thick points for planar brownian motion and the erdős taylor conjecture on random walk and conjectured that the limit exists and equals 1 tr a s the importance of determining the value of this limit is clarified in 1 3 below where this value appears in the power laws governing the local time of the walk the erd6s taylor conjecture was quoted in the book by r v sz 19 w but to the best of our knowledge the bounds in 1 1 were not improved prior to the present paper as it turns out an important step towards our solution of the erd6s taylor conjecture was the formulation by perkins and taylor 17 of an analogous problem on the maximal occupation measure that planar brownian motion run for unit time can

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

random walk models for multifractals reversible random walk models for bernoulli multifractals in dimension 2 together with marginal versions of these are first studied to account for both geometrical and analytical properties of such objects using large deviation theory it is also shown how the transience recurrence of a more complex reversible random walk is related to the minimum energy flow convergence divergence property new results on the asymptotics of such a walker are then derived in both transient recurrent situations

a subdiffusive behaviour of recurrent random walk in random environment on a regular tree we are interested in the random walk in random environment on an infinite tree lyons and pemantle ann probab 20 125 136 1992 give a precise recurrence transience criterion our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk x n in random environment on a regular tree which is closely related to mandelbrot s c r acad sci paris 278 289 292 1974 multiplicative cascade we prove under some general assumptions upon the distribution of the environment the existence of a new exponent nu in big 0 1 over 2 big such that max 0 le i le n x i behaves asymptotically like n nu the value of is explicitly formulated in terms of the distribution of the environment

multifractality of entangled random walks and non uniform hyperbolic spaces multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a non symmetric lattice of obstacles are investigated using the equivalence of the model to random walks on a locally non symmetric tree statistical properties of topological invariants such as drift and return probabilities have been studied by means of a renormalization group rg technique the comparison of the analytical rg results with numerical simulations as well as with the rigorous results of gerl and woess demonstrates clearly the validity of our approach it is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space which has an exponentially large number of states has been broken

the exact hausdorff dimension of a branching set we obtain a critical function for which the hausdorff measure of a branching set generated by a simple galton watson process is positive and finite

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

fatcop a fault tolerant condor pvm mixed integer programming solver we describe fatcop a new parallel mixed integer program solver written in pvm the implementation uses the condor resource management system to provide a virtual machine composed of otherwise idle computers the solver differs from previous parallel branch and bound codes by implementing a general purpose parallel mixed integer programming algorithm in an opportunistic multiple processor environment as opposed to a conventional dedicated environment it shows how to make effective use of resources as they become available while ensuring the program tolerates resource retreat the solver performs well on test problems arising from real applications and is particularly useful for solving long running hard mixed integer programming problems

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

stability conditions for discontinuous vector fields abstract in this work we describe the qualitative behavior of generic discontinuous vector fields around a point p r 3

distinguishing infections on different graph topologies the history of infections and epidemics holds famous examples where understanding containing and ultimately treating an outbreak began with understanding its mode of spread influenza hiv and most computer viruses spread person to person device to device and through contact networks cholera cancer and seasonal allergies on the other hand do not in this paper we study two fundamental questions of detection first given a snapshot view of a perhaps vanishingly small fraction of those infected under what conditions is an epidemic spreading via contact e g influenza distinguishable from a random illness operating independently of any contact network e g seasonal allergies second if we do have an epidemic under what conditions is it possible to determine which network of interactions is the main cause of the spread the causative network without any knowledge of the epidemic other than the identity of a minuscule subsample of infected nodes the core therefore of this paper is to obtain an understanding of the diagnostic power of network information we derive sufficient conditions that networks must satisfy for these problems to be identifiable and produce efficient highly scalable algorithms that solve these problems we show that the identifiability condition we give is fairly mild and in particular is satisfied by two common graph topologies the d dimensional grid and the erdos renyi graphs

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

parallel branch cut and price for large scale discrete optimization in discrete optimization most exact solution approaches are based on branch and bound which is conceptually easy to parallelize in its simplest forms more sophisticated variants such as the so called branch cut and price algorithms are more difficult to parallelize because of the need to share large amounts of knowledge discovered during the search process in the first part of the paper we survey the issues involved in parallelizing such algorithms we then review the implementation of symphony and coin bcp two existing frameworks for implementing parallel branch cut and price these frameworks have limited scalability but are effective on small numbers of processors finally we briefly describe our next generation framework which improves scalability and further abstracts many of the notions inherent in parallel bcp making it possible to implement and parallelize more general classes of algorithms

slow movement of random walk in random environment on a regular tree we consider a recurrent random walk in random environment on a regular tree under suitable general assumptions upon the distribution of the environment we show that the walk exhibits an unusual slow movement the order of magnitude of the walk in the first n steps is log n 3

