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

isoperimetry for wreath products of markov chains and multiplicity of selfintersections of random walks we prove an isoperimetric inequality for wreath products of markov chains with variable fibers we use isoperimetric inequalities for wreath products to estimate the return probability of random walks on infinite groups and graphs drift of random loops the expected value e exp trn where rn is the number of distinct sites visited up to the moment n and more generally open image in new window where lz n is the number of visits of z up to the moment n and f x y is some non negative function

on converse and saturation results for tikhonov regularization of linear ill posed problems in this paper we prove some new converse and saturation results for tikhonov regularization of linear ill posed problems tx y where t is a linear operator between two hilbert spaces

error estimates for regularization methods in hilbert scales in this paper we study regularization methods to reconstruct the solution x of the linear ill posed problem ax y a x to y from noisy data y delta in y the regularization methods are of the general form x alpha delta bar x g alpha b s a a b s a y delta a bar x where b denotes an unbounded self adjoint strictly positive definite operator in the hilbert space x assuming ax sim x a and x bar x p le e for some bar x in x a ge 0 and p ge 0 x r b r 2 x is the norm in a hilbert scale x r r in rnus we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g alpha lambda the regularization parameter alpha and the parameter s are chosen properly

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

saturation of regularization methods for linear ill posed problems in hilbert spaces we prove the saturation of methods for solving linear ill posed problems in hilbert spaces for a wide class of regularization methods it turns out that under a certain convexity assumption saturation must necessarily occur we provide easy to verify assumptions which allow us to calculate the rate at which saturation occurs

multistability and stable asynchronous periodic oscillations in a multiple delayed neural system abstract we consider a network of three identical neurons with multiple discrete signal transmission delays the model for such a network is a system of nonlinear delay differential equations after some consideration of the absolute synchronization of the system and the global attractivity of the zero solution we present a detailed discussion about the boundaries of the stability region of the trivial solution this allows us to determine the possible codimension one bifurcations which occur in the system in particular we show the existence of standard hopf bifurcations giving rise to synchronized periodic solutions and of d 3 equivariant hopf bifurcations giving rise to three types of periodic solutions phase locked mirror reflecting and standing waves hopf hopf and hopf steady state bifurcations interactions are shown to exist and give rise to coexistence of stable synchronized and desynchronized solutions perturbation techniques coupled with the floquet theory are used to determine the stability of the phase locked oscillations

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

synchronization in an array of linearly stochastically coupled networks with time delays abstract in this paper the complete synchronization problem is investigated in an array of linearly stochastically coupled identical networks with time delays the stochastic coupling term which can reflect a more realistic dynamical behavior of coupled systems in practice is introduced to model a coupled system and the influence from the stochastic noises on the array of coupled delayed neural networks is studied thoroughly based on a simple adaptive feedback control scheme and some stochastic analysis techniques several sufficient conditions are developed to guarantee the synchronization in an array of linearly stochastically coupled neural networks with time delays finally an illustrate example with numerical simulations is exploited to show the effectiveness of the theoretical results

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

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

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

regularization of some linear ill posed problems with discretized random noisy data for linear statistical ill posed problems in hilbert spaces we introduce an adaptive procedure to recover the unknown so lution from indirect discrete and noisy data this procedure is shown to be order optimal for a large class of problems smooth ness of the solution is measured in terms of general source condi tions the concept of operator monotone functions turns out to be an important tool for the analysis

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

synchronization analysis of linearly coupled systems described by differential equations with a coupling delay when a transmission delay occurs in the interconnection of linearly coupled systems described by ordinary differential equations lcodes both synchronization and the final synchronized state will vary in this paper mathematical analysis is presented on the synchronization phenomena of lcodes with a single coupling delay criteria are derived for both local and global synchronization it is known that in addition to the dynamical behaviors of the underlying uncoupled system and the coupling configuration the coupling strength and the coupling delay also play key roles on the stability of synchronization both theoretical and numerical analysis indicate that under some conditions if the coupling strength is large enough the coupled system can be completely synchronized for any coupling delay on the other hand in some cases the coupled system can be synchronized if the coupling delay is small enough

