on the existence of accessibility in a tree indexed percolation model abstract we study the accessibility percolation model on infinite trees the model is defined by associating an absolute continuous random variable x v to each vertex v of the tree the main question to be considered is the existence or not of an infinite path of nearest neighbors v 1 v 2 v 3 such that x v 1 x v 2 x v 3 and which spans the entire graph the event defined by the existence of such path is called percolation we consider the case of the accessibility percolation model on a spherically symmetric tree with growth function given by f i i 1 where 0 is a given constant we show that there is a percolation threshold at c 1 such that there is percolation if 1 and there is absence of percolation if 1 moreover we study the event of percolation starting at any vertex as well as the continuity of the percolation probability function finally we provide a comparison between this model with the well known f record model we also discuss a number of open problems concerning the accessibility percolation model for further consideration in future research

on statistical aspects of deterministic tree like fractals we prove convergence of the partition function of the skewed mass distribution for a class of finite type tree like fractals

a new approach to the general minimum distance decoding problem the zero neighbors algorithm minimum distance decoding mdd for a general error correcting linear code is a hard computational problem that recently has been shown to be np hard the complexity of known decoding algorithms is determined by min 2 k 2 n k where n is the code length and k is the number of information digits two new algorithms are suggested that reduce substantially the complexity of mdd the algorithms use a new concept of zero neighbors a special set of codewords only these codewords which can be computed in advance should be stored and used in the decoding procedure the number of zero neighbors is shown to be very small compared with min 2 k 2 n k for n gg 1 and a wide range of code rates r k n for example for r approx 0 5 this number grows approximately as a square root of the number of codewords

a classification of taut stein surfaces with a proper r action we present a classification of 2 dimensional taut stein manifolds with a proper mathbb r action for such manifolds the globalization with respect to the induced local mathbb c action turns out to be stein as an application we determine all 2 dimensional taut non complete hartogs domains over a riemann surface

the structure of quasi multipliers of c algebras let a be a c algebra and a its enveloping w algebra let lm a be the left multipliers of a rm a the right multipliers of a and qm a the quasi multipliers of a a question was raised by akemann and pedersen 1 whether qm a lm a rm a mckennon 20 gave a nonseparable counterexample l brown 6 shows the answer is negative for stable separable c algebras also in this paper we mainly consider a unitial c algebras we give a criterion for qm a lm a rm a in the case that a is stable we give a necessary and sufficient condition for qm a lm a rm a we also give answers for other c algebras

robust reconstruction on trees is determined by the second eigenvalue consider a markov chain on an infinite tree t v e rooted at in such a chain once the initial root state is chosen each vertex iteratively chooses its state from the one of its parent by an application of a markov transition rule and all such applications are independent let j denote the resulting measure for j j is defined on configurations x x2v 2 a v where a is some finite set let n denote the restriction of to the sigma algebra generated by the variables x where x is at distance exactly n from letting n maxi j2a dtv n n where dtv denotes total variation distance we say that the reconstruction problem is solvable if liminfn 1 n 0 reconstruction solvability roughly means that the n th level of the tree contains a non vanishing amount of information on the root of the tree as n 1 in this paper we study the problem of robust reconstruction let be a non degenerate distribution on a and 0 let be chosen according to n and 0 be obtained from by letting for each node independently v 0 v with probability 1 and 0 v to be an independent sample from otherwise we denote by n the resulting measure on 0 the measure n is a perturbation of the measure n letting n maxi j2a dtv n n we say that the reconstruction problem is robust solvable if liminfn 1 n 0 for all 0 1 where 2 m is the second largest eigenvalue of m in absolute value then for all non degenerate the reconstruction problem is robust solvable we prove a converse and show that the reconstruction problem is not robust solvable if b 2 m 2 1 this proves a conjecture by the second author and peres we also consider other models of noise and general trees

percolation threshold is not a decreasing function of the average coordination number it is commonly believed that the percolation critical probability is a monotonically decreasing function of the average coordination number for periodic lattice graphs in the same dimension this paper provides counterexamples a pair of planar lattices for which the bond percolation critical probabilities and average coordination numbers are in the same order and a pair for which the site percolation critical probabilities and average coordination numbers are in the same order these counterexamples confirm the existence of this counterintuitive phenomenon which was observed in one case in numerical estimates by van der marck

weighted boundary limits of the generalized kobayashi royden metrics on weakly pseudoconvex domains the purpose of this paper is to study the existence of weighted boundary limits of the generalized kobayashi royden metrics on weakly pseudoconvex domains in cn and to explore the connections between the limits and the levi invariants the main result extends graham s result on strongly pseudoconvex domains to a large class of weakly pseudoconvex domains

