random walks of infinite moment on free semigroups we consider random walks on finitely or countably generated free semigroups and identify their poisson boundaries for classes of measures which fail to meet the classical entropy criteria namely measures with infinite entropy or infinite logarithmic moment

a multiplicative ergodic theorem and nonpositively curved spaces we study integrable cocycles u n x over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space y e g a cartan hadamard space or a uniformly convex banach space it is proved that for any y 2 y and almost all x there exist a 0 and a unique geodesic ray t x in y starting at y such that lim n 1 1 n d an x u n x y d 0 in the case where y is the symmetric space gln r on r and the cocycles take values in gln r this is equivalent to the multiplicative ergodic theorem of oseledec two applications are also described the first concerns the determination of poisson boundaries and the second concerns hilbert schmidt operators

the poisson boundary of teichmuller space abstract for a markov operator on teichmuller space commuting with the action of the mapping class group we prove convergence of sample paths of the associated markov chain in the thurston compactification and show that the poisson boundary of the markov operator can be identified with the space of projective measured foliations the approach consists in using the authors results on the poisson boundary of the mapping class group in combination with a discretization procedure based on a harnack inequality for markov operators on teichmuller space

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

nearly round spheres look convex we prove that a riemannian manifold m g close enough to the round sphere in the c 4 topology has uniformly convex injectivity domains so m appears uniformly convex in any exponential chart the proof is based on the ma trudinger wang nonlocal curvature tensor which originates from the regularity theory of optimal transport

sublinear deviation between geodesics and sample paths we give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic like properties as an application we prove sublinear tracking in teichmuller distance for random walks on mapping class groups and on cayley graphs of a large class of finitely generated groups

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

the poisson boundary of text out f_n let mu be a probability measure on text out f n with finite first logarithmic moment with respect to the word metric finite entropy and whose support generates a nonelementary subgroup of text out f n we show that almost every sample path of the random walk on text out f n mu when realized in culler and vogtmann s outer space converges to the simplex of a free arational tree we then prove that the space mathcal fi of simplices of free and arational trees equipped with the hitting measure is the poisson boundary of text out f n mu using bestvina reynolds and hamenstadt s description of the gromov boundary of the complex mathcal ff n of free factors of f n this gives a new proof of the fact due to calegari and maher that the realization in mathcal ff n of almost every sample path of the random walk converges to a boundary point we get in addition that partial mathcal ff n equipped with the hitting measure is the poisson boundary of text out f n mu

a structure theorem for actions of semisimple lie groups we consider a connected semisimple lie group g with finite center an admissible probability measure on g and an ergodic g u space x v we first note lemma 0 1 that x v has a unique maximal projective factor of the form g q vo where q is a parabolic subgroup of g and then prove 1 theorem 1 if every noncompact simple factor of g has real rank at least two then the maximal projective factor is nontrivial unless v is a g

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

series expansion of the percolation probability for the directed square lattice by extrapolation from finite lattices the authors extend the known series for the percolation probability on the directed square lattice from eight terms to 41 analysing the series they obtain the estimates qc 0 355 299 or 0 000 001 beta 0 2764 or 0 0001 for the critical probability and the critical exponent from this together with scaling relations and previous results on the moments of the pair connectedness function the authors conjecture that beta may be exactly 199 720

invasion percolation as a fractal growth problem we apply the fixed scale transformation fst method developed for laplacian fractal growth to compute the fractal dimension generated by invasion percolation ip with and without trapping ip is a growth process with quenched random variables we show how to define the corresponding stochastic growth process that corresponds to the quenched average of the growth process with respect to the random variables this stochastic process turns out to have a long memory and to depend on the entire history of the system one can then compute the fst matrix elements for open and closed boundary conditions and from these the fractal dimensions of the clusters up to fourth order terms we obtain d 1 8872 for ip the exact value is d 9148 1 8958 and d 1 8196 for ip with trapping simulations give d1 82 these results provide an important test for the validity of the fst method as a new tool to study fractal structures generated by both equilibrium and irreversible growth processes

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

boundary and entropy of space homogeneous markov chains we study the poisson boundary equiv representation of bounded harmonic functions of markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations this automorphism group is locally compact but not necessarily discrete or unimodular the main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups while however the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other various new examples are given as applications including a description of the poisson boundary for random walks on vertex transitive graphs with infinitely many ends and on the diestel leader graphs

regularity of the entropy for random walks on hyperbolic groups we consider non degenerate finitely supported random walks on a finitely generated gromov hyperbolic group we study how the entropy and the escape rate vary with the probability of constant support

estimates of the site percolation probability exponents for some directed lattices series expansion analysis has been used to obtain estimates of the percolation probability exponent beta for directed site percolation on the square triangular simple cubic and body centred cubic lattices these estimates are consistent with those obtained by please 1977 for the corresponding bond problems and the results for site and bond problems on all lattices considered are summarised by beta 0 28 or 0 01 d 2 and beta 0 59 or 0 02 d 3

