probab 29 1 1 65 2001 we also answer positively two additional questions of benjamini et al ann probab 29 1 1 65 2001 under the assumption of unimodularity we prove that on any unimodular random rooted network the fusf is either connected or has infinitely many connected components almost surely and if the fusf and wusf are distinct then every component of the fusf is transient and infinitely ended almost surely all of these results are new even for cayley graphs

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

spanning trees on two dimensional lattices with more than one type of vertex for a two dimensional lattice with n vertices the number of spanning trees nst grows asymptotically as exp nz in the thermodynamic limit we present an exact integral expression and a numerical value for the entropy asymptotic growth constant z for spanning trees on 19 two dimensional lattices with more than one type of vertex given in o keeffe and hyde 1980 philos trans r soc a 295 553 especially an exact closed form expression for the entropy is derived for net 14 and the entropies of net 27 and the triangle lattice have the simple relation z27 ztri ln 4 4 some integral identities are also obtained

exponential tail bounds for loop erased random walk in two dimensions let m n be the number of steps of the loop erasure of a simple random walk on 2 from the origin to the circle of radius n we relate the moments of m n to es n the probability that a random walk and an independent loop erased random walk both started at the origin do not intersect up to leaving the ball of radius n this allows us to show that there exists c such that for all n and all k 1 2 e m k n and hence to establish exponential moment bounds for m n this implies that there exists c 0 such that for all n and all 0 p m n e m n 2e c using similar techniques we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any 0 such that for all n and 0 p m n 1 e m n ce c

auxiliary sdes for homogenization of quasilinear pdes with periodic coefficients we study the homogenization property of systems of quasi linear pdes of parabolic type with periodic coefficients highly oscillating drift and highly oscillating nonlinear term to this end we propose a probabilistic approach based on the theory of forward backward stochastic differential equations and introduce the new concept of auxiliary sdes

singular homogenization with stationary in time and periodic in space coefficients we study the homogenization problem for a random parabolic operator with coefficients rapidly oscillating in both the space and time variables and with a large highly oscillating nonlinear potential in a general stationary and mixing random media which is periodic in space it is shown that a solution of the corresponding cauchy problem converges in law to a solution of a limit stochastic pde 2005 published by elsevier inc msc 35k57 60h15

a fast and accurate multilevel inversion of the radon transform a number of imaging technologies reconstruct an image function from its radon projection using the convolution backprojection method the convolution is an o n2 log n algorithm where the image consists of n x n pixels while the backprojection is an o n3 algorithm thus constituting the major computational burden of the convolution backprojection method an o n2 log n multilevel backprojection method is presented here when implemented with a fourier domain postprocessing technique also presented here the resulting image quality is similar or superior to the image quality of the classical backprojection technique

discrete systems and abelian sandpiles abstract computations on the sandpile model lead to questions concerning discrete behaviors it is shown that the original question has an answer which is not helpful for the study of sandpiles a variation of the question using periodic solutions leads to better answers these might have an interpretation for the study of sandpiles

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

on the poisson equation and diffusion approximation 3 three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one the first one states the smoothness of an invariant diffusion density with respect to a parameter the second establishes a similar smoothness of the solution of the poisson equation in rd the third one states a diffusion approximation result or in other words an averaging of singularly perturbed diffusion for fully coupled sde systems or sde systems with complete dependence

minimal spanning forests minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs these limits can be taken with free or wired boundary conditions and are denoted fmsf free minimal spanning forest and wmsf wired minimal spanning forest respectively the wmsf is also the union of the trees that arise from invasion percolation started at all vertices we show that on any cayley graph where critical percolation has no infinite clusters all the component trees in the wmsf have one end a s in z d this was proved by alexander ann probab 23 1995 87 104 but a different method is needed for the nonamenable case we also prove that the wmsf components are thin in a different sense namely on any graph each component tree in the wmsf has p c 1 a s where p c denotes the critical probability for having an infinite cluster in bernoulli percolation on the other hand the fmsf is shown to be thick on any connected graph the union of the fmsf and independent bernoulli percolation with arbitrarily small parameter is a s connected in conjunction with a recent result of gaboriau this implies that in any cayley graph the expected degree of the fmsf is at least the expected degree of the fsf the weak limit of uniform spanning trees we also show that the number of infinite clusters for bernoulli p u percolation is at most the number of components of the fmsf where p u denotes the critical probability for having a unique infinite cluster finally an example is given to show that the minimal spanning tree measure does not have negative associations

