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

the mathematics of microstructure and the design of new materials the pathological energy function e u u 2 for u 0 e 0 1 has no minimizer as u decreases to 0 the energy also decreases but there is no way to achieve the value 0 although examples like this might seem to be unimaginably far from scientific thought they are at the heart of a new approach 1 to understand the complex microstructure and macroscopic response of materials that undergo phase transformations the free energy of such materials typically has no minimizer and the observed microstructures complex fine scale patterns of domains of different atomic lattice structure as shown below in a micrograph of cualni by c chu and r d j fig 1 have their origin in the material s ultimately futile attempt to find the minimum energy state 2 the lack of a ground state prohibits prediction of the macroscopic response from microscopic data via the standard procedure determine the free energy find the minimizing state and evaluate its macroscopic properties emerging mathematical methods linked to profound work in the 1940s by l c young and recently surveyed in 3 nevertheless deliver well defined macroscopic quantities obtained via averaging over all low energy states one area where predictions obtained in this new way have played a role is the recent synthetization of a new magnetostrictive material 4 5 whose magnetostrictive strain is 50 times larger than that of giant magnetostrictive materials formerly those with the largest strain

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

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

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

percolation of level sets for two dimensional random fields with lattice symmetry let x x 2 be a random field which may be viewed as the potential of an incompressible flow for which the trajectories follow the level lines of percolation methods are used to analyze the sizes of the connected components of level sets x x h and sets x x h in several classes of random fields with lattice symmetry in typical cases there is a sharp transition at a critical value ofh from exponential boundedness for such components to the existence of an unbounded component in some examples however there is a nondegenerate interval of values ofh where components are bounded but not exponentially so and in other cases each level set may be a single infinite line which visits every region of the lattice

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

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

index concepts for linear mixed systems of differential algebraic and hyperbolic type equations for many technical systems the use of a refined network modeling approach leads to hyperbolic type initial boundary value problems of partial differential algebraic equations pdaes the boundary conditions of these systems are governed by time dependent differential algebraic equations daes that couple the pdae system with the network elements that are modeled by daes in time only in order to classify these systems we extend some index notions that have already been introduced to treat parabolic type problems a perturbation index is considered that reflects the sensitivity of the mixed system to slight perturbations in the right hand side of the pdaes as well as in the input signals of the dae systems and initial values in order to make an a posteriori analysis of semidiscretization in space and time we introduce additionally a space and a method of lines mol index here one is especially interested in whether the semidiscretized systems properly reflect the properties of the underlying systems we will show that these indices may detect an artificial sensitivity with respect to perturbations e g if the semidiscretization does not consider the information transport along the characteristics

singular perturbation of an elastic energy with a singular weight abstract we study the singular perturbation of an elastic energy with a singular weight the minimization of this energy results in a multi scale pattern formation we derive an energy scaling law in terms of the perturbation parameter and prove that although one cannot expect periodicity of minimizers the energy of a minimizer is uniformly distributed across the sample finally following the approach developed by alberti and muller 2001 we prove that a sequence of minimizers of the perturbed energies converges to a young measure supported on piecewise linear periodic functions of slope 1 whose period depends on the location in the domain and the weights in the energy

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

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

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

a differentiation index for partial differential algebraic equations a differentiation index for nonlinear partial differential algebraic equations is presented determination of the differentiation index with respect to a direction in the space of independent variables uncovers all equations that must be satisfied by cauchy data on the hyperplane orthogonal to that direction this index is thus a generalization of the differentiation index of differential algebraic equations the motivation for this work is consistent initialization of partial differential algebraic systems the analysis is demonstrated through several examples including some taken from fluid dynamics and chemical engineering

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

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

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

spectral dimension and random walks on the two dimensional uniform spanning tree we study the simple random walk on the uniform spanning tree on mathbb z 2 we obtain estimates for the transition probabilities of the random walk the distance of the walk from its starting point after n steps and exit times of both euclidean balls and balls in the intrinsic graph metric in particular we prove that the spectral dimension of the uniform spanning tree on mathbb z 2 is 16 13 almost surely

the loop erased random walk and the uniform spanning tree on the four dimensional discrete torus let x and y be points chosen uniformly at random from mathbb z n 4 the four dimensional discrete torus with side length n we show that the length of the loop erased random walk from x to y is of order n2 log n 1 6 resolving a conjecture of benjamini and kozma we also show that the scaling limit of the uniform spanning tree on mathbb z n 4 is the brownian continuum random tree of aldous our proofs use the techniques developed by peres and revelle who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d dimensional discrete torus for d 5 in combination with results of lawler concerning intersections of four dimensional random walks

towards a theory of negative dependence the fkg theorem says that the positive lattice condition an easily checkable hypothesis which holds for many natural families of events implies positive association a very useful property thus there is a natural and useful theory of positively dependent events there is as yet no corresponding theory of negatively dependent events there is however a need for such a theory this paper unfortunately contains no substantial theorems its purpose is to present examples that motivate a need for such a theory give plausibility arguments for the existence of such a theory outline a few possible directions such a theory might take and state a number of specific conjectures which pertain to the examples and to a wish list of theorems

