the trace of spatial brownian motion is capacity equivalent to the unit square we show that with probability 1 the trace b 0 1 of brownian motion in space has positive capacity with respect to exactly the same kernels as the unit square more precisely the energy of occupation measure on b 0 1 in the kernel f x y is bounded above and below by constant multiples of the energy of lebesgue measure on the unit square the constants are random but do not depend on the kernel as an application we give almost sure asymptotics for the probability that an stable process approaches within of b 0 1 conditional on b 0 1

associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

large deviations for random walks on galton watson trees averaging and uncertainty in the study of large deviations for random walks in random environment a key distinction has emerged between quenched asymptotics conditional on the environment and annealed asymptotics obtained from averaging over environments in this paper we consider a simple random walk x n on a galton watson tree t i e on the family tree arising from a supercritical branching process denote by x n the distance between the node x n and the root of t our main result is the almost sure equality of the large deviation rate function for x n n under the quenched measure conditional upon t and the rate function for the same ratio under the annealed measure averaging on t according to the galton watson distribution this equality hinges on a concentration of measure phenomenon for the momentum of the walk the momentum at level n for a specific tree t is the average over random walk paths of the forward drift at the hitting point of that level this concentration or certainty is a consequence of the uncertainty in the location of the hitting point we also obtain similar results when x n is a biased walk on a galton watson tree even though in that case there is no known formula for the asymptotic speed our arguments rely at several points on a ubiquity lemma for galton watson trees due to grimmett and kesten 1984

critical random walk in random environment on trees we study the behavior of random walk in random environment rwre on trees in the critical case left open in previous work representing the random walk by an electrical network we assume that the ratios of resistances of neighboring edges of a tree are i i d random variables whose logarithms have mean zero and finite variance then the resulting rwre is transient if simple random walk on is transient but not vice versa we obtain general transience criteria for such walks which are sharp for symmetric trees of polynomial growth in order to prove these criteria we establish results on boundary crossing by tree indexed random walks these results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean zero finite variance random walks in one dimension which are of independent interest

projection theorems for hitting probabilities and a theorem of littlewood abstract littlewood proc london math soc 2 28 1928 383 394 showed that a positive superharmonic function u on the unit disc has radial limits a e using techniques due to doob this result is extended to all rank one symmetric spaces in addition simplifications are obtained of doob s ann inst fourier grenoble 15 1965 113 135 proof of normal convergence a e of a positive superharmonic function on a half space the symmetric space analogue of this half space result is also obtained the methods used are shown to fail for the potential theory on r n associated with u u 4 0 it is an open question as to whether littlewood s theorem holds in this context

logical connectives of probabilistic sets a wide class of logical connectives such as e g triangular norms applied to probabilistic sets is discussed analytical results are derived and their applications to the description of decision making processes are also treated

on distribution function description of probabilistic sets and its application in decision making the paper deals with some problems of decision making described and solved by the use of the concept of a probabilistic set the distribution function description of probabilistic sets which is important from a theoretical point of view is discussed and its application in decision making is illustrated by means of numerical examples

leighton s bounds for sturm liouville eigenvalues abstract an error is pointed out in a method of w leighton for computing two sided bounds for the eigenvalues of the sturm liouville problem ry py 0 y a y b 0 the error is corrected the underlying theory is examined and the method is generalized

critical random graphs diameter and mixing time let mathcal c 1 denote the largest connected component of the critical erd s renyi random graph g n frac 1 n we show that typically the diameter of mathcal c 1 is of order n1 3 and the mixing time of the lazy simple random walk on mathcal c 1 is of order n the latter answers a question of benjamini kozma and wormald these results extend to clusters of size n2 3 of p bond percolation on any d regular n vertex graph where such clusters exist provided that p d 1 1 o n 1 3

phase transition and critical behavior in a model of organized criticality we study a model of organized criticality where a single avalanche propagates through an a priori static i e organized sandpile configuration the latter is chosen according to an i i d distribution from a borel probability measure on 0 1 the avalanche dynamics is driven by a standard toppling rule however we simplify the geometry by placing the problem on a directed rooted tree as our main result we characterize which are critical in the sense that they do not admit an infinite avalanche but exhibit a power law decay of avalanche sizes our analysis reveals close connections to directed site percolation both in the characterization of criticality and in the values of the critical exponents

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

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

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

decision making in uncertain environments abstract in the paper some problems of decision making in fuzzy and probabilistic fuzzy environments are formulated the goals and constraints in such decision situations may be simultaneously fuzzy and random by their nature looking for a decision set which should be any reasonable function of goals and constraints it leads to operations on probabilistic sets which are also of interest in that work some numerical examples of decision making are illustrating the theoretical considerations presented in this paper

