a general decomposition theory for random cascades this announcement describes a probabilistic approach to cascades which in addition to providing an entirely probabilistic proof of the kahane peyriere theorem for independent cascades readily applies to general dependent cascades moreover this unifies various seemingly disparate cascade decompositions including kahane s t martingale decomposition and dimension disintegration 1 brief history of the problem a theory of positive t martingales was introduced in k3 as the general framework for independent multiplicative cascades and random coverings this includes spatial distributions of interest in both probability theory and in the physical sciences e g ccd ck de dg df dm f gw ms mw tls pw s for basic definitions let t be a locally compact metric space with borel sigmafield b and let f p be a probability space together with an increasing sequence fn n 1 2 of sub sigmafields of f a positive t martingale is a sequence qn of non negative functions on t such that i for each t t qn t n 1 2 is a martingale adapted to fn n 1 2 ii for p a s qn n 1 2 is a sequence of borel measurable nonnegative real valued functions on t let m t denote the space of positive borel measures on t and suppose that qn t is a positive t martingale for m t let qn denote the random measure defined by qn and dqn d t qn t t t denote the space of bounded continuous functions on t by cb t then essentially by the martingale convergence theorem one obtains a random borel measure q such that with probability one k3

the multifractal spectrum of brownian intersection local times let l be the projected intersection local time of two independent brownian paths in d for d 2 3 we determine the lower tail of the random variable ell mathbb u where mathbb u is the unit ball the answer is given in terms of intersection exponents which are explicitly known in the case of planar brownian motion we use this result to obtain the multifractal spectrum or spectrum of thin points for the intersection local times

hypermeander of spirals local bifurcations and statistical properties in both experimental studies and numerical simulations of waves in excitable media rigidly rotating spiral waves are observed to undergo transitions to complicated spatial dynamics with long term brownian like motion of the spiral tip this phenomenon is known as hypermeander in this paper we review a number of recent results on dynamics with noncompact group symmetries and make the case that hypermeander may occur at a codimension two bifurcation from a rigidly rotating spiral wave our predictions are based on center bundle reduction sandstede scheel and wulff and on central limit theorems and invariance principles for group extensions of hyperbolic dynamical systems these predictions are confirmed by numerical simulations of the center bundle equations

the large n limit of the threshold values in mandelbrot s fractal percolation process the authors consider mandelbrot s fractal percolation process characterised by a density parameter p in 0 1 and an integer subdivision index n 1 for each n the process is known to have a percolation transition at density pc n in 0 1 they prove that lim pc n pc where pc is the threshold value of the ordinary square lattice site percolation model

bifurcation from rotating waves we consider abstract evolution equations which are invariant under a lie group we discuss bifurcation from trajectories that lie in the orbit under the lie group of a single point into trajectories in the orbit of a closed curve

normal forms resonances and meandering tip motions near relative equilibria of euclidean group actions the equivariant dynamics near relative equilibria to actions of noncompact finite dimensional lie groups g can be described by a skew product flow on a center manifold dot g g textbf a v dot v varphi v with g in g with v in a slice transverse to the group action and a v in the lie algebra of g we present a normal form theory near relative equilibria varphi v 0 0 in this general case for the specific case of the euclidean groups se n the skew product takes the form dot r r textbf r v qquad dot s r textbf s v qquad dot v varphi v with textbf r v in so n textbf s v in mathr n we give a precise meaning to the intuitive idea of tip motion of a meandering spiral it corresponds to the dynamics of s t this clarifies the notion of meander radii and drift resonance in the plane n 2 for illustration we discuss the unbounded tip motions associated with a weak focus in v on the verge of hopf bifurcation in the case of resonant hopf and rotation frequencies of the spiral and study resonant relative hopf bifurcation we also encounter random brownian tip motions for trajectories v t rightarrow gamma which become homoclinic for t rightarrow infty we conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the bogdanov takens bifurcation which turn out to be small beyond any finite order