the fixed points of the multivariate smoothing transform let mathbf t 1 mathbf t 2 ldots be a sequence of random d times d matrices with nonnegative entries and let q be a random vector with nonnegative entries consider random vectors x with nonnegative entries satisfying begin aligned x mathop limits mathcal l sum i ge 1 mathbf t i x i q end aligned where mathop limits mathcal l denotes equality of the corresponding laws x i i ge 1 are i i d copies of x and independent of q mathbf t 1 mathbf t 2 ldots for d 1 this equation known as fixed point equation of the smoothing transform has been intensively studied under assumptions similar to the one dimensional case we obtain a complete characterization of all solutions x to in the non critical case and existence results in the critical case

stable limit laws for randomly biased walks on supercritical trees we consider a random walk on a supercritical galton watson tree with leaves where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree the biases are chosen independently on distinct edges each one according to a given law that satisfies a logarithmic nonlattice condition we determine the condition under which the walk is sub ballistic and in the sub ballistic regime we find a formula for the exponent 0 1 such that the distance x n moved by the walk in time n is of the order of n we prove a stable limiting law for walker distance at late time proving that the rescaled walk n x n converges in distribution to an explicitly identified function of the stable law of index this paper is a counterpart to ben arous et al ann probab 40 2012 280 338 in which it is proved that in the model where the biases on edges are taken to be a given constant there is a logarithmic periodicity effect that prevents the existence of a stable limit law for scaled walker displacement it is randomization of edge biases that is responsible for the emergence of the stable limit in the present article while also introducing further correlations into the model in comparison with the constant bias case the derivation requires the development of a detailed understanding of trap geometry and the interplay between traps and backbone the paper may be considered as a sequel to ben arous and hammond comm pure appl math 65 2012 1481 1527 since it makes use of a result on the regular tail of the total conductance of a randomly biased subcritical galton watson tree

martin capacity for markov chains the probability that a transient markov chain or a brownian path will ever visit a given set is classically estimated using the capacity of with respect to the green kernel g x y we show that replacing the green kernel by the martin kernel g x y g 0 y yields improved estimates which are exact up to a factor of 2 these estimates are applied to random walks on lattices and also to explain a connection found by lyons between capacity and percolation on trees

randomly biased walks on subcritical trees as a model of trapping by biased motion in random structure we study the time taken for a biased random walk to return to the root of a subcritical galton watson tree we do so for trees in which these biases are randomly chosen independently for distinct edges according to a law that satisfies a logarithmic non lattice condition the mean return time of the walk is in essence given by the total conductance of the tree we determine the asymptotic decay of this total conductance finding it to have a pure power law decay in the case of the conductance associated to a single vertex at maximal depth in the tree this asymptotic decay may be analysed by the classical defective renewal theorem due to the non lattice edge bias assumption however the derivation of the decay for total conductance requires computing an additional constant multiple outside the power law that allows for the contribution of all vertices close to the base of the tree this computation entails a detailed study of a convenient decomposition of the tree under conditioning on the tree having high total conductance as such our principal conclusion may be viewed as a development of renewal theory in the context of random environments for randomly biased random walk on a supercritical galton watson tree with positive extinction probability our main results may be regarded as a description of the slowdown mechanism caused by the presence of subcritical trees adjacent to the backbone that may act as traps that detain the walker indeed this conclusion is exploited in 18 to obtain a stable limiting law for walker displacement in such a tree

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

on the convergence of a class of random geometric series with application to random walks and percolation theory we consider a class of random geometric series with an underlying tree like structure that has a number of applications in statistical physics convergence criteria for these series are discussed and consistency with different criteria known to hold in the one dimensional limit is established a multiplicative two component model of percolation on a cayley tree is defined and analysed the order of the percolation transition and certain critical exponents are altered compared to conventional bond percolation

tree indexed random walks on groups and first passage percolation suppose that i i d random variables are attached to the edges of an infinite tree when the tree is large enough the partial sumss along some of its infinite paths will exhibit behavior atypical for an ordinary random walk this principle has appeared in works on branching random walks first passage percolation and rwre on trees we establish further quantitative versions of this principle which are applicable in these settings in particular different notions of speed for such a tree indexed walk correspond to different dimension notions for trees finally if the labeling variables take values in a group then properties of the group e g polynomial growth or a nontrivial poisson boundary are reflected in the sample path behavior of the resulting tree indexed walk

an optimization framework for product design an important problem in the product design and development process is to use the part worths preferences of potential customers to design a new product such that market share is maximized the authors present a new optimization framework for this problem the nested partitions np method this method is globally convergent and may utilize existing heuristic methods to speed its convergence we incorporate several known heuristics into this framework and demonstrate through numerical experiments that using the np method results in superior product designs our numerical results suggest that the new framework is particularly useful for designing complex products with many attributes