a tree based scaling exponent for random cluster models for models of random trees in the hypercubic lattice or more general random clusters to which tree structure may be added the authors introduce a novel scaling exponent for the particular model of uniform random spanning trees in two and three dimensions its value is estimated by monte carlo simulation

uniqueness of maximal entropy measure on essential spanning forests an essential spanning forest of an infinite graph g is a spanning forest of g in which all trees have infinitely many vertices let gn be an increasing sequence of finite connected subgraphs of g for which gn g pemantle s arguments imply that the uniform measures on spanning trees of gn converge weakly to an aut g invariant measure g on essential spanning forests of g we show that if g is a connected amenable graph and aut g acts quasitransitively on g then g is the unique invariant measure on essential spanning forests of g for which the specific entropy is maximal this result originated with burton and pemantle who gave a short but incorrect proof in the case d lyons discovered the error and asked about the more general statement that we prove

large deviations of convex hulls of self avoiding random walks a global picture of a random particle movement is given by the convex hull of the visited points we obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self avoiding random walks namely the classical self avoiding walk the smart kinetic self avoiding walk and the loop erased random walk to obtain a comprehensive description of the measured random quantities we applied sophisticated large deviation techniques which allowed us to obtain the distributions over a large range of support down to probabilities far smaller than p 10 ensuremath 100 we give an approximate closed form of the so called large deviation rate function mathrm ensuremath phi which generalizes above the upper critical dimension to the previously studied case of the standard random walk further we show correlations between the two observables also in the limits of atypical large or small values

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

a liapounov bound for solutions of the poisson equation in this paper we consider psi irreducible markov processes evolving in discrete or continuous time on a general state space we develop a liapounov function criterion that permits one to obtain explicit bounds on the solution to the poisson equation and in particular obtain conditions under which the solution is square integrable these results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the markov process as a second consequence of the bounds obtained a perturbation theory for markov processes is developed which gives conditions under which both the solution to the poisson equation and the invariant probability for the process are continuous functions of its transition kernel the techniques are illustrated with applications to queueing theory and autoregressive processes

stationarity of lagrangian velocity in compressible environments we study the transport of a passive tracer particle by a random d dimensional gaussian compressible velocity field it is well known since the work of lumley see 13 and port and stone see 20 that the observations of the velocity field from the moving particle the so called lagrangian velocity process are statistically stationary when the field itself is incompressible in this paper we study the question of stationarity of lagrangian observations in compressible environments we show that given sufficient temporal decorrelation of the velocity statistics there exists a transformation of the original probability measure under which the lagrangian velocity process is time stationary the transformed probability is equivalent to the original measure as an application of this result we prove the law of large numbers for the particle trajectory

percolation beyond ℤd the contributions of oded schramm oded schramm 1961 2008 influenced greatly the development of percolation theory beyond the usual d setting in particular the case of nonamenable lattices here we review some of his work in this field

long range properties of spanning trees we compute some large scale properties of the uniform spanning tree process on bounded regions in z2 in particular we compute the distribution of the meeting point of the branches of the tree issued from three boundary points we also compute the crossing probabilities of branches of the tree on rectangular and annular regions as well as the winding number of the branches of the tree

dimension of the loop erased random walk in three dimensions we measure the fractal dimension of loop erased random walk lerw in three dimensions and estimate that it is 1 624 00 0 000 05 lerw is closely related to the uniform spanning tree and the abelian sandpile model we simulated lerw on both the cubic and face centered cubic lattices the corrections to scaling are slightly smaller for the face centered cubic lattice

