the true self repelling motion we construct and study a continuous real valued random process which is of a new type it is self interacting self repelling but only in a local sense it only feels the self repellance due to its occu pation time measure density in the immediate neighbourhood of the point it is just visiting we focus on the most natural process with these properties that we call true self repelling motion this is the continuous counterpart to the integer valued true self avoiding walk which had been studied among others by the first author one of the striking properties of true self repelling motion is that although the coupleoxt occupation time measure of x at time tu is a continuous markov process x is not driven by a stochastic diaerential equation and is not a semi martingale it turns out for instance that it has a finite variation of order 3 2 which contrasts with the finite quadratic variation of semi martingales one of the key tools in the construction of x is a continuous system of coalescing brownian motions similar to those that have been constructed by arratia a1 a2 we derive various properties of x existence and properties of the occupation time densities ltoxu local variation etc and an identity that shows that the dynamics of x can be very loosely speaking described as follows dxt is equal to the gradient in space of ltoxu in a gener alized sense even though x7 ltoxu is not diaerentiable

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

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

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

critical random walk in random environment on trees we study the behavior of random walk in random environment rwre on trees in the critical case left open in previous work representing the random walk by an electrical network we assume that the ratios of resistances of neighboring edges of a tree are i i d random variables whose logarithms have mean zero and finite variance then the resulting rwre is transient if simple random walk on is transient but not vice versa we obtain general transience criteria for such walks which are sharp for symmetric trees of polynomial growth in order to prove these criteria we establish results on boundary crossing by tree indexed random walks these results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean zero finite variance random walks in one dimension which are of independent interest

estimating kappa from binocular data a common error in statistical analysis of ophthalmic data is the lack of accounting for the positive correlation generally present between observations made in fellow eyes the alternative of data analysis from only one eye in each patient may lead to loss of power and unrealistically large confidence intervals this paper discusses a method to estimate kappa a measure of agreement between two graders when both graders rate the same set of pairs of eyes the method assumes that the true left eye and right eye kappa values are equal and makes use of the correlated binocular data to estimate confidence intervals for the common kappa simulations show that the new estimators are better than the estimator based on only one eye new confidence intervals had the correct coverage probability but were usually only about 70 per cent as wide as single eye intervals the general methodology described here applies to analysis of grader agreement in rating other paired body structures

a measurable group theoretic solution to von neumann s problem we give a positive answer in the measurable group theory context to von neumann s problem of knowing whether a non amenable countable discrete group contains a non cyclic free subgroup we also get an embedding result of the free group von neumann factor into restricted wreath product factors

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

conformal invariance of domino tiling let u be a multiply connected region in r 2 with smooth boundary let p be a polyomino in z 2 approximating u as 0 we show that for certain boundary conditions on p the height distribution on a random domino tiling dimer covering of p is conformally invariant in the limit as tends to 0 in the sense that the distribution of heights of boundary components or rather the difference of the heights from their mean values only depends on the conformal type of u the mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way the mean height and all the moments are explicitly evaluated

agreement between two independent groups of raters we propose a coefficient of agreement to assess the degree of concordance between two independent groups of raters classifying items on a nominal scale this coefficient defined on a population based model extends the classical cohen s kappa coefficient for quantifying agreement between two raters weighted and intraclass versions of the coefficient are also given and their sampling variance is determined by the jackknife method the method is illustrated on medical education data which motivated the research

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

the dirichlet problem for orthodiagonal maps abstract we prove that the discrete harmonic function corresponding to smooth dirichlet boundary conditions on orthodiagonal maps that is plane graphs having quadrilateral faces with orthogonal diagonals converges to its continuous counterpart as the mesh size goes to 0 this provides a convergence statement for discrete holomorphic functions similar to the one obtained by chelkak and smirnov 4 for isoradial graphs we observe that by the double circle packing theorem 3 any finite simple 3 connected planar map admits an orthodiagonal representation our result improves the work of skopenkov 41 and werness 47 by dropping all regularity assumptions required in their work and providing effective bounds in particular no bound on the vertex degrees is required thus the result can be applied to models of random planar maps that with high probability admit orthodiagonal representation with mesh size tending to 0 in a companion paper 21 we show that this can be done for the discrete mating of trees random map model of duplantier gwynne miller and sheffield 15 25