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

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

random walk on barely supercritical branching random walk let mathcal t be a supercritical galton watson tree with a bounded offspring distribution that has mean mu 1 conditioned to survive let varphi mathcal t be a random embedding of mathcal t into mathbb z d according to a simple random walk step distribution let mathcal t p be percolation on mathcal t with parameter p and let p c mu 1 be the critical percolation parameter we consider a random walk x n n ge 1 on mathcal t p and investigate the behavior of the embedded process varphi mathcal t p x n as n rightarrow infty and simultaneously mathcal t p becomes critical that is p p n searrow p c we show that when we scale time by n p n p c 3 and space by sqrt p n p c n the process varphi mathcal t p x n n ge 1 converges to a d dimensional brownian motion we argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments

a macroscopic multifractal analysis of parabolic stochastic pdes it is generally argued that the solution to a stochastic pde with multiplicative noise such as dot u frac12 u u xi where xi denotes space time white noise routinely produces exceptionally large peaks that are macroscopically multifractal see for example gibbon and doering arch ration mech anal 177 115 150 2005 gibbon and titi proc r soc a 461 3089 3097 2005 and zimmermann et al phys rev lett 85 17 3612 3615 2000 a few years ago we proved that the spatial peaks of the solution to the mentioned stochastic pde indeed form a random multifractal in the macroscopic sense of barlow and taylor j phys a 22 13 2621 2626 1989 proc lond math soc 3 64 125 152 1992 the main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio temporal peaks of the solution form infinitely many different multifractals on infinitely many different scales which we sometimes refer to as stretch factors a simpler though still complex such structure is shown to also exist for the constant coefficient version of the said stochastic pde

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

monotonicity of uniqueness for percolation on cayley graphs all infinite clusters are born simultaneously consider site or bond percolation with retention parameter p on an infinite cayley graph in response to questions raised by grimmett and newman 1990 and benjamini and schramm 1996 we show that the property of having almost surely a unique infinite open cluster is increasing in p moreover in the standard coupling of the percolation models for all parameters a s for all p2 p1 pc each infinite p2 cluster contains an infinite p1 cluster this yields an extension of alexander s 1995 simultaneous uniqueness theorem as a corollary we obtain that the probability v p that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p pc all our results extend to quasi transitive infinite graphs with a unimodular automorphism group

biased random walks on directed trees we define directed rooted labeled and unlabeled trees and find measures on the space of directed rooted unlabeled trees which are invariant with respect to transition probabilities corresponding to a biased random walk on a directed rooted labeled tree we use these to calculate the speed of a biased random walk on directed rooted labeled trees the results are mainly applied to directed trees with recurrent subtrees where the random walker cannot escape

geometry of the uniform spanning forest transitions in dimensions 4 8 12 the uniform spanning forest usf in d is the weak limit of random uniformly chosen spanning trees in n n d pemantle 11 proved that the usf consists a s of a single tree if and only if d 4 we prove that any two components of the usf in d are adjacent a s if 5 d 8 but not if d 9 more generally let n x y be the minimum number of edges outside the usf in a path joining x and y in d then max n x y x y in mathbb z d left lfloor d 1 4 right rfloor textrm a s

biased random walks on galton watson trees we consider random walks with a bias toward the root on the family tree t of a supercritical galton watson branching process and show that the speed is positive whenever the walk is transient the corresponding harmonic measures are carried by subsets of the boundary of dimension smaller than that of the whole boundary when the bias is directed away from the root and the extinction probability is positive the speed may be zero even though the walk is transient the critical bias for positive speed is determined

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

day s implementation of the pruess method for sturm liouville eigenvalues abstract an error in w day s approach to finding two sided bounds for eigenvalues of sturm liouville problems is pointed out the algorithm is corrected and generalized and numerical results are given which include an application involving the eigenvalue parameter in the boundary condition

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

bounds and higher order estimates for sturm liouville eigenvalues in 2 an algorithm is described for computing upper and lower bounds for the eigenvalues of the regular sturm liouville problem on a finite interval a b the method which deals easily with coefficients having jump discontinuities replaces the coefftcients by constants on each subinterval of a partition rc a x0 x x b see i for a survey of the related literature the bounds converge to the true eigenvalues as h maxi hi 0 where hi xi xi1 the rate of convergence being of the first order in h 1 21 however it was noted in 2 that numerical results suggested that the arithmetic mean of the upper and lower bounds exhibits second order convergence in section 3 we give a proof of this second order convergence and also explain a remarkable numerical result noted in 2 although the main purpose of the algorithm of 2 is to give bounds the fact that it simultaneously gives higher order estimates is an important bonus