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

subsequential scaling limits of simple random walk on the two dimensional uniform spanning tree the first main result of this paper is that the law of the rescaled two dimensional uniform spanning tree is tight in a space whose elements are measured rooted real trees continuously embedded into euclidean space various properties of the intrinsic metrics measures and embeddings of the subsequential limits in this space are obtained with it being proved in particular that the hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8 5 in addition the tightness result is applied to deduce that the annealed law of the simple random walk on the two dimensional uniform spanning tree is tight under a suitable rescaling for the limiting processes which are diffusions on random real trees embedded into euclidean space detailed transition density estimates are derived

homogenization of periodic linear degenerate pdes it is well known under the name of periodic homogenization that under a centering condition of the drift a periodic diffusion process on r d converges under diffusive rescaling to a d dimensional brownian motion existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion in this paper we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set as a consequence it is no longer the case that the effective diffusivity matrix is necessarily non degenerate it turns out that provided that some very weak regularity conditions are met the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion in particular this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank we also discuss the application of our results to the homogenization of a class of elliptic and parabolic pdes c 2008 elsevier inc all rights reserved

indistinguishability of the components of random spanning forests we prove that the infinite components of the free uniform spanning forest fusf of a cayley graph are indistinguishable by any invariant property given that the forest is different from its wired counterpart similar result is obtained for the free minimal spanning forest fmsf we also show that with the above assumptions there can only be 0 1 or infinitely many components which solves the problem for the fusf of caylay graphs completely these answer questions by benjamini lyons peres and schramm for cayley graphs which have been open up to now our methods apply to a more general class of percolations those satisfying weak insertion tolerance and work beyond cayley graphs in the more general setting of unimodular random graphs

the scaling limit of loop erased random walk in three dimensions we show that the scaling limit exists and is invariant under dilations and rotations we give some tools that might be useful to show universality

harmonic models and spanning forests of residually finite groups abstract we prove a number of identities relating the sofic entropy of a certain class of non expansive algebraic dynamical systems the sofic entropy of the wired spanning forest and the tree entropy of cayley graphs of residually finite groups we also show that homoclinic points and periodic points in harmonic models are dense under general conditions

exploration of the non asymptotic bias and variance of stochastic gradient langevin dynamics applying standard markov chain monte carlo mcmc algorithms to large data sets is computationally infeasible the recently proposed stochastic gradient langevin dynamics sgld method circumvents this problem in three ways it generates proposed moves using only a subset of the data it skips the metropolis hastings accept reject step and it uses sequences of decreasing step sizes in teh et al 2014 we provided the mathematical foundations for the decreasing step size sgld including consistency and a central limit theorem however in practice the sgld is run for a relatively small number of iterations and its step size is not decreased to zero the present article investigates the behaviour of the sgld with fixed step size in particular we characterise the asymptotic bias explicitly along with its dependence on the step size and the variance of the stochastic gradient on that basis a modified sgld which removes the asymptotic bias due to the variance of the stochastic gradients up to first order in the step size is derived moreover we are able to obtain bounds on the finite time bias variance and mean squared error mse the theory is illustrated with a gaussian toy model for which the bias and the mse for the estimation of moments can be obtained explicitly for this toy model we study the gain of the sgld over the standard euler method in the limit of large data sets

approach to criticality in sandpiles a popular theory of self organized criticality predicts that the stationary density of the abelian sandpile model equals the threshold density of the corresponding fixed energy sandpile we recently announced that this density conjecture is false when the underlying graph is any of z2 the complete graph kn the cayley tree the ladder graph the bracelet graph or the flower graph in this paper we substantiate this claim by rigorous proof and extensive simulations we show that driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink nevertheless we do find and prove a relationship between the two models the threshold density of the fixed energy sandpile is the point at which the driven dissipative sandpile begins to lose a macroscopic amount of sand to the sink

transfer current and pattern fields in spanning trees when a simply connected domain d subset mathbb r d d ge 2 is approximated in a good way by embedded connected weighted graphs we prove that the transfer current matrix defined on the edges of the graph viewed as an electrical network converges up to a local weight factor to the differential of green s function on d this observation implies that properly rescaled correlations of the spanning tree model and correlations of minimal subconfigurations in the abelian sandpile model have a universal and conformally covariant limit we further show that on a periodic approximation of the domain all pattern fields of the spanning tree model as well as the minimal pattern e g zero height fields of the sandpile converge weakly in distribution to gaussian white noise

