the structure of the allelic partition of the total population for galton watson processes with neutral mutations we consider a sub critical galton watson process with neutral mutations infinite alleles model and decompose the entire population into clusters of individuals carrying the same allele we specify the law of this allelic partition in terms of the distribution of the number of clone children and the number of mutant children of a typical individual the approach combines an extension of harris representation of galton watson processes and a version of the ballot theorem some limit theorems related to the distribution of the allelic partition are also given

the dual tree of a recursive triangulation of the disk in the recursive lamination of the disk one tries to add chords one after another at random a chord is kept and inserted if it does not intersect any of the previously inserted ones curien and le gall ann probab vol 39 pp 2224 2270 2011 have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process m based on a new approach resembling ideas from the so called contraction method in function spaces we prove that when properly rescaled the planar dual of the discrete lamination converges almost surely in the gromov hausdorff sense to a limit real tree t which is encoded by m this confirms a conjecture of curien and le gall 1 introduction and main results in 18 curien and le gall introduce the model of random recursive triangulations of the disk the construction goes as follows atn 1 two points are sampled independently with uniform distribution on the circle they are connected by a chord a straight line which splits the disk into two fragments later on at each step two independent points are sampled uniformly at random on the circle and are connected by a chord if the latter does not intersect any of the previously inserted chords in other words the two points are connected by a chord if they both fall in the same fragment this gives rise to a sequence of laminations of the disk for us a lamination will be a collection of chords which may only intersect at their end points at timen the lamination ln consists of the union of the chords inserted up to timen as an increasing closed subset of the disk ln converges and it is proved in 18 that

the scaling limit of the incipient infinite cluster in high dimensional percolation ii integrated super brownian excursion for independent nearest neighbor bond percolation on zd with d 6 we prove that the incipient infinite cluster s two point function and three point function converge to those of integrated super brownian excursion ise in the scaling limit the proof is based on an extension of the new expansion for percolation derived in a previous paper and involves treating the magnetic field as a complex variable a special case of our result for the two point function implies that the probability that the cluster of the origin consists of n sites at the critical point is given by a multiple of n 3 2 plus an error term of order n 3 2 e with e 0 this is a strong version of the statement that the critical exponent is given by 2

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

oscillation and nonoscillation of two dimensional difference systems this paper is concerned with the oscillation and nonoscillation behavior of solution of the nonlinear two dimensional difference system xn ang yn yn 1 f n xn n n n0 n0 n0 1 some necessary and sufficient conditions are given for the system to admit the existence of oscillatory and nonoscillatory solutions with special asymptotic properties some important results in the literatures are generalized examples to illustrate the result are included

oscillation criteria for a class of second order emden fowler delay dynamic equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for the second order emden fowler delay dynamic equations x t p t x t 0 on a time scale t here is a quotient of odd positive integers with p t real valued positive rd continuous functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales our results in this paper not only extend the results given in r p agarwal m bohner s h saker oscillation of second order delay dynamic equations can appl math q 13 1 2005 1 18 but also unify the oscillation of the second order emden fowler delay differential equation and the second order emden fowler delay difference equation

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

oscillation criteria for two dimensional dynamic systems on time scales we establish some oscillation criteria for the two dimensional dynamic system x d t b t g y s t y d t a t f x t on a time scale t our results not only unify the oscillation of two dimensional differential systems and difference systems but include the oscillation results for differential systems and provide new oscillation criteria for difference systems several examples are considered to illustrate the main results

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

invariance principles for random bipartite planar maps it is conjectured in the physics literature that properly rescaled random planar maps when conditioned to have a large number of faces should converge to a limiting surface whose law does not depend up to scaling factors on details of the class of maps that are sampled previous works on the topic starting with chassaing schaeffer have shown that the radius of a random quadrangulation with n faces converges in distribution once rescaled by n 1 4 to the diameter of the brownian snake up to a scaling constant using a bijection due to bouttier di francesco guitter between bipartite planar maps and a family of labeled trees we show the corresponding invariance principle for a class of random maps that follow a boltzmann distribution the radius of such maps conditioned to have n faces or n vertices and under a criticality assumption converges in distribution once rescaled by n 1 4 to a scaled version of the diameter of the brownian snake convergence results for the so called profile of maps are also provided as well as convergence of rescaled bipartite maps to the brownian map as introduced by marckert mokkadem the proofs of these results rely on a new invariance principle for two type spatial galton watson trees