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

on a class of skewed self similar and hyperbolic fractals probabilistic recurrent renewal theory is used herein to describe the geometry of a new class of self similar distributions in a skewed multiplicative mass splitting procedure it is shown how this model class can account for a deterministic and conservative treelike structure in connection with recent developments on anomalous multiplicative multifractals coexistence of both micro and macroscopic scales in such objects is a consequence of the underlying renewal phenomena being transient as is shown in a second part of the article various entropylike measures of the intrinsic disorder prevailing in these structures are proposed

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

quenched survival of bernoulli percolation on galton watson trees we explore the survival function for percolation on galton watson trees letting g t p represent the probability a tree t survives bernoulli percolation with parameter p we establish several results about the behavior of the random function g mathbf t cdot where mathbf t is drawn from the galton watson distribution these include almost sure smoothness in the supercritical region an expression for the k text th order taylor expansion of g mathbf t cdot at criticality in terms of limits of martingales defined from mathbf t this requires a moment condition depending on k and a proof that the k text th order derivative extends continuously to the critical value each of these results is shown to hold for almost every galton watson tree

fundamental approximate identities and quasi multipliers of simple af c algebras abstract we study c algebras with fundamental approximate identities as a generalization of stable c algebras and show that the only separable simple af c algebras for which the quasi multipliers equal the left plus right multipliers are unital or elementary

the maximum likelihood decoding threshold for cycle codes of graphs for a class mathcal c of binary linear codes we write theta mathcal c colon 0 1 to 0 1 2 for the maximum likelihood decoding threshold function of mathcal c the function whose value at r in 0 1 is the largest bit error rate p that the codes in mathcal c can tolerate with a negligible probability of maximum likelihood decoding error across a binary symmetric channel we show that if mathcal c is the class of cycle codes of graphs then theta mathcal c r le 1 sqrt r 2 2 1 r for each r and show that equality holds only when r is asymptotically achieved by the cycle codes of regular graphs

a decomposition theory for binary linear codes the decomposition theory of matroids initiated by paul seymour in the 1980s has had an enormous impact on research in matroid theory this theory when applied to matrices over the binary field yields a powerful decomposition theory for binary linear codes in this paper we give an overview of this code decomposition theory and discuss some of its implications in the context of the recently discovered formulation of maximum likelihood ml decoding of a binary linear code over a binary input discrete memoryless channel as a linear programming problem we translate matroid theoretic results of grotschel and truemper from the combinatorial optimization literature to give examples of nontrivial families of codes for which the ml decoding problem can be solved in time polynomial in the length of the code one such family is that consisting of codes for which the codeword polytope is identical to the koetter vontobel fundamental polytope derived from the entire dual code cperp however we also show that such families of codes are not good in a coding theoretic sense either their dimension or their minimum distance must grow sublinearly with code length as a consequence we have that decoding by linear programming when applied to good codes cannot avoid failing occasionally due to the presence of pseudocode words

support algebras of σ unital c algebras and their quasi multipliers we study certain dense hereditary subalgebras of n unital c algebras and their relations with the pedersen ideals the quasi multipliers of the dense hereditary subalgebras are also studied

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

unpredictable paths and percolation we construct a nearest neighbor process s n on z that is less predictable than simple random walk in the sense that given the process until time n the conditional probability that s n k x is uniformly bounded by ck for some 1 2 from this process we obtain a probability measure on oriented paths in z 3 such that the number of intersections of two paths chosen independently according to has an exponential tail for d 4 the uniform measure on oriented paths from the origin in z d has this property we show that on any graph where such a measure on paths exists oriented percolation clusters are transient if the retention parameter p is close enough to 1 this yields an extension of a theorem of grimmett kesten and zhang who proved that supercritical percolation clusters in z d are transient for all d 3