a note on the kappa statistic for clustered dichotomous data the kappa statistic is widely used to assess the agreement between two raters motivated by a simulation based cluster bootstrap method to calculate the variance of the kappa statistic for clustered physician patients dichotomous data we investigate its special correlation structure and develop a new simple and efficient data generation algorithm for the clustered physician patients dichotomous data based on the delta method and its special covariance structure we propose a semi parametric variance estimator for the kappa statistic an extensive monte carlo simulation study is performed to evaluate the performance of the new proposal and five existing methods with respect to the empirical coverage probability root mean square error and average width of the 95 confidence interval for the kappa statistic the variance estimator ignoring the dependence within a cluster is generally inappropriate and the variance estimators from the new proposal bootstrap based methods and the sampling based delta method perform reasonably well for at least a moderately large number of clusters e g the number of clusters k 50 the new proposal and sampling based delta method provide convenient tools for efficient computations and non simulation based alternatives to the existing bootstrap based methods moreover the new proposal has acceptable performance even when the number of clusters is as small as k 25 to illustrate the practical application of all the methods one psychiatric research data and two simulated clustered physician patients dichotomous data are analyzed

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

kappa statistic for clustered matched pair data kappa statistic is widely used to assess the agreement between two procedures in the independent matched pair data for matched pair data collected in clusters on the basis of the delta method and sampling techniques we propose a nonparametric variance estimator for the kappa statistic without within cluster correlation structure or distributional assumptions the results of an extensive monte carlo simulation study demonstrate that the proposed kappa statistic provides consistent estimation and the proposed variance estimator behaves reasonably well for at least a moderately large number of clusters e g k 50 compared with the variance estimator ignoring dependence within a cluster the proposed variance estimator performs better in maintaining the nominal coverage probability when the intra cluster correlation is fair 0 3 with more pronounced improvement when is further increased to illustrate the practical application of the proposed estimator we analyze two real data examples of clustered matched pair data copyright 2014 john wiley sons ltd

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

the analysis of repeated categorical measurements subject to nonignorable nonresponse abstract conditional likelihood methods have been proposed for analyzing the repeated measurement of categorical responses this article extends the methods to include partially classified subjects the extension allows for an ignorable or nonignorable nonresponse mechanism and uses standard statistical software for the computations two examples of incomplete longitudinal categorical data illustrate the method

indistinguishability of percolation clusters we show that when percolation produces infinitely many infinite clusters on a cayley graph one cannot distinguish the clusters from each other by any invariantly defined property this implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity a k a long range order we then derive applications concerning uniqueness in kazhdan groups and in wreath products and inequalities for p u

conformal invariance of voronoi percolation it is proved that in the voronoi model for percolation in dimension 2 and 3 the crossing probabilities are asymptotically invariant under conformal change of metric

infinite canonical super brownian motion and scaling limits we construct a measure valued markov process which we call infinite canonical super brownian motion and which corresponds to the canonical measure of super brownian motion conditioned on non extinction infinite canonical super brownian motion is a natural candidate for the scaling limit of various random branching objects on mathbb z d when these objects are critical mean field and infinite we prove that icsbm is the scaling limit of the spread out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension we conjecture that it also arises as the scaling limit in various other models above the upper critical dimension such as the incipient infinite lattice tree above 8 dimensions the incipient infinite cluster for unoriented percolation above 6 dimensions uniform spanning trees above 4 dimensions and invasion percolation above 6 dimensions

random walks capacity and percolation on trees a collection of several different probabilistic processes involving trees is shown to have an unexpected unity this makes possible a fruitful interplay of these probabilistic processes the processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well our work has five specific aims first an exact correspondence between random walks and percolation on trees is proved extending and sharpening previous work of the author this is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes second we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation this capacitary formulation extends and sharpens work of fan on random interval coverings third we show how this formulation also applies to generalize work of evans on random labelling of trees fourth the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a s in particular we resolve a conjecture of griffeath on the necessity of the nash williams criterion fifth for this last purpose we establish several new basic results on branching processes in varying environments

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

isoperimetric inequalities and markov chains soit a i j i j n la matrice stochastique d une chaine de markov reversible sur n 1 2 soit n 2 tel que f r c f d f c 0 n ou r 2n n 2 ou c 0 est independant de f et ou r indique la norme dans l r n d pour la mesure i i alors on a sup i j j 1 a i j 0 n 2