a mating of trees approach for graph distances in random planar maps we introduce a general technique for proving estimates for certain random planar maps which belong to the gamma liouville quantum gravity lqg universality class for gamma in 0 2 the family of random planar maps we consider are those which can be encoded by a two dimensional random walk with i i d increments via a mating of trees bijection and includes the uniform infinite planar triangulation uipt gamma sqrt 8 3 and planar maps weighted by the number of different spanning trees gamma sqrt 2 bipolar orientations gamma sqrt 4 3 or schnyder woods gamma 1 that can be put on the map using our technique we prove estimates for graph distances in the above family of random planar maps in particular we obtain non trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the watabiki prog theor phys suppl 114 1 17 1993 prediction for the hausdorff dimension of gamma lqg and we establish the existence of an exponent for certain distances in the map the basic idea of our approach is to compare a given random planar map m to a mated crt map a random planar map constructed from a correlated two dimensional brownian motion using a strong coupling zaitsev in esaim probab stat 2 41 108 1998 of the encoding walk for m and the brownian motion used to construct the mated crt map this allows us to deduce estimates for graph distances in m from the estimates for graph distances in the mated crt map which we proved using continuum theory in a previous work in the special case when gamma sqrt 8 3 we instead deduce estimates for the sqrt 8 3 mated crt map from known results for the uipt the arguments of this paper do not directly use sle lqg and can be read without any knowledge of these objects

rayleigh processes real trees and root growth with re grafting the real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree aldous s brownian continuum random tree the random tree like object naturally associated with a standard brownian excursion may be thought of as a random compact real tree the continuum random tree is a scaling limit as n of both a critical galton watson tree conditioned to have total population size n as well as a uniform random rooted combinatorial tree with n vertices the aldous broder algorithm is a markov chain on the space of rooted combinatorial trees with n vertices that has the uniform tree as its stationary distribution we construct and study a markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as n of the aldous broder chain a key technical ingredient in this work is the use of a pointed gromov hausdorff distance to metrize the space of rooted compact real trees berkeley statistics technical report no 654 february 2004 revised october 2004 to appear in probability theory and related fields

oscillation of second order delay differential equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for a second order delay differential equation on time scales in terms of the coefficients

pointwise estimates for a class of non homogeneous kolmogorov equations we consider a class of ultraparabolic differential equations that satisfy the hormander s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions the method extends the moser s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption here we remove that assumption by proving some potential estimates and some ad hoc sobolev type inequalities for solutions

a fragmentation process connected to brownian motion let b s s 0 be a standard brownian motion and t 1 its first passage time at level 1 for every t 0 we consider ladder time set t of the brownian motion with drift t b t s b s ts and the decreasing sequence f t f 1 t f 2 t of lengths of the intervals of the random partition of 0 t 1 induced by t the main result of this work is that f t t 0 is a fragmentation process in the sense that for 0 t t f t is obtained from f t by breaking randomly into pieces each component of f t according to a law that only depends on the length of this component and independently of the others we identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by aldous and pitman 3

central limit theorems for the spectra of classes of random fractals we discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal the continuum random tree in the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit we will show examples from a class of random fractals generated from dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem the brownian continuum random tree can also be viewed as a random fractal generated by a dirichlet distribution the first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this though the positivity of the variance arising in the central limit theorem is left open in both cases these fractals can be described through a general crump mode jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known

on a functional contraction method methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space our approach is an extension of the so called contraction method to the space c 0 1 of continuous functions endowed with uniform topology and the space d 0 1 of cadlag functions with the skorokhod topology the contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences it is based on stochastic fixed point equations where probability metrics can be used to obtain contraction properties and allow the application of banach s fixed point theorem we develop the use of the zolotarev metrics on the spaces c 0 1 and d 0 1 in this context applications are given in particular a short proof of donsker s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed

growth of the brownian forest trees in brownian excursions have been studied since the late 1980s forests in excursions of brownian motion above its past minimum are a natural extension of this notion in this paper we study a forest valued markov process which describes the growth of the brownian forest the key result is a composition rule for binary galton watson forests with i i d exponential branch lengths we give elementary proofs of this composition rule and explain how it is intimately linked with williams decomposition for brownian motion with drift

volume growth and heat kernel estimates for the continuum random tree in this article we prove global and local point wise volume and heat kernel bounds for the continuum random tree we demonstrate that there are almost surely logarithmic global fluctuations and log logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r 0 we also show that the on diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t 0 almost surely finally we prove that this quenched almost sure behaviour contrasts with the local annealed averaged over all realisations of the tree volume and heat kernel behaviour which is smooth