projections of planar mandelbrot random measures abstract let be a planar mandelbrot measure and its orthogonal projection on one of the principal axes we study the thermodynamic and geometric properties of we first show that is exact dimensional with dim min dim dim where is the bernoulli product measure obtained as the expectation of we also prove that is absolutely continuous with respect to if and only if dim dim our results provides a new proof of dekking grimmett falconer formula for the hausdorff and box dimension of the topological support of as well as a new variational interpretation we obtain the free energy function of on a wide subinterval 0 q c of r for q 0 1 it is given by a variational formula which sometimes yields phase transitions of order larger than 1 for q 1 it is given by min which can exhibit first order phase transitions this is in contrast with the analyticity of over 0 q c also we prove the validity of the multifractal formalism for at each q c 0

phase diagram and correlation length bounds for mandelbrot aerogels the authors study a variant of the mandelbrot percolation process which is of current use as a model of aerogels the model has two parameters one of them q is the usual multiscale parameter of the mandelbrot percolation process and the other p is a bernoulli percolation parameter that is reserved for the last step of the construction they investigate the phase diagram of this model in the q p plane there are two phases a sol phase and a gel phase classified according to whether a limit of crossing probabilities vanishes or is nonzero in the sol phase they define a correlation length via the rate of decay of a rescaled connectivity function they show that this length scale diverges at the phase boundary furthermore they demonstrate that if the phase boundary is approached with q fixed and p tending up to its critical value pg q then up to logarithmic corrections the divergence is at least as fast as mod pg p mod 2dh where dh 2 mod log q log n mod can be identified as the hausdorff dimension of the background medium

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

sets avoided by brownian motion a fixed two dimensional projection of a three dimensional brownian motion is almost surely neighborhood recurrent is this simultaneously true of all the two dimensional projections with probability 1 equivalently three dimensional brownian motion hits any infinite cylinder with probability 1 does it hit all cylinders this papers shows that the answer is no brownian motion in three dimensions avoids random cylinders and in fact avoids bodies of revolution that grow almost as fast as cones

multiple points in the sample paths of a levy process we obtain a sufficient condition for the sample paths of a levy process to contain multiple points our condition is close to one conjectured by hendricks and taylor

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

new approach to percolation lattice system results results of an integral equation approach to pair connectedness are given for lattice models with emphasis on site percolation the lowest order approximation is expressible in terms of the lattice green function and is isomorphic to a polya random walk it yields percolation exponents gamma 2 nu 1 for d 3 and gamma 1 nu 1 2 for d or 4 d dimension for simple hypercubic lattices it also yields usefully sharp estimates for the site percolation threshold probability pp a first correction is described

global analysis of self similarity features of cellular automata selected examples abstract the analysis of self similarity properties of the evolution patterns of linear cellular automata is investigated using the theory of hierarchical iterated function systems and substitution systems we present several selected examples

intersections of random walks in four dimensions ii letf n be the probability that the paths of two simple random walks of lengthn starting at the origin in 4 have no intersection it has previously been shown thatf n c logn 1 2 here it is proved that for allr 1 2 mathop lim limits n to infty log n r f n infty

on the structure of mandelbrot s percolation process and other random cantor sets we consider generalizations of mandelbrot s percolation process for the process which we call the random sierpinski carpet we show that it passes through several different phases as its parameter increases from zero to one the final section treats the percolation phase

galton watson trees with the same mean have the same polar sets evans defined a notion of what it means for a set b to be polar for a process indexed by a tree the main result herein is that a tree picked from a galton watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate this implies that deterministic and nondeterministic trees behave identically in a variety of probability models mapping subsets of euclidean space to trees and polar sets to capacity criteria it follows that certain random cantor sets are capacity equivalent to each other and to deterministic cantor sets an extension to branching processes in varying environment is also obtained

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

the dimension of projections of fractal percolations fractal percolation or mandelbrot percolation is one of the most well studied families of random fractals in this paper we study some of the geometric measure theoretical properties dimension of projections and structure of slices of these random sets although random the geometry of those sets is quite regular our results imply that denoting by e subset mathbb r 2 a typical realization of the fractal percolation on the plane if dim rm h e 1 then for all lines l the orthogonal projection el of e to l has the same hausdorff dimension as e if dim rm h e 1 then for any smooth real valued function f which is strictly increasing in both coordinates the image f e contains an interval the second statement is quite interesting considering the fact that e is almost surely a cantor set a random dust for a large part of the parameter domain see chayes et al probab theory relat fields 77 3 307 324 1988 finally we solve a related problem about the existence of an interval in the algebraic sum of d 2 one dimensional fractal percolations