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

bias in a binary risk behaviour model subject to inconsistent reports and dropout in a south african high school cohort study we describe a methodology for analysing self reported risk behaviour transitional patterns in a binary outcome variable subject to misclassification and a large loss to follow up the motivation stems from the analysis of self reported transitional patterns in responses to the question have you ever smoked a whole cigarette in a cohort of south african school children the partially complete records analysis pcra introduced estimates the transitional probability as the ratio of the joint probability of the response at two time points based on the complete records for this time sequence over the marginal probabilities of the response based on the complete records at the first time point and assumes a non informative missing pattern a comparison was made using un weighted complete records and inverse probability weighted logistic regression the estimates of the probabilities of reporting ever having smoked a cigarette obtained from the three methods were similar for a particular transition the pcra method lacked precision compared with the inverse probability weighted logistic regression a simulation study indicated an association between bias and reporting error in all three methods the pcra method can be considered as a method for the estimation of transition probabilities in a cohort study where there is consistency in the self reported risk behaviour pattern and the sample size is large at baseline the inverse probability weighting approach is more precise and is suitable for this setting in order to determine risk factors for the incidence of self reported substance used in a cohort with a high dropout rate

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

random cluster representation of the blume capel model the so called diluted random cluster model may be viewed as a random cluster representation of the blume capel model it has three parameters a vertex parameter a an edge parameter p and a cluster weighting factor q stochastic comparisons of measures are developed for the vertex marginal when q 1 2 and the edge marginal when q 1 taken in conjunction with arguments used earlier for the random cluster model these permit a rigorous study of part of the phase diagram of the blume capel model

kazhdan groups have cost 1 we prove that every countably infinite group with kazhdan s property t has cost 1 answering a well known question of gaboriau it remains open if they have fixed price 1

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

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

approaching criticality via the zero dissipation limit in the abelian avalanche model the discrete height abelian sandpile model was introduced by bak tang wiesenfeld and dhar as an example for the concept of self organized criticality when the model is modified to allow grains to disappear on each toppling it is called bulk dissipative we provide a detailed study of a continuous height version of the abelian sandpile model called the abelian avalanche model which allows an arbitrarily small amount of dissipation to take place on every toppling we prove that for non zero dissipation the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure we show that in the whole non zero dissipation regime the model is not critical i e spatial covariances of local observables decay exponentially we then study the zero dissipation limit and prove that the self organized critical model is recovered both for the stationary measure and for the dynamics we obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality we rigorously establish the mean field value of this exponent for d 4

infinite dimensional stochastic differential equations and tail σ fields we present general theorems solving the long standing problem of the existence and pathwise uniqueness of strong solutions of infinite dimensional stochastic differential equations isdes called interacting brownian motions these isdes describe the dynamics of infinitely many brownian particles moving in mathbb r d with free potential varphi and mutual interaction potential varpsi we apply the theorems to essentially all interaction potentials of ruelle s class such as the lennard jones 6 12 potential and riesz potentials and to logarithmic potentials appearing in random matrix theory we solve isdes of the ginibre interacting brownian motion and the sine beta interacting brownian motion with beta 1 2 4 we also use the theorems in separate papers for the airy and bessel interacting brownian motions one of the critical points for proving the general theorems is to establish a new formulation of solutions of isdes in terms of tail sigma fields of labeled path spaces consisting of trajectories of infinitely many particles these formulations are equivalent to the original notions of solutions of isdes and more feasible to treat in infinite dimensions

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

testing the equality of two dependent kappa statistics procedures are developed and compared for testing the equality of two dependent kappa statistics in the case of two raters and a dichotomous outcome variable such problems may arise when each of a sample of subjects are rated under two distinct settings and it is of interest to compare the observed levels of inter observer and intra observer agreement the procedures compared are extensions of previously developed procedures for comparing kappa statistics computed from independent samples the results of a monte carlo simulation show that adjusting for the dependency between samples tends to be worthwhile only if the between setting correlation is comparable in magnitude to the within setting correlations in this case a goodness of fit procedure that takes into account the dependency between samples is recommended

estimating kappa from binocular data and comparing marginal probabilities suppose that two graders classify all eyes in a sample of patients for the presence or absence of a specified abnormality in the statistical analysis of the data possible correlation between the observations in the right and left eyes should be taken into account recently general methods have been developed to analyse grader agreement in rating paired body structures in the present paper an alternative procedure is proposed that utilizes existing formulae for the computation of weighted kappa and its standard error in addition the mcnemar test is generalized to compare the probabilities of a positive judgement by the two graders