oscillation and nonoscillation criteria for linear dynamic systems on time scales in this paper we establish oscillation and nonoscillation criteria for the linear dynamic system u d pv v d qu s here we assume that p and q are nonnegative rd continuous functions on a time scale t such that supt indeed we extend some known oscillation results for differential systems and difference systems to the so called dynamic systems

new kamenev type oscillation criteria for second order differential equations on a measure chain we establish new kamenev type criteria for oscillation of the second order linear differential equation p t x d t d q t x s t 0 on a measure chain our results are extensions of those for differential equations and provide new oscillation criteria for difference equations several examples are given to show the significance of the results

some new oscillation results for a nonlinear dynamic system on time scales we study the oscillation of a system of two first order nonlinear equations on time scales this form includes the classical emden fowler differential equation and many of its extensions we generalize some well known results of atkinson bohner erbe peterson and others we illustrate the results by several examples including a superlinear emden fowler dynamic system

ray knight representation of flows of branching processes with competition by pruning of levy trees we introduce flows of branching processes with competition which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term this generalizes the logistic branching processes studied by lambert ann appl probab 15 2 1506 1535 2005 following the approach developed by dawson and li ann probab 40 2 813 857 2012 we first construct such processes as the solutions of certain flow of stochastic differential equations we then propose a novel genealogical description for branching processes with competition based on interactive pruning of levy trees and establish a ray knight representation result for these processes in terms of the local times of suitably pruned forests

oscillation of certain two dimensional nonlinear difference systems abstract several new oscillation criteria for two dimensional nonlinear difference systems are established examples which dwell upon the importance of our results are also included

the contour of splitting trees is a levy process splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution to i i d copies of themselves the width process of a splitting tree is then a binary homogeneous crump mode jagers cmj process and is not markovian unless the lifetime distribution is exponential or a dirac mass at here we allow the birth rate to be infinite that is pairs of birth times and life spans of newboms form a poisson point process along the lifetime of their mother with possibly infinite intensity measure a splitting tree is a random so called chronological tree each element of a chronological tree is a so called existence point of some individual vertex in a discrete tree where is a nonnegative real number called chronological level time we introduce a total order on existence points called linear order and a mapping from the tree into the real line which preserves this order the inverse of is called the exploration process and the projection of this inverse on chronological levels the contour process for splitting trees truncated up to level r we prove that a thus defined contour process is a levy process reflected below and killed upon hitting 0 this allows one to derive properties of i splitting trees conceptual proof of le gall le jan s theorem in the finite variation case exceptional points coalescent point process and age distribution ii cmj processes one dimensional marginals conditionings limit theorems and asymptotic numbers of individuals with infinite versus finite descendances

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

tree valued resampling dynamics martingale problems and applications the measure valued fleming viot process is a diffusion which models the evolution of allele frequencies in a multi type population in the neutral setting the kingman coalescent is known to generate the genealogies of the individuals in the population at a fixed time the goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies we encode the genealogy of the population as an isometry class of an ultra metric space which is equipped with a probability measure the space of ultra metric measure spaces together with the gromov weak topology serves as state space for tree valued processes we use well posed martingale problems to construct the tree valued resampling dynamics of the evolving genealogies for both the finite population moran model and the infinite population fleming viot diffusion we show that sufficient information about any ultra metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled individuals we give explicit formulas for the evolution of the laplace transform of the distribution of finite subtrees under the tree valued fleming viot dynamics

lp compactness of inhomogeneous kinetic equation with rough force abstract we are concerned with the inhomogeneous kinetic equations with rough force l p compactness of the inhomogeneous kinetic equation with rough force is constructed by using phase space analysis and the fourier multiplier method

compact brownian surfaces i brownian disks we show that under certain natural assumptions large random plane bipartite maps with a boundary converge after rescaling to a one parameter family mathrm bd l 0 l infty of random metric spaces homeomorphic to the closed unit disk of mathbb r 2 the space mathrm bd l being called the brownian disk of perimeter l and unit area these results can be seen as an extension of the convergence of uniform plane quadrangulations to the brownian map which intuitively corresponds to the limit case where l 0 similar results are obtained for maps following a boltzmann distribution in which the perimeter is fixed but the area is random

self similar fragmentations derived from the stable tree i the basic object we consider is a certain model of continuum random tree called the stable tree we construct a fragmentation process f t t 0 out of this tree by removing the vertices located under height t thanks to a self similarity property of the stable tree we show that the fragmentation process is also self similar the semigroup and other features of the fragmentation are given explicitly asymptotic results are given as well as a couple of related results on continuous state branching processes