normal form theory for relative equilibria and relative periodic solutions we show that in the neighbourhood of relative equilibria and relative periodic solutions coordinates can be chosen so that the equations of motion in normal form admit certain additional equivariance conditions up to arbitrarily high order in particular normal forms for relative periodic solutions effectively reduce to normal forms for relative equilibria enabling the calculation of the drift of solutions bifurcating from relative periodic solutions

capacity and principal eigenvalues the method of enlargement of obstacles revisited we describe a coarse graining method which provides lower bounds on the principal dirichlet eigenvalue of the laplacian in regions receiving small obstacles and sharpens the previous method of enlargement of obstacles based on a quantitative wiener criterion one replaces the actual obstacles by obstacles of a much larger size controls on the shift of principal eigenvalues and capacity estimates on the locus where the wiener criterion breaks down are derived the results are written in a self contained fashion

on the connectivity properties of the complementary set in fractal percolation models we study the connectivity properties of the complementary set in poisson multiscale percolation model and in mandelbrot s percolation model in arbitrary dimension by using a result about majorizing dependent random fields by bernoulli fields we prove that if the selection parameter is less than certain critical value then by choosing the scaling parameter large enough we can assure that there is no percolation in the complementary set

a new fractal dimension the topological hausdorff dimension we introduce a new concept of dimension for metric spaces the so called topological hausdorff dimension it is defined by a very natural combination of the definitions of the topological dimension and the hausdorff dimension the value of the topological hausdorff dimension is always between the topological dimension and the hausdorff dimension in particular this new dimension is a non trivial lower estimate for the hausdorff dimension we examine the basic properties of this new notion of dimension compare it to other well known notions determine its value for some classical fractals such as the sierpinski carpet the von koch snowflake curve kakeya sets the trail of the brownian motion etc as our first application we generalize the celebrated result of chayes chayes and durrett about the phase transition of the connectedness of the limit set of mandelbrot s fractal percolation process they proved that certain curves show up in the limit set when passing a critical probability and we prove that actually thick families of curves show up where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large hausdorff dimension the proof of this is basically a lower estimate of the topological hausdorff dimension of the limit set for the sake of completeness we also give an upper estimate and conclude that in the non trivial cases the topological hausdorff dimension is almost surely strictly below the hausdorff dimension finally as our second application we show that the topological hausdorff dimension is precisely the right notion to describe the hausdorff dimension of the level sets of the generic continuous function in the sense of baire category defined on a compact metric space

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

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

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

the critical point of fractal percolation in three and more dimensions the fractal percolation model is constructed by the progressive replacement of each cube by a random collection of sub cubes each sub cube being retained with probability p there is a critical probability pc m d depending on the dimension d and the substitution ratio m the authors point out that for large m the value of pc m d is near to that of site percolation on a lattice derived from the d dimensional hypercubic lattice by the addition of certain bonds

harmonic analysis of additive levy processes let x1 xn denote n independent d dimensional levy processes and consider the n parameter random field mathfrak x t x 1 t 1 cdots x n t n first we demonstrate that for all nonrandom borel sets f subseteq bf r d the minkowski sum mathfrak x bf r n oplus f of the range mathfrak x bf r n of mathfrak x with f can have positive d dimensional lebesgue measure if and only if a certain capacity of f is positive this improves our earlier joint effort with yuquan zhong by removing a certain condition of symmetry in khoshnevisan et al ann probab 31 2 1097 1141 2003 moreover we show that under mild regularity conditions our necessary and sufficient condition can be recast in terms of one potential densities this rests on developing results in classical non probabilistic harmonic analysis that might be of independent interest as was shown in khoshnevisan et al ann probab 31 2 1097 1141 2003 the potential theory of the type studied here has a large number of consequences in the theory of levy processes presently we highlight a few new consequences