harmonic functions on planar and almost planar graphs and manifolds via circle packings the circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant harmonic bounded dirichlet function if p is a bounded circle packing in bbb r 2 whose contacts graph is a bounded valence triangulation of a disk then with probability 1 the simple random walk on p converges to a limit point moreover in this situation any continuous function on the limit set of p extends to a continuous harmonic function on the closure of the contacts graph of p that is this dirichlet problem is solvable we define the notions of almost planar graphs and manifolds and show that under the assumptions of transience and bounded local geometry these possess non constant harmonic bounded dirichlet functions let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph

random forests and networks analysis wilson proceedings of the twenty eight annual acm symposium on the theory of computing pp 296 303 1996 in the 1990s described a simple and efficient algorithm based on loop erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph this algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking in recent works avena and gaudilliere in a proof of the transfer current theorem in absence of reversibility in stat probab lett 142 17 22 2018 avena and gaudilliere in j theor probab 2017 https doi org 10 1007 s10959 017 0771 3 avena et al in approximate and exact solutions of intertwining equations though random spanning forests 2017 arxiv 1702 05992v1 avena et al in intertwining wavelets or multiresolution analysis on graphs through random forests 2017 arxiv 1707 04616 to appear in acha 2018 we focused on applications of spanning rooted forests on finite graphs the resulting main conclusions are reviewed in this paper by collecting related theorems algorithms heuristics and numerical experiments a first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications 1 a random walk based notion of well distributed points in a graph 2 a framework to describe metastable like dynamics in finite settings by means of markov intertwining dualities 3 coarse graining schemes for networks and associated processes 4 wavelets like pyramidal algorithms for graph signals

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

exceptional times and invariance for dynamical random walks consider a sequence open image in new window of i i d random variables associate to each xi 0 an independent mean one poisson clock every time a clock rings replace that x variable by an independent copy and restart the clock in this way we obtain i i d stationary processes xi t t 0 i 1 2 whose invariant distribution is the law of x1 0

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

exact solutions for kpz type growth processes random matrices and equilibrium shapes of crystals abstract three models from statistical physics can be analyzed by employing space time determinantal processes 1 crystal facets in particular the statistical properties of the facet edge and equivalently tilings of the plane 2 one dimensional growth processes in the kardar parisi zhang universality class and directed last passage percolation 3 random matrices multi matrix models and dyson s brownian motion we explain the method and survey results of physical interest

corner percolation on ℤ2 and the square root of 17 we consider a four vertex model introduced by balint toth a dependent bond percolation model on 2 in which every edge is present with probability 1 2 and each vertex has exactly two incident edges perpendicular to each other we prove that all components are finite cycles almost surely but the expected diameter of the cycle containing the origin is infinite moreover we derive the following critical exponents the tail probability diameter of the cycle of the origin n n and the expectation mathbb e length of a typical cycle with diameter n n with gamma 5 sqrt 17 4 0 219 ldots and delta sqrt 17 1 4 1 28 ldots the value of comes from a singular sixth order ode while the relation 3 2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive brownian motion whose level sets have hausdorff dimension 3 2 we also include many open problems for example on the conformal invariance of certain linear entropy models

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

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

maximum likelihood estimation and model selection in contingency tables with missing data abstract in many studies the values of one or more variables are missing for subsets of the original sample this article focuses on the problem of obtaining maximum likelihood estimates mle for the parameters of log linear models under this type of incomplete data the appropriate systems of equations are presented and the expectation maximization em algorithm dempster laird and rubin 1977 is suggested as one of the possible methods for solving them the algorithm has certain advantages but other alternatives may be computationally more effective tests of fit for log linear models in the presence of incomplete data are considered the data from the protective services project for older persons blenkner bloom and nielsen 1971 blenkner bloom and weber 1974 are used to illustrate the procedures discussed in the article