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

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

analysis of profile functions for general linear regularization methods the stable approximate solution of ill posed linear operator equations in hilbert spaces requires regularization tight bounds for the noise free part of the regularization error are constitutive for bounding the overall error norm bounds of the noise free part which decrease to zero along with the regularization parameter are called profile functions and are the subject of our analysis the interplay between properties of the regularization and certain smoothness properties of solution sets which we shall describe in terms of sourcewise representations is crucial for the decay of associated profile functions on the one hand we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness on the other hand if smoothness fits the regularization then decay rates are easily obtained if smoothness does not fit then we will measure this in terms of some distance function tight bounds for these allow us to obtain profile functions finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition in many parts the analysis is done on geometric basis extending classical concepts of linear regularization theory in hilbert spaces we emphasize intrinsic features of linear ill posed problems which are frequently hidden in the classical analysis of such problems

synchronization in complex delayed dynamical networks with nonsymmetric coupling abstract a new general complex delayed dynamical network model with nonsymmetric coupling is introduced and then we investigate its synchronization phenomena several synchronization criteria for delay independent and delay dependent synchronization are provided which generalize some previous results the matrix jordan canonical formalization method is used instead of the matrix diagonalization method so in our synchronization criteria the coupling configuration matrix of the network does not required to be diagonalizable and may have complex eigenvalues especially we show clearly that the synchronizability of a delayed dynamical network is not always characterized by the second largest eigenvalue even though all the eigenvalues of the coupling configuration matrix are real furthermore the effects of time delay on synchronizability of networks with unidirectional coupling are studied under some typical network structures the results are illustrated by delayed networks in which each node is a two dimensional limit cycle oscillator system consisting of a two cell cellular neural network numerical simulations show that these networks can realize synchronization with smaller time delay and will lose synchronization when the time delay increase larger than a threshold

on random walks on wreath products wreath products are a type of semidirect product they play an important role in group theory this paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting somewhat exotic behaviors the crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group

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

anchored expansion percolation and speed benjamini lyons and schramm random walks and discrete potential theory 1999 56 84 considered properties of an infinite graph g and the simple random walk on it that are preserved by random perturbations in this paper we solve several problems raised by those authors the anchored expansion constant is a variant of the cheeger constant its positivity implies positive lower speed for the simple random walk as shown by virag geom funct anal 10 2000 1588 1605 we prove that if g has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter p sufficiently close to 1 a better estimate for the parameters p where this holds is in the appendix we also show that positivity of the anchored expansion constant is preserved under a random stretch if and only if the stretching law has an exponential tail we then study a simple random walk in the infinite percolation cluster in cayley graphs of certain amenable groups known as lamplighter groups we prove that zero speed for a random walk on a lamplighter group implies zero speed for random walk on an infinite cluster for any supercritical percolation parameter p for p large enough we also establish the converse

exponential synchronization of hybrid coupled networks with delayed coupling this paper investigates exponential synchronization of coupled networks with hybrid coupling which is composed of constant coupling and discrete delay coupling there is only one transmittal delay in the delayed coupling the fact is that in the signal transmission process the time delay affects only the variable that is being transmitted from one system to another then it makes sense to assume that there is only one single delay contributing to the dynamics some sufficient conditions for synchronization are derived based on lyapunov functional and linear matrix inequality lmi in particular the coupling matrix may be asymmetric or nondiagonal moreover the transmittal delay can be different from the one in the isolated system a distinctive feature of this work is that the synchronized state will vary in comparison with the conventional synchronized solution especially the degree of the nodes and the inner delayed coupling matrix heavily influence the synchronized state finally a chaotic neural network is used as the node in two regular networks to show the effectiveness of the proposed criteria

the ising model on diluted graphs and strong amenability say that a graph has persistent transition if the ising model on the graph can exhibit a phase transition nonuniqueness of gibbs measures in the presence of a nonzero external field we show that for nonamenable graphs for bernoulli percolation with p close to 1 all the infinite clusters have persistent transition on the other hand we show that for transitive amenable graphs the infinite clusters for any stationary percolation do not have persistent transition this extends a result of georgii for the cubic lattice a geometric consequence of this latter fact is that the infinite clusters are strongly amenable i e their anchored cheeger constant is 0 finally we show that the critical temperature for the ising model with no external field on the infinite clusters of bernoulli percolation with parameter p on an arbitrary bounded degree graph is a continuous function of p

synchronization analysis in an array of asymmetric neural networks with time varying delays and nonlinear coupling in this paper the global exponential synchronization is investigated for an array of asymmetric neural networks with time varying delays and nonlinear coupling assuming neither the differentiability for time varying delays nor the symmetry for the inner coupling matrices by employing a new lyapunov krasovskii functional applying the theory of kronecker product of matrices and the technique of linear matrix inequality lmi a delay dependent sufficient condition in lmis form for checking global exponential synchronization is obtained the proposed result generalizes and improves the earlier publications an example with chaotic nodes is given to show the effectiveness of the obtained result