corner percolation on ℤ2 and the square root of 17 we consider a four vertex model introduced by balint toth a dependent bond percolation model on 2 in which every edge is present with probability 1 2 and each vertex has exactly two incident edges perpendicular to each other we prove that all components are finite cycles almost surely but the expected diameter of the cycle containing the origin is infinite moreover we derive the following critical exponents the tail probability diameter of the cycle of the origin n n and the expectation mathbb e length of a typical cycle with diameter n n with gamma 5 sqrt 17 4 0 219 ldots and delta sqrt 17 1 4 1 28 ldots the value of comes from a singular sixth order ode while the relation 3 2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive brownian motion whose level sets have hausdorff dimension 3 2 we also include many open problems for example on the conformal invariance of certain linear entropy models

the augmented multiplicative coalescent bounded size rules and critical dynamics of random graphs random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component one of the standard models are the achlioptas random graph processes on a fixed set of n vertices here at each step one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion an important class of such rules are the bounded size rules where for a fixed k ge 1 all components of size greater than k are treated equally while a great deal of work has gone into analyzing the subcritical and supercritical regimes the nature of the critical scaling window the size and complexity deviation from trees of the components in the critical regime and nature of the merging dynamics has not been well understood in this work we study such questions for general bounded size rules our first main contribution is the construction of an extension of aldous s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components we show that this process referred to as the standard augmented multiplicative coalescent amc is nearly feller with a suitable topology on the state space our second main result proves the convergence of suitably scaled component size and surplus vector for any bounded size rule to the standard amc this result is new even for the classical erd s renyi setting the key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from bhamidi et al the barely subcritical regime arxiv preprint 2012 on the size of the largest component in the barely subcritical regime

on the distribution of betweenness centrality in random trees abstract betweenness centrality is a quantity that is frequently used to measure how central a vertex v is it is defined as the sum over pairs of vertices other than v of the proportions of shortest paths that pass through v in this paper we study the distribution of the betweenness centrality in random trees and related subcritical graph families specifically we prove that the betweenness centrality of the root vertex in a simply generated tree is usually of linear order but has a mean of order n 3 2 we also show that a randomly chosen vertex typically also has linear order betweenness centrality and that the maximum betweenness centrality in a simply generated tree is of order n 2 we obtain limiting distributions for the betweenness centrality of the root vertex and of a randomly chosen vertex as well as for the maximum betweenness centrality and we also show that the centroid has positive probability in the limit to be the vertex of maximum betweenness centrality some similar results also hold for subcritical graph classes which will be briefly discussed finally we study random recursive trees and other families of increasing trees where the situation is quite different here the root betweenness centrality is of quadratic order as is the maximum betweenness centrality the betweenness centrality of a random vertex on the other hand is again of linear order again we also have limiting distributions upon suitable normalisation

the standard additive coalescent regard an element of the set x 1 x 2 x 1 x 2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent of evans and pitman is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j they showed that a version x t t of this process arises as a n weak limit of the process started at time 1 2 log n with n clusters of mass 1 n we show this standard additive coalescent may be constructed from the continuum random tree of aldous by poisson splitting along the skeleton of the tree we describe the distribution of x t on at a fixed time t we show that the size of the cluster containing a given atom as a process in t has a simple representation in terms of the stable subordinator of index 1 2 as t we establish a gaussian limit for centered and normalized cluster sizes and study the size of the largest cluster

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

recursive self similarity for random trees random triangulations and brownian excursion recursive self similarity for a random object is the property of being decomposable into independent rescaled copies of the original object certain random combinatorial objects trees and triangulations possess approximate versions of recursive self similarity and then their continuous limits possess exact recursive self similarity in particular since the limit continuum random tree can be identified with brownian excursion we get a nonobvious recursive self similarity property for brownian excursion

compactness in kinetic transport equations and hypoellipticity abstract we establish improved hypoelliptic estimates on the solutions of kinetic transport equations using a suitable decomposition of the phase space our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions f x v l 1 satisfying some appropriate transport relation v x f 1 x 2 1 v 2 g may be inferred solely from additional integrability and compactness with respect to v in a forthcoming work the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the boltzmann equation with a rough force field arsenio and saint raymond in preparation 4