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

dimensional properties of the harmonic measure for a random walk on a hyperbolic group this paper deals with random walks on isometry groups of gromov hyperbolic spaces and more precisely with the dimension of the harmonic measure nu associated with such a random walk we first establish a link of the form dim nu leq h l between the dimension of the harmonic measure the asymptotic entropy h of the random walk and its rate of escape l then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure

novel position space renormalization group for bond directed percolation in two dimensions abstract a new position space renormalization group approach is investigated for bond directed percolation in two dimensions the threshold value for the bond occupation probabilities is found to be p c 0 6443 correlation length exponents on time parallel and space transverse directions are found to be 1 719 and 1 076 respectively which are in very good agreement with the best known series expansion results

percolation as a fractal growth problem abstract we consider percolation in two dimensions as a fractal growth problem and apply to it the theory of fractal growth based on the fixed scale transformation approach developed for diffusion limited aggregation and the dielectric breakdown model this represents an important test for this new theoretical method based on an additional invariance property with respect to the renormalization group we compute the fractal dimension of the percolating cluster including terms up to third order the result is d 1 8830 for the square lattice and d 1 8650 for the triangular lattice these values are in excellent agreement with the universal exact result d 91 48 1 8958 and also show the potential of this new method for standard problems

on laws of large numbers for random walks we prove a general noncommutative law of large numbers this applies in particular to random walks on any locally finite homogeneous graph as well as to brownian motion on riemannian manifolds whi

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

real space renormalisation group for directed systems in arbitrary dimensions a real space renormalisation group approach to intrinsically anisotropic systems i e systems having two different correlation lengths is defined the problems of directed self avoiding walks dsaw and directed percolation in d dimensions are discussed in detail for dsaw one obtains the exact critical fugacity in all dimensions for directed percolation the percolation threshold pc obtained is very good in two dimensions asymptotically exact for large dimensions and quite good in between the parallel and perpendicular correlation length exponents nu sub and nu perpendicular to are computed for the two problems

ergodic properties of boundary actions and the nielsen schreier theory we study the basic ergodic properties ergodicity and conservativity of the action of a subgroup h of a free group f on the boundary pt f with respect to the uniform measure our approach is geometrical and combinatorial and it is based on choosing a system of nielsen schreier generators in h associated with a geodesic spanning tree in the schreier graph x h bs f we give several mod 0 equivalent descriptions of the hopf decomposition of the boundary into the conservative and the dissipative parts further we relate conservativity and dissipativity of the action with the growth of the schreier graph x and of the subgroup h equiv cogrowth of x respectively on the other hand our approach sheds a new light on entirely algebraic properties of subgroups of a free group we also construct numerous examples illustrating the connections between various relevant notions

real space renormalisation group for directed systems a new real space renormalisation group method is developed to treat systems with directionally dependent critical behaviour here two effective lengths s sub b and sperpendicular to b are defined for a given cell of linear size b the renormalised lattice is constructed from these effective lengths and is deformed from the original lattice it is the anisotropic rescaling of these effective lengths which give the anisotropic exponents nu sub and nu perpendicular to this method gives the exact result for the case of fully directed saws on a square lattice in the b to infinity limit for directed bond and directed site lattice animals the results obtained here are also in good agreement with the other known values

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

fixed scale transformation for ising and potts clusters the fractal dimension of ising and potts clusters are determined via the fixed scale transformation approach which exploits both the self similarity and the dynamical invariance of these systems at criticality the results are easily extended to droplets a discussion of the interrelationships between the present approach and renormalization group methods as well as glauber type dynamics is provided

entropy and drift in word hyperbolic groups the fundamental inequality of guivarc h relates the entropy and the drift of random walks on groups it is strict if and only if the random walk does not behave like the uniform measure on balls we prove that in any nonelementary hyperbolic group which is not virtually free endowed with a word distance the fundamental inequality is strict for symmetric measures with finite support uniformly for measures with a given support this answers a conjecture of s lalley for admissible measures this is proved using previous results of ancona and blachere haissinsky mathieu for non admissible measures this follows from a counting result interesting in its own right we show that in any infinite index subgroup the number of non distorted points is exponentially small the uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup

the lipschitz continuity of the distance function to the cut locus let n be a closed submanifold of a complete smooth riemannian manifold m and uv the total space of the unit normal bundle of n for each v c uv let p v denote the distance from n to the cut point of n on the geodesic y with the velocity vector tyv 0 v the continuity of the function p on uv is well known in this paper we prove that p is locally lipschitz on which p is bounded in particular if m and n are compact then p is globally lipschitz on uy therefore the canonical interior metric 6 may be introduced on each connected component of the cut locus of n and this metric space becomes a locally compact and complete length space let n be an immersed submanifold of a complete cc riemannian manifold m and 7 uv n the unit normal bundle of n for each positive integer k and vector v c uv let a number k v denote the parameter value of fy where 7yv denotes the geodesic for which the velocity vector is v at t 0 such that yv ak v is the k th focal point conjugate point for the case in which n is a point of n along y counted with focal or conjugate multiplicities a unit speed geodesic segment 0 a m emanating from n is called an n segment if t d n y t on 0 a by the first variation formula an n segment is orthogonal to n a point 7y to on an n segment 7y v e uv is called a cut point of n if there is no n segment properly containing y o to for each v c uv let p v denote the distance from n to the cut point on y of n whitehead 27 investigated the structure of the conjugate locus and the cut locus of a point on a real analytic finsler manifold he determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood cf also 25 and proved the continuity of the function p in compact symmetric spaces t sakai 19 and m takeuchi 23 determined the detailed structure of the cut locus of a point the detailed structure of the cut locus of a point in a 2 dimensional riemannian manifold has been investigated by poincare myers and others 7 11 13 hartman first tried to show the absolute continuity of the function p when m is 2 dimensional he proved in 8 that if p is of bounded variation then p is absolutely continuous recently hebda 11 and the first named author 13 independently proved ambrose s problem by showing that p is absolutely continuous on a closed arc on which p is bounded when n is a point in a 2 dimensional riemannian manifold therefore the cut locus of a point in a compact 2 dimensional received by the editors october 14 1998 and in revised form april 13 1999 2000 mathematics subject classification primary 53c22 secondary 28a78 supported in part by a grant in aid for scientific research from the ministry of education science sports and culture japan 2000 american mathematical society

theory of laplacian fractals diffusion limited aggregation and dielectric breakdown model abstract we describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the laplace equation and a stochastic field this new theory provides a systematic method for the calculation of the fractal dimension d and of the multifractal spectrum of the growth probability a detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented our approach exploits the scale invariance of the laplace equation that implies that the structure is self similar both under growth and scale transformation this allows one to introduce a fixed scale transformation instead of coarse graining as in the renormalization group theory that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process for the calculation of the matrix elements of this transformation one has to consider an infinite but rapidly convergent number of processes that occurs outside a considered diagram

boundary behavior for groups of subexponential growth in this paper we introduce a method for partial description of the poisson boundary for a certain class of groups acting on a segment as an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric infinitely supported measure of finite entropy h this implies that the entropy h of the corresponding random walk is finite and positive as another application we exhibit certain discontinuity for the recurrence property of random walks finally as a corollary of our results we get new estimates from below for the growth function of a certain class of grigorchuk groups in particular we exhibit the first example of a group generated by a finite state automaton such that the growth function is subexponential but grows faster than exp n for any 1 we show that in some of our examples the growth function satisfies exp n ln

are damage spreading transitions generically in the universality class of directed percolation we present numerical evidence for the fact that the damage spreading transition in the domany kinzel automaton found by martinset al is in the same universality class as directed percolation we conjecture that also other damage spreading transitions should be in this universality class unless they coincide with other transitions as in the ising model with glauber dynamics and provided the probability for a locally damaged state to become healed is not zero

sur la dynamique unidimensionnelle en regularite intermediaire using probabilistic methods we prove new rigidity results for groups and pseudo groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class i e between c 1 and c 2 in particular we show some generalizations of denjoy theorem and the classical kopell lemma for abelian groups these techniques are also applied to the study of codimension 1 foliations for instance we obtain several generalized versions of sacksteder theorem in class c 1 we conclude with some remarks about the stationary measure

avalanches and correlations in driven interface depinning we study the critical behavior of a driven interface in a medium with random pinning forces by analyzing spatial and temporal correlations in a lattice model recently proposed by sneppen phys rev lett bf 69 3539 1992 the static and dynamic behavior of the model is related to the properties of directed percolation we show that due to the interplay of local and global growth rules the usual method of dynamical scaling has to be modified we separate the local from the global part of the dynamics by defining a train of causal growth events or avalanche which can be ascribed a well defined dynamical exponent z loc 1 zeta c simeq 1 63 where zeta c is the roughness exponent of the interface we observe that the avalanche size distribution obeys a power law decay with an exponent kappa simeq 1 25

the poisson formula for groups with hyperbolic properties the poisson boundary of a group g with a probability measure is the space of ergodic components of the time shift in the path space of the associated random walk via a generalization of the classical poisson formula it gives an integral representation of bounded harmonic functions on g in this paper we develop a new method of identifying the poisson boundary based on entropy estimates for conditional random walks it leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the poisson boundary with natural topological boundaries for several classes of groups word hyperbolic groups and discontinuous groups of isometries of gromov hyperbolic spaces groups with inflnitely many ends cocompact lattices in cartan hadamard manifolds discrete subgroups of semisimple lie groups