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

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

zeta functions and solutions of falconer type problems for self similar subsets of ℤn this paper uses the method of zeta functions to solve falconer type problems about sets of k simplices whose endpoints belong simultaneously to a self similar subset f of z k n and a disc b x of a large radius x assuming that the similarity transformations pairwise commute we study four euclidean metric invariants of the simplices the most basic and frequently studied of which is the distance between endpoints of a 1 simplex for each we introduce a zeta function derive its functional properties and apply such information to derive a lower bound on the upper minkowski dimension of f which guarantees that the number of distinct metric invariants must be unbounded as x mathematics subject classifications 11m41 28a80 52c10 11j69 11m32

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

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

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

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

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

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

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

the ising model and percolation on trees and tree like graphs we calculate the exact temperature of phase transition for the ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field in the same setting we calculate the critical temperature for spin percolation the same problems are solved for the diluted models and for more general random interaction strengths in the case of no interaction we generalize to percolation on certain tree like graphs this last calculation supports a general conjecture on the coincidence of two critical probabilities in percolation theory

an approach to multi criteria decision making problems using probabilistic set theory abstract the method of multi criteria decision making under uncertainty and imprecision is presented in this paper the criteria represented by probabilistic sets may be aggregated using various operators such aggregation leads to the final decision set from which the essential information as mean value variance etc can be obtained applying different choice criteria the best alternative can be chosen the considerations are illustrated by means of numerical examples

cover times blanket times and majorizing measures we exhibit a strong connection between cover times of graphs gaussian processes and talagrand s theory of majorizing measures in particular we show that the cover time of any graph g is equivalent up to universal constants to the square of the expected maximum of the gaussian free field on g scaled by the number of edges in g this allows us to resolve a number of open questions we give a deterministic polynomial time algorithm that computes the cover time to within an o 1 factor for any graph answering a question of aldous and fill 1994 we also positively resolve the blanket time conjectures of winkler and zuckerman 1996 showing that for any graph the blanket and cover times are within an o 1 factor the best previous approximation factor for both these problems was o loglogn 2 for n vertex graphs due to kahn kim lovasz and vu 2000

the scaling limit of loop erased random walk in three dimensions we show that the scaling limit exists and is invariant under dilations and rotations we give some tools that might be useful to show universality

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

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

on the compatibility of binary sequences an ordered pair of semi infinite binary sequences is said to be compatible if there is a way of removing a certain number possibly infinite of ones from and zeroes from that would map both sequences to the same semi infinite sequence this notion was introduced by peter winkler who also posed the following question and being independent i i d bernoulli sequences with parameters p and p respectively does there exist p p so that the set of compatible pairs has positive measure it is known that this does not happen for p and p very close to in the positive direction we construct for any 0 a deterministic binary sequence whose set of zeroes has hausdorff dimension larger than 1 and such that p is compatible 0 for p small enough where p stands for the product bernoulli measure with parameter p 2014 wiley periodicals inc

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

kakeya sets and directional maximal operators in the plane we completely characterize the boundedness of planar directional maximal operators on lp more precisely if is a set of directions we show that m the maximal operator associated to line segments in the directions is unbounded on lp for all p 1

intermittency and multifractality a case study via parabolic stochastic pdes let denote space time white noise and consider the following stochastic partial dierential equations i u 1 u 00 u started identically at one and ii z 1 z 00 started identically at zero it is well known that the solution to i is intermittent whereas the solution to ii is not and the two equations are known to be in dierent universality classes we prove that the tall peaks of both systems are multifractals in a natural large scale sense some of this work is extended to also establish the multifractal behavior of the peaks of stochastic pdes on r r d with d 2 g lawler has asked us if intermittency is the same as multifractality the present work gives a negative answer to this question as a byproduct of our methods we prove also that the peaks of the brownian motion form a large scale monofractal whereas the peaks of the ornstein uhlenbeck process on r are multifractal throughout we make extensive use of the macroscopic fractal theory of m t barlow and s j taylor 3 4 we expand on aspects of the barlow taylor theory as well

biased random walks on galton watson trees with leaves we consider a biased random walk xn on a galton watson tree with leaves in the sub ballistic regime we prove that there exists an explicit constant 0 1 depending on the bias such that xn is of order n denoting n the hitting time of level n we prove that n n1 is tight moreover we show that n n1 does not converge in law at least for large values of we prove that along the sequences n k k n n1 converges to certain infinitely divisible laws key tools for the proof are the classical harris decomposition for galton watson trees a new variant of regeneration times and the careful analysis of triangular arrays of i i d heavy tailed random variables

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