the packing measure of the range of super brownian motion we prove that the total range of super brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function g r r 4 log log1 r 3 in super critical dimensions d geq 5 more precisely we prove that the total occupation measure of super brownian motion is equal to the g packing measure restricted to its range up to a deterministic multiplicative constant that only depends on space dimension d

subsequential scaling limits of simple random walk on the two dimensional uniform spanning tree the first main result of this paper is that the law of the rescaled two dimensional uniform spanning tree is tight in a space whose elements are measured rooted real trees continuously embedded into euclidean space various properties of the intrinsic metrics measures and embeddings of the subsequential limits in this space are obtained with it being proved in particular that the hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8 5 in addition the tightness result is applied to deduce that the annealed law of the simple random walk on the two dimensional uniform spanning tree is tight under a suitable rescaling for the limiting processes which are diffusions on random real trees embedded into euclidean space detailed transition density estimates are derived

inhomogeneous continuum random trees and the entrance boundary of the additive coalescent regard an element of the set of ranked discrete distributions x1 x2 x1 x2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j aldous and pitman 1998 showed that a version of this process starting from time with infinitesimally small clusters can be constructed from the brownian continuum random tree of aldous 1991 1993 by poisson splitting along the skeleton of the tree in this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees

oscillation of two dimensional difference systems abstract the authors consider the nonlinear two dimensional difference system x n b n g y n y n 1 a n f x n n n o no no 1 where n 0 n 1 2 a n and b n n n n 0 are real sequences and f g r r are continuous with uf u 0 and ug u 0 for u 0 a solution x n y n of the system is oscillatory if both components are oscillatory the authors obtain sufficient conditions for all solutions of the system to be oscillatory some of their results allow a n to oscillate examples to illustrate the results are included

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

the depth first processes of galton watson trees converge to the same brownian excursion in this paper we show a strong relation between the depth first processes associated to galton watson trees with finite variance conditioned by the total progeny the depth first walk the depth first queue process the height process a consequence is that these processes suitably normalized converge to the same brownian excursion this provides an alternative proof of aldous one of the convergence of the depth first walk to the brownian excursion which does not use the existence of a limit tree the methods that we introduce allow one to compute some functionals of trees or discrete excursions for example we compute the limit law of the process of the height of nodes with a given out degree and the process of the height of nodes root of a given subtree

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

a limit theorem for the contour process of condidtioned galton watson trees in this work we study asymptotics of the genealogy of galton watson processes conditioned on the total progeny we consider a fixed aperiodic and critical offspring distribution such that the rescaled galton watson processes converges to a continuous state branching process csbp with a stable branching mechanism of index 1 2 we code the genealogy by two different processes the contour process and the height process that le gall and le jan recently introduced we show that the rescaled height process of the corresponding galton watson family tree with one ancestor and conditioned on the total progeny converges in a functional sense to a new process the normalized excursion of the continuous height process associated with the stable csbp we deduce from this convergence an analogous limit theorem for the contour process in the brownian case 2 the limiting process is the normalized brownian excursion that codes the continuum random tree the result is due to aldous who used a different method

random stable laminations of the disk we study large random dissections of polygons we consider random dissections of a regular polygon with n sides which are chosen according to boltzmann weights in the domain of attraction of a stable law of index 2 1 2 as n goes to infinity we prove that these random dissections converge in distribution towards a random compact set called the random stable lamination if 2 we recover aldous brownian triangulation however if 2 1 2 large faces remain in the limit and a dierent random compact set appears we show that the random stable lamination can be coded by the continuous time height function associated to the normalized excursion of a strictly stable spectrally positive levy process of index using this coding we establish that the hausdor dimension of the stable random lamination is almost surely 2 1

local convergence for permutations and local limits for uniform rho avoiding permutations with rho 3 we set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of consecutive pattern occurrences we also characterize random limiting objects for this new topology introducing a notion of shift invariant property corresponding to the notion of unimodularity for random graphs we then study two models in the framework of random pattern avoiding permutations we compute the local limits of uniform rho avoiding permutations for rho 3 when the size of the permutations tends to infinity the core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern for this result we use bijections between rho avoiding permutations and rooted ordered trees local limit results for galton watson trees the second moment method and singularity analysis

statistical tests for large tree structured data we develop a general statistical framework for the analysis and inference of large tree structured data with a focus on developing asymptotic goodness of fit tests we first propose a consistent statistical model for binary trees from which we develop a class of invariant tests using the model for binary trees we then construct tests for general trees by using the distributional properties of the continuum random tree which arises as the invariant limit for a broad class of models for tree structured data based on conditioned galton watson processes the test statistics for the goodness of fit tests are simple to compute and are asymptotically distributed as 2 and f random variables we illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer we use a novel approach with tree based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients

sle as a mating of trees in euclidean geometry the mating of trees approach to schramm loewner evolution sle in the random geometry of liouville quantum gravity lqg has been recently developed by duplantier et al liouville quantum gravity as a mating of trees 2014 arxiv 1409 7055 in this paper we consider the mating of trees approach to sle in euclidean geometry let eta be a whole plane space filling sle with parameter kappa 4 parameterized by lebesgue measure the main observable in the mating of trees approach is the contour function a two dimensional continuous process describing the evolution of the minkowski content of the left and right frontier of eta we prove regularity properties of the contour function and show that as in the lqg case it encodes all the information about the curve eta we also prove that the uniform spanning tree on mathbb z 2 converges to sle8 in the natural topology associated with the mating of trees approach

stochastic analysis of tree like data structures the purpose of this article is to present two types of data structures binary search trees and usual combinatorial binary trees although they constitute the same set of rooted trees they are constructed via completely different rules and thus the underlying probabilistic models are different too both kinds of data structure can be analysed by probabilistic and stochastic tools binary search trees more or less with martingales and binary trees which can be considered as a special case of galton watson trees with stochastic processes it is also an aim of this article to demonstrate the strength of analytical methods in specific parts of probability theory related to combinatorial problems and we especially make use of the concept of generating functions one reason for the use of generating functions is that recursive combinatorial descriptions can be translated to relations for generating functions and another is that analytical properties of these generating functions can be used to derive asymptotic probabilistic relations

kamenev type and interval oscillation criteria for second order linear differential equations on a measure chain abstract we establish kamenev type criteria and interval criteria for oscillation of the second order scalar differential equation p t x t q t x t 0 on a measure chain our results cover those for differential equations and provide new oscillation criteria for difference equations several examples are given to show the significance of the results

continuum sites stepping stone models coalescing exchangeable partitions and random trees analogues of stepping stone models are considered where the sitespace is continuous the migration process is a general markov process and the type space is infinite such processes were defined in previous work of the second author by specifying a feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing markov processes an alternative representation is obtained here in terms of a limit of interacting particle systems it is shown that under a mild condition on the migration process the continuum sites stepping stone process has continuous sample paths the case when the migration process is brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating brownian motion this duality relation is also used to show that a tree like random compact metric space that is naturally associated to an in nite family of coalescing brownian motions on the circle has hausdorff and packing dimension both almost surely equal to 1 2 and moreover this space is capacity equivalent to the middle 1 2 cantor set and hence also to the brownian zero set

oscillation and asymptotic behavior of two dimensional difference systems this paper is concerned with the oscillation behavior of solutions of the nonlinear two dimesional difference system dx n b ng y n dy n 1 a nf x n n n n 0 n 0 n 0 1 some necessary and sufficient conditions are given for the oscillation of all solution of the system our result improve and generalize some results in the literature examples to illustrate the result are included

oscillation criteria for second order nonlinear delay dynamic equations abstract in this paper we consider the second order nonlinear delay dynamic equation r t x t p t f x t 0 on a time scale t by employing the generalized riccati technique we will establish some new sufficient conditions which ensure that every solution oscillates or converges to zero the obtained results improve the well known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations some applications and examples are considered to illustrate the main results

geodesics in large planar maps and in the brownian map we study geodesics in the random metric space called the brownian map which appears as the scaling limit of large planar maps in particular we completely describe geodesics starting from the distinguished point called the root and we characterize the set s of all points that are connected to the root by more than one geodesic the set s is dense in the brownian map and homeomorphic to a non compact real tree furthermore for every x in s the number of distinct geodesics from x to the root is equal to the number of connected components of s x in particular points of the brownian map can be connected to the root by at most three distinct geodesics our results have applications to the behavior of geodesics in large planar maps

scaling limit for the ant in a simple high dimensional labyrinth we prove that after suitable rescaling the simple random walk on the trace of a large critical branching random walk in mathbb z d converges to the brownian motion on the integrated super brownian excursion when d 14

brownian motion on the continuum tree we construct brownian motion on a continuum tree a structure introduced as an asymptotic limit to certain families of finite trees we approximate the dirichlet form of brownian motion on the continuum tree by adjoining one dimensional brownian excursions we study the local times of the resulting diffusion using time change methods we find explicit expressions for certain hitting probabilities and the mean occupation density of the process