diffusion approximation for the convection diffusion equation with random drift we consider the asymptotic behavior of the solutions ofscaled convection diffusion equations t u t x x t x 1 v t 2 x x u t x with the initial condition u 0 x u 0 x as the parameter 0 under the assumptions that 0 and v t x t x r d is a d dimensional stationary zero mean incompressible gaussian random field markovian and mixing in t we show that the laws of u t t 0 in an appropriate functional space converge weakly as 0 to a type measureconcentrated on a solution of a certain constant coefficient heat equation

an averaging principle for two time scale stochastic functional differential equations abstract delays are ubiquitous pervasive and entrenched in everyday life thus taking it into consideration is necessary dupire recently developed a functional ito formula which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications based on the new development this work examines functional diffusions with two time scales in which the slow varying process includes path dependent functionals and the fast varying process is a rapidly changing diffusion the gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example this paper establishes mixed functional ito formulas and the corresponding martingale representation then it develops an averaging principle using weak convergence methods by treating the fast varying process as a random noise under appropriate conditions it is shown that the slow varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow varying process with respect to the invariant measure of the fast varying process

loop erased random walk as a spin system observable the determination of the hausdorff dimension of the scaling limit of loop erased random walk is closely related to the study of the one point function of loop erased random walk i e the probability a loop erased random walk passes through a given vertex recent work in the theoretical physics literature has investigated the hausdorff dimension of loop erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one point function of loop erased random walk inspired by this we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one point function of loop erased random walk

the smoluchowski kramers limit of stochastic differential equations with arbitrary state dependent friction we study a class of systems of stochastic differential equations describing diffusive phenomena the smoluchowski kramers approximation is used to describe their dynamics in the small mass limit our systems have arbitrary state dependent friction and noise coefficients we identify the limiting equation and in particular the additional drift term that appears in the limit is expressed in terms of the solution to a lyapunov matrix equation the proof uses a theory of convergence of stochastic integrals developed by kurtz and protter the result is sufficiently general to include systems driven by both white and ornstein uhlenbeck colored noises we discuss applications of the main theorem to several physical phenomena including the experimental study of brownian motion in a diffusion gradient

limit behavior of two time scale diffusions revisited abstract this work is concerned with diffusions with two time scales or singularly perturbed diffusions asymptotic expansions of the solution of the associated cauchy problem for parabolic partial differential equation are obtained and the desired error bounds are derived these asymptotic expansions are then used to analyze related limit distributions of normalized integral functionals

variance reduction using nonreversible langevin samplers a standard approach to computing expectations with respect to a given target measure is to introduce an overdamped langevin equation which is reversible with respect to the target distribution and to approximate the expectation by a time averaging estimator as has been noted in recent papers 30 37 61 72 introducing an appropriately chosen nonreversible component to the dynamics is beneficial both in terms of reducing the asymptotic variance and of speeding up convergence to the target distribution in this paper we present a detailed study of the dependence of the asymptotic variance on the deviation from reversibility our theoretical findings are supported by numerical simulations

loop erased walks and total positivity we consider matrices whose elements enumerate weights of walks in planar directed weighted graphs not necessarily acyclic these matrices are totally nonnegative more precisely all their minors are formal power series in edge weights with nonnegative coefficients a combinatorial explanation of this phenomenon involves loop erased walks applications include total positivity of hitting matrices of brownian motion in planar domains

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

the component graph of the uniform spanning forest transitions in dimensions 9 10 11 ldots 9 10 11 we prove that the uniform spanning forests of mathbb z d and mathbb z ell have qualitatively different connectivity properties whenever ell d ge 4 in particular we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex which we call the component graph we introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8 to separate dimensions 5 6 7 and 8 we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph our result sharpens a theorem of benjamini kesten peres and schramm who proved that the diameter of the component graph increases by one every time the dimension increases by four