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

strong law of large numbers for the capacity of the wiener sausage in dimension four we prove a strong law of large numbers for the newtonian capacity of a wiener sausage in the critical dimension four where a logarithmic correction appears in the scaling the main step of the proof is to obtain precise asymptotics for the expected value of the capacity this requires a delicate analysis of intersection probabilities between two independent wiener sausages

intersection properties of simple random walks a renormalization group approach we study estimates for the intersection probability g m of two simple random walks on lattices of dimensiond 4 4 e as a problem in euclidean field theory we rigorously establish a renormalization group flow equation forg m and bounds on the function which show that ind 4 g m tends to zero logarithmically as the killing rate mass m tends to zero and that the fixed point g ind 4 e is bounded by const e g conste our methods also yield estimates on the intersection probability of three random walks ind 3 3 e for e 0 these results were first obtained by lawler 1

stable processes on the boundary of a regular tree we define a class of processes on the boundary of a regular tree that can be viewed as stable levy processes on z n 0 z n we show that the range of these processes can be compared with a bernoulli percolation as in peres which easily leads to various results on the intersection properties we develop an alternative approach based on the comparison with a branching random walk by this method we establish the existence of points of infinite multiplicity when the index of the process equals the dimension of the state space as for planar brownian motion

what is the probability of intersecting the set of brownian double points we give potential theoretic estimates for the probability that a set a contains a double point of planar brownian motion run for unit time unlike the probability for a to intersect the range of a markov process this cannot be estimated by a capacity of the set a instead we introduce the notion of a capacity with respect to two gauge functions simultaneously we also give a polar decomposition of a into a set that never intersects the set of brownian double points and a set for which intersection with the set of brownian double points is the same as intersection with the brownian path

direct estimates on intersection probabilities of random walks a study is made of intersection properties of independent random walks in d dimensional lattice space z sup d a simple method is developed which makes it possible to estimate intersection probabilities of two random walks with killing rate m directly it is expected that the method can be generalized and extended to other issues

brownian motion on a random recursive sierpinski gasket we introduce a random recursive fractal based on the sierpinski gasket and construct a diffusion upon the fractal via a dirichlet form this form and its symmetrizing measure are determined by the electrical resistance of the fractal the effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion the main result is to obtain uniform upper and lower bounds for the transition density of the brownian motion on the fractal in terms of this metric the bounds are not tight as there are logarithmic corrections due to the randomness in the environment and the behavior of the shortest paths in the effective resistance metric is not well understood the results are deduced from the study of a suitable general branching process

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

level sets of additive levy processes we provide a probabilistic interpretation of a class of natural capacities on euclidean space in terms of the level sets of a suitably chosen multiparameter additive levy process x we also present several probabilistic applications of the aforementioned potential theoretic connections they include areas such as intersections of levy processes and level sets as well as hausdorff dimension computations

phase transition in dependent percolation in this paper we discuss two different models of dependent percolation on the graph 2 these models can be thought of as percolation in a random environment they were inspired by the work of mccoy and wu 7 8 on the ising model in a random environment as well as other models of particle systems in a random environment 9 5 6 3 we show that both models of dependent percolation exhibit phase transitions this proves a version of stability for percolation on 2 and proves a conjecture of jonasson mossel and peres 4 who proved a similar result on 3

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

fractal percolation and branching cellular automata branching cellular automata bca are introduced as generalisations of fractal percolation by admitting neighbour dependence we associate sequences of random sets with bca s and study their convergence in case of convergence we derive the hausdorff dimension of the limit set and of its boundary to accomplish the latter we proof that the boundary of a set generated by a bca is again a set generated by a bca

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

brownian sheet and capacity the main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the brownian sheet as applications we determine the escape rates of the brownian sheet and also obtain a local intersection equivalence between the brownian sheet and the additive brownian motion other applications concern quasisure properties in wiener space

continuous symmetry reduction and return maps for high dimensional flows we present two continuous symmetry reduction methods for reducing highdimensional dissipative flows to local return maps in the hilbert polynomial basis approach the equivariant dynamics is rewritten in terms of invariant coordinates in the method of moving frames or method of slices the state space is sliced locally in such a way that each group orbit of symmetry equivalent points is represented by a single point in either approach numerical computations can be performed in the original state space representation and the solutions are then projected onto the symmetry reduced state space the two methods are illustrated by reduction of the complex lorenz system a 5 dimensional dissipative flow with rotational symmetry while the hilbert polynomial basis approach appears unfeasible for high dimensional flows symmetry reduction by the method of moving frames offers hope