multi parameter regularization and its numerical realization in this paper we propose and analyse a choice of parameters in the multi parameter regularization of tikhonov type a modified discrepancy principle is presented within the multi parameter regularization framework an order optimal error bound is obtained under the standard smoothness assumptions we also propose a numerical realization of the multi parameter discrepancy principle based on the model function approximation numerical experiments on a series of test problems support theoretical results finally we show how the proposed approach can be successfully implemented in laplacian regularized least squares for learning from labeled and unlabeled examples

synchronization analysis for nonlinearly coupled complex networks with an asymmetrical coupling matrix abstract in this paper the global synchronization for an array of nonlinearly coupled identical chaotic systems is investigated a distinctive feature of this work is to address synchronization issues for nonlinearly coupled complex networks with an asymmetrical coupling matrix by projecting the nonlinear coupling function onto a linear one and assuming the difference between them as a disturbing function we give some criteria for the global synchronization in virtual of the left eigenvector corresponding to the zero eigenvalue of the coupling matrix numerical examples are also provided to demonstrate the effectiveness of the theory

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

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

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

a novel approach to synchronization of nonlinearly coupled network systems with delays in this investigation a novel approach to establishing the global synchronization of coupled network systems is presented under this approach individual subsystems can be non autonomous and the coupling configuration is rather general the coupling terms can be non diffusive nonlinear time dependent asymmetric and with time delays with an iteration scheme the problem of synchronization is transformed into solving a corresponding linear system of algebraic equations subsequently delay dependent and delay independent criteria for global synchronization can be established we implement the present approach to analyze synchronization of the fitzhugh nagumo systems under delayed and nonlinear sigmoidal coupling two examples are presented to demonstrate new dynamical scenarios where oscillatory behavior and multistability emerge or are suppressed as the coupled neurons synchronize under the synchronization criterion in addition asynchrony induced by the coupling strength or coupling delay occurs while the synchronization criterion is violated

generic behavior of master stability functions in coupled nonlinear dynamical systems master stability functions msfs are fundamental to the study of synchronization in complex dynamical systems for example for a coupled oscillator network a necessary condition for synchronization to occur is that the msf at the corresponding normalized coupling parameters be negative to understand the typical behaviors of the msf for various chaotic oscillators is key to predicting the collective dynamics of a network of these oscillators we address this issue by examining systematically msfs for known chaotic oscillators our computations and analysis indicate that it is generic for msfs being negative in a finite interval of a normalized coupling parameter a general scheme is proposed to classify the typical behaviors of msfs into four categories these results are verified by direct simulations of synchronous dynamics on networks of actual coupled oscillators

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

an optimal order regularization method which does not use additional smoothness assumptions this paper defines an optimal method to reconstruct the solutions of operator equations of the first kind only the case of compact operators is considered the method is in principle a discrepancy method it does not require any additional knowledge about the solution and is optimal for all standard smoothness assumptions in order to analyze the properties of the new regularization method variable hilbert scales are introduced and several well known results for hilbert scales are generalized convergence theorems for classes of optimal and suboptimal methods are derived from a generalization of the interpolation inequality

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

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

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

rate of escape of random walks on wreath products and related groups this article examines the rate of escape for a random walk on g z and proves laws of the iterated logarithm for both the inner and outer radius of escape the class of g for which these results hold includes finite g as well as groups of the form h z so the construction can be iterated laws of the iterated logarithm are also found for random walk on baumslag solitar groups and a discrete version of the sol geometry

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

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

dynamics of fitzhugh nagumo excitable systems with delayed coupling small lattices of n nearest neighbor coupled excitable fitzhugh nagumo systems with time delayed coupling are studied and compared with systems of fitzhugh nagumo oscillators with the same delayed coupling bifurcations of equilibria in an n 2 case are studied analytically and it is then numerically confirmed that the same bifurcations are relevant for the dynamics in the case ng2 bifurcations found include inverse and direct hopf and fold limit cycle bifurcations typical dynamics for different small time lags and coupling intensities could be excitable with a single globally stable equilibrium asymptotic oscillatory with symmetric limit cycle bistable with stable equilibrium and a symmetric limit cycle and again coherent oscillatory but nonsymmetric and phase shifted for an intermediate range of time lags inverse sub critical hopf and fold limit cycle bifurcations lead to the phenomenon of oscillator death the phenomenon does not occur in the case of fitzhugh nagumo oscillators with the same type of coupling

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