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

on combinatorial testing problems we study a class of hypothesis testing problems in which upon observing the realization of an n dimensional gaussian vector one has to decide whether the vector was drawn from a standard normal distribution or alternatively whether there is a subset of the components belonging to a certain given class of sets whose elements have been contaminated that is have a mean different from zero we establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk the combinatorial and geometric structure of the class of sets is shown to play a crucial role the bounds are illustrated on various examples

uniqueness of infinite rigid components in percolation models the case of nonplanar lattices we prove uniqueness of the infinite rigid component for standard bond percolation on periodic lattices in d dimensional euclidean space for arbitrary d and more generally when the lattice is a quasi transitive and amenable graph our approach to uniqueness of the infinite rigid component improves earlier ones that were confined to planar settings

percolation in the hyperbolic plane the purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi isometric to the hyperbolic plane

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

quasi symmetries of determinantal point processes the main result of this paper is that determinantal point processes on mathbb r corresponding to projection operators with integrable kernels are quasi invariant in the continuous case under the group of diffeomorphisms with compact support theorem 1 4 in the discrete case under the group of all finite permutations of the phase space theorem 1 6 the radon nikodym derivative is computed explicitly and is given by a regularized multiplicative functional theorem 1 4 applies in particular to the sine process as well as to determinantal point processes with the bessel and the airy kernels theorem 1 6 to the discrete sine process and the gamma kernel process the paper answers a question of grigori olshanski

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

confidence interval estimation of the intraclass correlation coefficient for binary outcome data we obtain closed form asymptotic variance formulae for three point estimators of the intraclass correlation coefficient that may be applied to binary outcome data arising in clusters of variable size our results include as special cases those that have previously appeared in the literature fleiss and cuzick 1979 applied psychological measurement 3 537 542 bloch and kraemer 1989 biometrics 45 269 287 altaye donner and klar 2001 biometrics 57 584 588 simulation results indicate that confidence intervals based on the estimator proposed by fleiss and cuzick provide coverage levels close to nominal over a wide range of parameter combinations two examples are presented

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 self avoiding random walk in two and three dimensions if n is the position of the self avoiding random walk in d obtained by erasing loops from simple random walk then it is proved that the mean square displacemente n 2 grows at least as fast as the flory predictions for the usual saw i e at least as fast asn3 2 ford 2 andn6 5 ford 3 in particular if the mean square displacement of the usual saw grows liken1 18 ind 3 as expected then the loop erased process is in a different universality class

a comparison of methods for calculating a stratified kappa investigators use the kappa coefficient to measure chance corrected agreement among observers in the classification of subjects into nominal categories the marginal probability of classification may depend however on one or more confounding variables we consider assessment of interrater agreement with subjects grouped into strata on the basis of these confounders we assume overall agreement across strata is constant and consider a stratified index of agreement or stratified kappa based on weighted summations of the individual kappas we use three weighting schemes 1 equal weighting 2 weighting by the size of the table and 3 weighting by the inverse of the variance in a simulation study we compare these methods under differing probability structures and differing sample sizes for the tables we find weighting by sample size moderately efficient under most conditions we illustrate the techniques by assessing agreement between surgeons and graders of fundus photographs with respect to retinal characteristics with stratification by initial severity of the disease

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

random walks and harmonic functions on infinite planar graphs using square tilings we study a wide class of transient planar graphs through a geometric model given by a square tiling of a cylinder for many graphs the geometric boundary of the tiling is a circle and is easy to describe in general the simple random walk on the graph converges with probability 1 to a point in the geometric boundary we obtain information on the harmonic measure and estimates on the rate of convergence this allows us to extend results we previously proved for triangulations of a disk

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

the z invariant massive laplacian on isoradial graphs we introduce a one parameter family of massive laplacian operators delta m k k in 0 1 defined on isoradial graphs involving elliptic functions we prove an explicit formula for the inverse of delta m k the massive green function which has the remarkable property of only depending on the local geometry of the graph and compute its asymptotics we study the corresponding statistical mechanics model of random rooted spanning forests we prove an explicit local formula for an infinite volume boltzmann measure and for the free energy of the model we show that the model undergoes a second order phase transition at k 0 thus proving that spanning trees corresponding to the laplacian introduced by kenyon invent math 150 2 409 439 2002 are critical we prove that the massive laplacian operators delta m k k in 0 1 provide a one parameter family of z invariant rooted spanning forest models when the isoradial graph is moreover mathbb z 2 periodic we consider the spectral curve of the massive laplacian we provide an explicit parametrization of the curve and prove that it is harnack and has genus 1 we further show that every harnack curve of genus 1 with z w leftrightarrow z 1 w 1 symmetry arises from such a massive laplacian