phase transition for the once reinforced random walk on mathbb z d like trees in this short paper we consider the once reinforced random walk with reinforcement parameter a on trees with bounded degree which are transient for the simple random walk on each of these trees we prove that there exists an explicit critical parameter a 0 such that the once reinforced random walk is almost surely recurrent if a a 0 and almost surely transient if a a 0 this provides the first examples of phase transition for the once reinforced random walk

scaling limit of the recurrent biased random walk on a galton watson tree we show that the trace of the null recurrent biased random walk on a galton watson tree properly renormalized converges to the brownian forest our result extends to the setting of the random walk in random environment on a galton watson tree

transient random walks in random environment on a galton watson tree we consider a transient random walk xn in random environment on a galton watson tree under fairly general assumptions we give a sharp and explicit criterion for the asymptotic speed to be positive as a consequence situations with zero speed are revealed to occur in such cases we prove that xn is of order of magnitude n with lambda in 0 1 we also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed which improves a recent result of collevecchio probab theory related 136 1 81 101 2006

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

the maximum of a branching random walk with semiexponential increments we consider an infinite galton watson tree and label the vertices with a collection of i i d random variables y in the case where the upper tail of the distribution of y is semiexponential we then determine the speed of the corresponding tree indexed random walk in contrast to the classical case where the random variables y have finite exponential moments the normalization in the definition of the speed depends on the distribution of y interpreting the random variables y as displacements of the offspring from the parent y describes a branching random walk the result on the speed gives a limit theorem for the maximum of the branching random walk that is for the position of the rightmost particle in our case this maximum grows faster than linear in time

almost sure convergence for stochastically biased random walks on trees we are interested in the biased random walk on a supercritical galton watson tree in the sense of lyons ann probab 18 931 958 1990 and lyons pemantle and peres probab theory relat fields 106 249 264 1996 and study a phenomenon of slow movement in order to observe such a slow movement the bias needs to be random the resulting random walk is then a tree valued random walk in random environment we investigate the recurrent case and prove under suitable general integrability assumptions that upon the system s non extinction the maximal displacement of the walk in the first n steps divided by log n 3 converges almost surely to a known positive constant

on the transience of processes defined on galton watson trees we introduce a simple technique for proving the transience of certain processes defined on the random tree mathcal g generated by a supercritical branching process we prove the transience for once reinforced random walks on mathcal g that is a generalization of a result of durrett kesten and limic probab theory related fields 122 2002 567 592 moreover we give a new proof for the transience of a family of biased random walks defined on mathcal g other proofs of this fact can be found in ann probab 16 1988 1229 1241 and ann probab 18 1990 931 958 as part of more general results a similar technique is applied to a vertex reinforced jump process a by product of our result is that this process is transient on the 3 ary tree davis and volkov probab theory related fields 128 2004 42 62 proved that a vertex reinforced jump process defined on the b ary tree is transient if b 4 and recurrent if b 1 the case b 2 is still open

parallel branch and bound algorithms for general mixed integer programming on the cm 5 this proof of concept paper describes parallel solution of general mixed integer programs by a branch and bound algorithm on the cm 5 multiprocessing system it goes beyond prior parallel branch and bound work by implementing a reasonably realistic general purpose mixed integer programming algorithm as opposed to a specialized method for a narrow class of problems it shows how to use the capabilities of the cm 5 to produce an efficient parallel implementation employing centrally controlled search achieving near linear speedups using 64 128 processors on a variety of difficult problems derived from real applications in concrete terms a problem requiring half an hour to solve on a sparc 2 workstation might be solved in 15 20 seconds and a problem originally taking a week might be reduced to about an hour central search control does have limitations and some final computational experiments begin to address the merits of more decentralized options

domination between trees and application to an explosion problem we define a notion of stochastic domination between trees where one tree dominates another if when the vertices of each are labeled with independent identically distributed random variables one tree is always more likely to contain a path with a specified property sufficient conditions for this kind of domination are 1 more symmetry and 2 earlier branching we apply these conditions to the problem of determining how fast a tree must grow before first passage percolation on the tree exhibits an explosion that is to say infinitely many vertices are reached in finite time for a tree in which each vertex at distance n 1 from the root has f n offspring f nondecreasing an explosion occurs with exponentially distributed passage times if and only if f n 1

phase transition in reinforced random walk and rwre on trees a random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future using exchangeability theory the process is shown to be equivalent to a random walk in a random environment rwre that is to say a mixture of markov chains criteria are given to determine whether a rwre is transient or recurrent these criteria apply to show that the reinforced random walk can vary from transient to recurrent depending on the value of an adjustable parameter measuring the strength of the feedback the value of the parameter at the phase transition is calculated