localization transition of biased random walks on random networks we study random walks on large random graphs that are biased towards a randomly chosen but fixed target node we show that a critical bias strength bc exists such that most walks find the target within a finite time when b bc for b bc a finite fraction of walks drift off to infinity before hitting the target the phase transition at b bc is a critical point in the sense that quantities such as the return probability p t show power laws but finite size behavior is complex and does not obey the usual finite size scaling ansatz by extending rigorous results for biased walks on galton watson trees we give the exact analytical value for bc and verify it by large scale simulations

on the intractability of permuting a block code to minimize trellis complexity an important problem in the theory and application of block code trellises is to find a coordinate permutation of a given code to minimize the trellis complexity we show that the problem of finding a coordinate permutation that minimizes the number of vertices at a given depth in the minimal trellis for a binary linear block code is np complete

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

globalizations fiber bundles and envelopes of holomorphy we describe a relationship between globalizations of local holomorphic actions on stein manifolds induced by global actions of certain non compact lie groups and holomorphic fiber bundles with smooth stein base and fiber and connected structure group to this end we prove a univalence result for particular stein riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms we then derive some consequences on the existence of stein globalizations

on the speed of random walks on graphs lyons pemantle and peres asked whether the asymptotic lower speed in an infinite tree is bounded by the asymptotic speed in the regular tree with the same average number of branches in the more general setting of random walks on graphs we establish a bound on the expected value of the exit time from a vertex set in terms of the size and distance from the origin of its boundary and prove this conjecture we give sharp bounds for limiting speed or when applicable sublinear rate of escape in terms of growth properties of the graph for trees we get a bound for the speed in terms of the hausdorff dimension of the harmonic measure on the boundary as a consequence two conjectures of lyons pemantle and peres are resolved and a new bound is given for the dimension of the harmonic measure defined by the biased random walk on a galton watson tree

conditional independence and chain event graphs graphs provide an excellent framework for interrogating symmetric models of measurement random variables and discovering their implied conditional independence structure however it is not unusual for a model to be specified from a description of how a process unfolds i e via its event tree rather than through relationships between a given set of measurements here we introduce a new mixed graphical structure called the chain event graph that is a function of this event tree and a set of elicited equivalence relationships this graph is more expressive and flexible than either the bayesian network equivalent in the symmetric case or the probability decision graph various separation theorems are proved for the chain event graph these enable implied conditional independencies to be read from the graph s topology we also show how the topology can be exploited to tease out the interesting conditional independence structure of functions of random variables associated with the underlying event tree

diffusions and random shadows in negatively curved manifolds abstract let m be a d dimensional complete simply connected negatively curved manifold there is a natural notion of hausdorff dimension for its boundary at infinity this is shown to provide a notion of global curvature or average rate of growth in two probabilistic senses first on surfaces d 2 it is twice the critical drift separating transience from recurrence for brownian motion with constant length radial drift equivalently it is twice the critical for the existence of a green function for the operator 2 r second for any d it is the critical intensity for almost sure coverage of the boundary by random shadows cast by balls appropriately scaled produced from a constant intensity poisson point process

the random cluster model on a homogeneous tree the random cluster model on a homogeneous tree is defined and studied it is shown that for 1 q 2 the percolation probability in the maximal random cluster measure is continuous inp while forq 2 it has a discontinuity at the critical valuep p c q it is also shown that forq 2 there is nonuniqueness of random cluster measures for an entire interval of values ofp the latter result is in sharp contrast to what happens on the integer lattice z d

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

limit set of a weakly supercritical contact process on a homogeneous tree a conjecture of liggett 9 concerning the regime of weak survival for the contact process on a homogeneous tree is proved the conjecture is shown to imply that the hausdor dimension of the limit set of such a contact process is no larger than half the hausdor dimension of the space of ends of the tree the conjecture is also shown to imply that at the boundary between weak survival and strong survival the contact process survives only weakly a theorem previously proved by zhang 14 finally a stronger form of a theorem of hawkes and lyons concerning the hausdor dimension of a galton watson tree is proved