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

index and characteristic analysis of linear pdae systems automatable analyses of the index and well posedness of two general classes of linear pdaes are presented for use in equation based dynamic simulation packages these analyses determine whether a problem possesses a unique solution trajectory the potential effect of noise in the initial and boundary conditions on the calculated solution and the index of the model equations measures of the degeneracy of the system rather than a differentiation index bound the order of derivatives of forcing functions that appear in the solution

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

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

the components of the wired spanning forest are recurrent we show that a s all of the connected components of the wired spanning forest are recurrent proving a conjecture of benjamini lyons peres and schramm our analysis relies on a simple martingale involving the effective conductance between the endpoints of an edge in a uniform spanning tree we believe that this martingale is of independent interest and will find further applications in the study of uniform spanning trees and forests

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

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

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

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

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

dynamic stability of non minimizing phase mixtures we study the dynamics of pattern formation in the one dimensional partial differential equation u tt w u x x u xxt u 0 proposed recently by ball holmes pego and swart as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 u x 2 1 2 this problem possesses infinitely many equilibrium solutions corresponding to different microstructures or phase mixtures all locally energetically unstable in the natural norms however many are locally dynamically stable theorem 1 1 and the paper proves this theorem and investigates other properties of these equilibria

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

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

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

differential algebraic equations indices and integral algebraic equations in a recent report hairer lubich and roche report ch 1211 dept de mathematiques universite de geneve geneve switzerland 1988 define the index of differential algebraic equations daes by considering the effect of perturbations of the equations on the solutions this index which will be called the perturbation index p i is one more than the number of derivatives of the perturbation that must appear in any estimate of the bound of the change in the solution an earlier form of index used by a number of authors including gear and petzold siam j numer anal 21 1984 pp 716 728 is determined by the number of differentiations of the daes that are required to generate an ordinary differential equation ode satisfied by the solution this will be called the differential index d i hairer lubich and roche give an example whose differential index is one and perturbation index is two and other examples where they are identical it will be shown that d i leqq p i leqq d i 1 and th

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

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

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

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

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

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

minimum spanning trees and random resistor networks in d dimensions we consider minimum cost spanning trees both in lattice and euclidean models in d dimensions for the cost of the optimum tree in a box of size l we show that there is a correction of order l ensuremath theta where ensuremath theta ensuremath leqslant 0 is a universal d dependent exponent there is a similar form for the change in optimum cost under a change in boundary condition at nonzero temperature t there is a crossover length ensuremath xi ensuremath sim t ensuremath ensuremath nu such that on length scales larger than ensuremath xi the behavior becomes that of uniform spanning trees there is a scaling relation ensuremath theta ensuremath 1 ensuremath nu and we provide several arguments that show that ensuremath nu and ensuremath 1 ensuremath theta both equal ensuremath nu mathrm perc the correlation length exponent for ordinary percolation in the same dimension d in all dimensions d ensuremath geqslant 1 the arguments all rely on the close relation of kruskal s greedy algorithm for the minimum spanning tree percolation and for some arguments random resistor networks the scaling of the entropy and free energy at small nonzero t and hence of the number of near optimal solutions is also discussed we suggest that the steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions as is the traveling salesman problem in two dimensions hence all will have the same value of ensuremath theta ensuremath 3 4 in two dimensions

correlations in the n 0 limit of the dense o n loop model the two dimensional dense o n loop model for n 1 is equivalent to the bond percolation and for n 0 to the dense polymers or spanning trees we consider the boundary correlations on the half space and calculate the probability pb that a cluster of bonds has a single common point with the boundary in the limit n 0 we find an analytical expression for pb using the generalized kirchhoff theorem

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

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

ends in free minimal spanning forests we show that for a transitive unimodular graph the number of ends is the same for every tree of the free minimal spanning forest this answers a question of lyons peres and schramm

history dependent quantum random walks as quantum lattice gas automata quantum random walks qrw were first defined as one particle sectors of quantum lattice gas automata qlga recently they have been generalized to include history dependence either on previous coin internal i e spin or velocity states or on previous position states these models have the goal of studying the transition to classicality or more generally changes in the performance of quantum walks in algorithmic applications we show that several history dependent qrw can be identified as one particle sectors of qlga this provides a unifying conceptual framework for these models in which the extra degrees of freedom required to store the history information arise naturally as geometrical degrees of freedom on the lattice

the index of general nonlinear daes in the last few years there has been considerable research on differential algebraic equations daes f t y y 0 where f y is identically singular much of the mathematical effort has focused on computing a solution that is assumed to exist more recently there has been some discussion of solvability of daes there has historically been some imprecision in the use of the two key concepts of solvability and index for daes the index is also important in control and systems theory but with different terminology the consideration of increasingly complex nonlinear daes makes a clear and correct development necessary this paper will try to clarify several points concerning the index after establishing some new and more precise terminology that we need some inaccuracies in the literature will be corrected the two types of indices most frequently used the differentiation index and the perturbation index are defined with respect to solutions of unperturbed problems examples are given to show that these indices can be very different for the same problem we define new maximum indices which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other these indices are also related to an index defined in terms of jacobians