superbranching processes and projections of random cantor sets we study sequences x 0 x 1 of random variables taking values in the positive integers which grow faster than branching processes in the sense that x m n geqq sum limits i 1 x m x n m i for m n 0 where the x n m i are distributed as x n and have certain properties of independence we prove that under appropriate conditions x 1 almost surely and in l 1 where sup e x n 1 n our principal application of this result is to study the lebesgue measure and hausdorff dimension of certain projections of sets in a class of random cantor sets being those obtained by repeated random subdivisions of the m adic subcubes of 0 1 d we establish a necessary and sufficient condition for the lebesgue measure of a projection of such a random set to be non zero and determine the box dimension of this projection

mixing time of a kinetically constrained spin model on trees power law scaling at criticality on the rooted k ary tree we consider a 0 1 kinetically constrained spin model in which the occupancy variable at each node is re sampled with rate one from the bernoulli p measure iff all its children are vacant for this process the following picture was conjectured to hold as long as p is below the percolation threshold p c 1 k the process is ergodic with a finite relaxation time while for p p c the process on the infinite tree is no longer ergodic and the relaxation time on a finite regular sub tree becomes exponentially large in the depth of the tree at the critical point p p c the process on the infinite tree is still ergodic but with an infinite relaxation time moreover on finite sub trees the relaxation time grows polynomially in the depth of the tree the conjecture was recently proved by the second and forth author except at criticality here we analyse the critical and quasi critical case and prove for the relevant time scales i power law behavior in the depth of the tree at p p c and ii power law scaling in p c p 1 when p approaches p c from below our results which are very close to those obtained recently for the ising model at the spin glass critical point represent the first rigorous analysis of a kinetically constrained model at criticality

non existence of phase transition of oriented percolation on sierpinski carpet lattices a percolation problem on sierpinski carpet lattices is considered it is obtained that the critical probability of oriented percolation is equal to 1 in contrast it was already shown that the critical probability pc of percolation is strictly less than 1 in kumagai 9 this result shows a difference between fractal like lattice and d lattice

a geometric characterisation of resonance in hopf bifurcation from relative equilibria we give a new characterisation of resonance in hopf bifurcation from relative equilibria in systems with compact symmetry group this characterisation provides a full geometric explanation of the resonance phenomenon in addition we develop techniques based on normal form theory to give a complete solution to the associated bifurcation problem

wiener s test for super brownian motion and the brownian snake we prove a wiener type criterion for super brownian motion and the brownian snake if f is a borel subset of d and x d we provide a necessary and sufficientcondition for super brownian motion started at x to immediately hit the set f equivalently this condition is necessary and sufficient for the hitting time of f by thebrownian snake with initial point x to be 0 a key ingredient of the proof isan estimate showing that the hitting probability of f is comparable up to multiplicative constants to the relevant capacity of f this estimate which is of independent interest refines previous results due to perkins and dynkin an important role is played by additivefunctionals of the brownian snake which are investigated here via the potentialtheory of symmetric markov processes as a direct application of our probabilisticresults we obtain a necessary and sufficient condition for the existence in a domain d of a positivesolution of the equation u u 2 which explodes at a given point of d

hopf bifurcations from relative equilibria in spherical geometry abstract resonant and nonresonant hopf bifurcations from relative equilibria posed in two spatial dimensions in systems with euclidean se 2 symmetry have been extensively studied in the context of spiral waves in a plane and are now well understood we investigate hopf bifurcations from relative equilibria posed in systems with compact so 3 symmetry where so 3 is the group of rotations in three dimensions on a sphere unlike the se 2 case the skew product equations cannot be solved directly and we use the normal form theory due to fiedler and turaev to simplify these systems we show that the normal form theory resolves the nonresonant case but not the resonant case new methods developed in this paper combined with the normal form theory resolves the resonant case

the intersection of brownian paths as a case study of a renormalization group method for quantum field theory a new approach is presented for the study of the probability that the random paths generated by two independent brownian motions in d intersect or more generally are within a short distance a of each other the well known behavior of that function of a above below and at the critical dimension d 4 as well as further corrections are derived here by means of a single renormalization group equation the equation s derivation is expected to shed some light on the function of the o d 4 quantum field theory

phase transitions in mandelbrot s percolation process in three dimensions we study the phase structure and transitions in three dimensional mandelbrot percolation a process which generates random fractal sets we establish the existence of three distinct phase transitions and we show that two of these transitions corresponding to percolation across the initial set by paths and sheets are discontinuous