phase transitions in semidefinite relaxations statistical inference problems arising within signal processing data mining and machine learning naturally give rise to hard combinatorial optimization problems these problems become intractable when the dimensionality of the data is large as is often the case for modern datasets a popular idea is to construct convex relaxations of these combinatorial problems which can be solved efficiently for large scale datasets semidefinite programming sdp relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs it has been observed several times that when the statistical noise is small enough sdp relaxations correctly detect the underlying combinatorial structures in this paper we develop asymptotic predictions for several detection thresholds as well as for the estimation error above these thresholds we study some classical sdp relaxations for statistical problems motivated by graph synchronization and community detection in networks we map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions our results clarify the effectiveness of sdp relaxations in solving high dimensional statistical problems

on the complexity of some coding problems corresp it is shown that the problem of finding a codeword with least weight and whose weight is not a multiple of k in a binary linear code belongs to the class of nondeterministic polynomial np hard problems for any k geq 2 some other related problems are shown to belong to the same class these results were motivated by a conjecture due to berlekamp mceliece and van tilborg that the problem of finding the hamming distance of a code is np hard

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

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

hausdorff dimension of the contours of symmetric additive levy processes let x 1 x n denote n independent symmetric levy processes on r d the corresponding additive levy process is defined as the following n parameter random field on r d mathfrak x t x 1 t 1 cdots x n t n quad t in bf r n khoshnevisan and xiao ann probab 30 1 62 100 2002 have found a necessary and sufficient condition for the zero set mathfrak x 1 0 of mathfrak x to be non trivial with positive probability they also provide bounds for the hausdorff dimension of mathfrak x 1 0 which hold with positive probability in the case that mathfrak x 1 0 can be non void here we prove that the hausdorff dimension of mathfrak x 1 0 is a constant almost surely on the event mathfrak x 1 0 neq varnothing moreover we derive a formula for the said constant this portion of our work extends the well known formulas of horowitz israel j math 6 176 182 1968 and hawkes j lond math soc 8 517 525 1974 both of which hold for one parameter levy processes more generally we prove that for every nonrandom borel set f in 0 n the hausdorff dimension of mathfrak x 1 0 cap f is a constant almost surely on the event mathfrak x 1 0 cap f neq varnothing this constant is computed explicitly in many cases

causal analysis with chain event graphs as the chain event graph ceg has a topology which represents sets of conditional independence statements it becomes especially useful when problems lie naturally in a discrete asymmetric non product space domain or when much context specific information is present in this paper we show that it can also be a powerful representational tool for a wide variety of causal hypotheses in such domains furthermore we demonstrate that as with causal bayesian networks cbns the identifiability of the effects of causal manipulations when observations of the system are incomplete can be verified simply by reference to the topology of the ceg we close the paper with a proof of a back door theorem for cegs analogous to pearl s back door theorem for cbns

the complexity of computing the covering radius of a code the problem of finding the covering radius of a binary linear code is shown to be nondeterministic polynomial np hard in fact a problem that is complete for the class prod 2 p in the polynomial hierarchy is shown to be reducible to the covering radius problem so that finding the covering radius is strictly harder than any np complete problem unless the polynomial hierarchy collapses with np prod 2 p

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

lyons pemantle peres monotonicity problem for high biases the speed v of a biased random walk on a galton watson tree without leaves is increasing for 1160 2014 wiley periodicals inc

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

brownian motion and thermal capacity let w denote d dimensional brownian motion we find an explicit formula for the essential supremum of hausdorff dimension of w e f where e 0 and f rd are arbitrary nonrandom compact sets our formula is related intimately to the thermal capacity of watson proc lond math soc 3 37 1978 342 362 we prove also that when d 2 our formula can be described in terms of the hausdorff dimension of e f where e f is viewed as a subspace of space time

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 the existence of asymptotically good linear codes in minor closed classes let c c 1 c 2 be a sequence of codes such that each c i is a linear n i k i d i code over some fixed finite field f where n i is the length of the code words k i is the dimension and d i is the minimum distance we say that c is asymptotically good if for some e 0 and for all i 0 we have n i i and min k i n i d i n i e sequences of asymptotically good codes exist we prove that if c is a class of gf p n linear codes where p is prime and n 1 closed under puncturing and shortening and if c contains an asymptotically good sequence then c must contain all gf p linear codes our proof relies on a powerful new result from matroid structure theory

diffusion limited aggregation on a tree we study the following growth model on a regular d ary tree points at distance n adjacent to the existing subtree are added with probabilities proportional to n where 1 is a positive real parameter the heights of these clusters are shown to increase linearly with their total size this complements known results that show the height increases only logarithmically when 1 results are obtained using stochastic monotonicity and regeneration results which may be of independent interest our motivation comes from two other ways in which the model may be viewed as a problem in first passage percolation and as a version of diffusion limited aggregation dla adjusted so that fingering occurs

branching random walk with exponentially decreasing steps and stochastically self similar measures we consider a branching random walk on r whose step size decreases by a fixed factor 0 1 2 the limit measure is almost surely a s absolutely continuous with respect to the lebesgue measure but for pisot 1 it is a s singular 2 for all a 5 1 2 the support of the measure is a s the closure of its interior 3 for pisot 1 the support of the measure is fractured it is a s disconnected and the components of the complement are not isolated on both sides

criteria for evaluation of universal formulas for percolation thresholds several universal formulas that predict approximate values for percolation thresholds of all periodic graphs have been proposed in the physics and engineering literature the existing universal formulas have been found to have substantial errors in their predictions for some lattices this paper proposes a set of desirable criteria for universal formulas to satisfy and investigates which criteria are satisfied by two bond threshold formulas and two site threshold formulas most cited in the literature the analysis is limited to lattices in two dimensions

dynamical percolation on general trees haggstrom et al ann inst h poincare probab stat 33 4 497 528 1997 have introduced a dynamical version of percolation on a graph g when g is a tree they derived a necessary and sufficient condition for percolation to exist at some time t in the case that g is a spherically symmetric tree peres and steif in probab theory relat fields 111 1 141 165 1998 derived a necessary and sufficient condition for percolation to exist at some time t in a given target set d the main result of the present paper is a necessary and sufficient condition for the existence of percolation at some time t in d in the case that the underlying tree is not necessary spherically symmetric this answers a question of yuval peres personal communication we present also a formula for the hausdorff dimension of the set of exceptional times of percolation

factorization properties of lattices over the integers abstract let a be a nonsingular n n matrix over the integers l l a denotes the lattice whose elements are combinations with integer coefficients of the rows of a l is cyclic if it can be defined in the modular form l x x i a i x i 0 mod d where the a i s and d are integers and 0 a i d let l l 1 l 2 b be lattices over the integers l l 1 l 2 is a factorization of l if every element of l is a combination of the rows of b such that the vector of combination coefficients is in l 1 and b is a nonsingular n n matrix the following results are proved every lattice can be expressed as a product of cyclic factors in polynomial time every cyclic lattice can be factored into simple term explained in the text factors in polynomial time every simple lattice can be factored into prime factors in polynomial time if a prime factorization of the determinant of its basis is given in addition we provide polynomial algorithms for the following problems transform a cyclic lattice given by a basis into a modular form and vice versa find a basis of a finite modular lattice given in modular form

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

a central limit theorem for biased random walks on galton watson trees let mathcal t be a rooted galton watson tree with offspring distribution p k that has p 0 0 mean m kp k 1 and exponential tails consider the biased random walk x n n 0 on mathcal t this is the nearest neighbor random walk which when at a vertex v with d v offspring moves closer to the root with probability d v and moves to each of the offspring with probability 1 d v it is known that this walk has an a s constant speed tt v lim n x n n where x n is the distance of x n from the root with tt v 0 for 0 m and tt v 0 for m for all m we prove a quenched clt for x n n tt v for m the walk is positive recurrent and there is no clt the most interesting case by far is m where the clt has the following form for almost every mathcal t the ratio x nt sqrt n converges in law as n to a deterministic multiple of the absolute value of a brownian motion our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle previously such a measure was explicitly known only for 1 and the construction of appropriate harmonic coordinates