universality of symmetric one dimensional random flights the asymptotic behaviour of all the moments of a class of symmetric one dimensional random flights at the first few orders are calculated different symmetric random flights are considered and it is shown that they simply differ up to a scaling in the number of jumps and that important statistical quantities are universal

boundary trace of reflecting brownian motions we establish a uniform dimensional result for normally reflected brownian motion rbm in a large class of non smooth domains hausdorff dimensions for the boundary occupation time and the boundary trace of rbm are determined extensions to stable like jump processes and to symmetric reflecting diffusions are also given

no directed fractal percolation in zero area we consider the fractal percolation process on the unit square with fixed decimation parametern and level dependent retention parameters pk that is for allk 1 at thek th stage every retained square of side lengthn1 k is partitioned inton2 congruent subsquares and each of these is retained with probabilitypk independent of all others we show that if kpk 0 i e if the area of the limiting set vanishes a s then a s the limiting set contains no directed crossings of the unit square a directed crossing is a path that crosses the unit square from left to right and moves only up down and to the right

thick points for spatial brownian motion multifractal analysis of occupation measure let t x r denote the total occupation measure of the ball of radius r centered at x for brownian motion in ir we prove that sup x 1 t x r r log r 16 a s as r 0 thus solving a problem posed by taylor in 1974 furthermore for any a 0 16 the hausdorff dimension of the set of thick points x for which lim sup r 0 t x r r log r a is almost surely 2 a 8 this is the correct scaling to obtain a nondegenerate multifractal spectrum for brownian occupation measure analogous results hold for brownian motion in any dimension d 3 these results are related to the lil of ciesielski and taylor 1962 for the brownian occupation measure of small balls in the same way that levy s uniform modulus of continuity and the formula of orey and taylor 1974 for the dimension of fast points are related to the usual lil we also show that the lim inf scaling of t x r is quite different we exhibit non random c1 c2 0 such that c1 supx lim infr 0 t x r r 2 c2 a s in the course of our work we provide a general framework for obtaining lower bounds on the hausdorff dimension of random fractals of limsup type

the existence of phase v in the mandelbrot percolation process dekking and meester defined six phases for a subclass of random cantor sets consisting of those generated by bernoulli random substitutions they proved that the random sierpinski carpet passed through all these phases asp tended from 0 to 1 but the were not able to prove the existencne of phase v in the mandelbrot percolation process in this paper we accomplish the proof by improving their methods

on the phase transition to sheet percolation in random cantor sets thed dimensional random cantor set is a generalization of the classical middle thirds cantor set starting with the unit cube 0 1 d at every stage of the construction we divide each cube remaining intomd equal subcubes and select each of these at random with probabilityp the resulting limit set is a random fractal which may be crossed by paths or d 1 dimensional sheets we examine the critical probabilityps m d marking the existence of these sheet crossings and show that ps m d 1 pc md asm where pc md is the critical probability of site percolation on the lattice md obtained by adding the diagonal edges to the hypercubic lattice d this result is then used to show that at least for sufficiently large values ofm the phases corresponding to the existence of path and sheet crossings are distinct

random flights in euclidean space i general analysis and results for flights with prescribed hit expectance density about the origin we consider the problem of random flights in euclidean space defined by a series of displacements ri the magnitude and direction of each being independent of all the preceding ones the displacements are not restricted to prescribed lattice sites we begin with some new results generalizing a well known latticewalk relation between the probability of return to the origin and the expected number of times the origin is visited in the course of a random walk we go on to consider flights for which the hit expectancy is prescribed within a hypersphere of radiusr centered at the flight origin for uniform hit expectance density i e hit expectancy proportional to volume size within the sphere we solve the problem in three and five dimensions for a certain class of displacement probability densities that are prescribed only for displacement distancesr greater thanr for each such displacement probability r we find both the value of the hit expectance density and the form of the displacement probability density forr r that are dictated by the constraint of uniform hit expectance density within the sphere of radiusr in an appendix we show the way the common appearance of integral equations of ornstein zernike type in problems of random flight liquid and lattice gas structure and percolation theory yield certain corresponding results in all three areas