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

an extension of hawkes theorem on the hausdorff dimension of a galton watson tree let be the genealogical tree of a supercritical multitype galton watson process and let be the limit set of i e the set of all infinite self avoiding paths called ends through that begin at a vertex of the first generation the limit set is endowed with the metric d 2 n where n n is the index of the first generation where and differ to each end is associated the infinite sequence of types of the vertices of let be the space of all such sequences for any ergodic shift invariant probability measure on define to be the set of all generic sequences i e the set of all sequences such that each finite sequence v occurs in with limiting frequency v where v is the set of all that begin with the word v then the hausdorff dimension of 1 in the metric d is

speed of the biased random walk on a galton watson tree we give an expression of the speed of the biased random walk on a galton watson tree in the particular case of the simple random walk we recover the result of lyons et al erg theory dyn syst 15 593 619 1995 the proof uses a description of the invariant distribution of the environment seen from the particle

random walks on the lamplighter group kaimanovich and vershik described certain finitely generated groups of exponential growth such that simple random walk on their cayley graph escapes from the identity at a sublinear rate or equivalently all bounded harmonic functions on the cayley graph are constant here we focus on a key example called g 1 by kaimanovich and vershik and show that inward biased random walks on g 1 move outward faster than simple random walk indeed they escape from the identity at a linear rate provided that the bias parameter is smaller than the growth rate of g 1 these walks can be viewed as random walks interacting with a dynamical environment on mathbb z the proof uses potential theory to analyze a stationary environment as seen from the moving particle

causal identifiability via chain event graphs we present the chain event graph ceg as a complementary graphical model to the causal bayesian network for the representation and analysis of causally manipulated asymmetric problems our focus is on causal identifiability finding conditions for when the effects of a manipulation can be estimated from a subset of events observable in the unmanipulated system ceg analogues of email protected s back door and front door theorems are presented applicable to the class of singular manipulations which includes both email protected s basic do intervention and the class of functional manipulations possible on bayesian networks these theorems are shown to be more flexible than their bayesian network counterparts both in the types of manipulation to which they can be applied and in the nature of the conditioning sets which can be used

graphical codes revisited the set of all even subgraphs of a connected graph g on p vertices with q edges forms a binary linear code c c sub e g with parameters q q p 1 g where g is the girth of g such codes were studied systematically by bredeson and hakimi 1967 and hakimi and bredeson 1968 who were concerned with the problems of augmenting c to a larger q k g code and of efficiently decoding such augmented graphical codes we give a new approach to these problems by requiring the augmented codes to be graphical on one hand we present two construction methods which turn out to contain the methods proposed by hakimi and bredeson as special cases as we show this not only gives a better understanding of their construction it also results in augmenting codes of larger dimension we look at the case of 1 error correcting graphical codes in some detail in particular we show how to obtain the extended hamming codes as purely graphical codes by our approach on the other hand we follow a suggestion of ntafos and hakimi 1981 and use techniques from combinatorial optimization to give decoding procedures for graphical codes which turn out to be considerably more efficient than the approach via majority logic decoding proposed by bredeson and hakimi we also consider the decoding problem for the even graphical code based on the complete graph k sub 2n in more detail we discuss an efficient hardware implementation of an encoding decoding scheme for these codes and show that things may be arranged in such a way that one can also correct all adjacent double errors finally we discuss nonlinear graphical codes

the critical ising model on trees concave recursions and nonlinear capacity we consider the ising model on a general tree under various boundary conditions all plus free and spin glass in each case we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity we obtain exact capacity criteria that govern behavior at critical temperatures for plus boundary conditions an l 3 capacity arises in particular on a spherically symmetric tree that has n b n vertices at level n up to bounded factors we prove that there is a unique gibbs measure for the ferromagnetic ising model at the relevant critical temperature if and only if 1 2 our proofs are based on a new link between nonlinear recursions on trees and l p capacities

robust phase transitions for heisenberg and other models on general trees we study several statistical mechanical models on a general tree particular attention is devoted to the classical heisenberg models where the state space is the d dimensional unit sphere and the interactions are proportional to the cosines of the angles between neighboring spins the phenomenon of interest here is the classification of phase transition non uniqueness of the gibbs state according to whether it is robust in many cases including all of the heisenberg and potts models occurrence of robust phase transition is determined by the geometry branching number of the tree in a way that parallels the situation with independent percolation and usual phase transition for the ising model the critical values for robust phase transition for the heisenberg and potts models are also calculated exactly in some cases such as the q 3 potts model robust phase transition and usual phase transition do not coincide while in other cases such as the heisenberg models we conjecture that robust phase transition and usual phase transition are equivalent in addition we show that symmetry breaking is equivalent to the existence of a phase transition a fact believed but not known for the rotor model on z 2