scaling limit of the loop erased random walk green s function we consider loop erased random walk lerw running between two boundary points of a square grid approximation of a planar simply connected domain the lerw green s function is the probability that the lerw passes through a given edge in the domain we prove that this probability multiplied by the inverse mesh size to the power 3 4 converges in the lattice size scaling limit to a constant times an explicit conformally covariant quantity which coincides with the hbox sle 2 green s function the proof does not use sle techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a spinor observable for random walk started from one of the vertices of the marked edge

universality of high dimensional spanning forests and sandpiles we prove that the wired uniform spanning forest exhibits mean field behaviour on a very large class of graphs including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph several of our results are new even in the case of mathbb z d d ge 5 in particular we prove that every tree in the forest has spectral dimension 4 3 and walk dimension 3 almost surely and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1 2 respectively the past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest we obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth and is 1 on any bounded degree nonamenable graph we deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the abelian sandpile model are 2 and 1 2 respectively for any transitive graph with polynomial growth of dimension at least five and are 1 and 1 2 respectively for any bounded degree nonamenable graph in the case of mathbb z d d ge 5 some of our results regarding critical exponents recover earlier results of bhupatiraju et al electron j probab 22 85 51 2017 in this case we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant order multiplicative errors rather than the polylogarithmic order multiplicative errors present in that work

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

eigenvalue intervals for a two point boundary value problem on a measure chain we study the existence of eigenvalue intervals for the second order differential equation on a measure chain x t p t f x t 0 t t1 t2 satisfying the boundary conditions x t1 x t1 0 and x t2 x t2 0 where f is a positive function and p a nonnegative function that is allowed to vanish on some subintervals of t1 t2 of the measure chain the methods involve applications of a fixed point theorem for operators on a cone in a banach space

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

existence of three positive solutions for some second order boundary value problems in this paper a new fixed point theorem of functional type in a cone is established with using the new fixed point theorem and imposing growth conditions on the nonlinearity the existence of three positive solutions for the boundary value problemx t f t x t x t 0 0 0 is continuous finally an example is given to illustrate the importance of results obtained

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 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

spectral dimension and random walks on the two dimensional uniform spanning tree we study the simple random walk on the uniform spanning tree on mathbb z 2 we obtain estimates for the transition probabilities of the random walk the distance of the walk from its starting point after n steps and exit times of both euclidean balls and balls in the intrinsic graph metric in particular we prove that the spectral dimension of the uniform spanning tree on mathbb z 2 is 16 13 almost surely

subgradient methods for huge scale optimization problems we consider a new class of huge scale problems the problems with sparse subgradients the most important functions of this type are piece wise linear for optimization problems with uniform sparsity of corresponding linear operators we suggest a very efficient implementation of subgradient iterations which total cost depends logarithmically in the dimension this technique is based on a recursive update of the results of matrix vector products and the values of symmetric functions it works well for example for matrices with few nonzero diagonals and for max type functions we show that the updating technique can be efficiently coupled with the simplest subgradient methods the unconstrained minimization method by b polyak and the constrained minimization scheme by n shor similar results can be obtained for a new nonsmooth random variant of a coordinate descent scheme we present also the promising results of preliminary computational experiments

an adaptive three term conjugate gradient method based on self scaling memoryless bfgs matrix abstract due to its simplicity and low memory requirement conjugate gradient methods are widely used for solving large scale unconstrained optimization problems in this paper we propose a three term conjugate gradient method the search direction is given by a symmetrical perry matrix which contains a positive parameter the value of this parameter is determined by minimizing the distance of this matrix and the self scaling memoryless bfgs matrix in the frobenius norm the sufficient descent property of the generated directions holds independent of line searches the global convergence of the given method is established under wolfe line search for general non convex functions numerical experiments show that the proposed method is promising

existence of solutions for a two point boundary value problem on time scales let t be a time scale let f 0 s 1 xr 2 r be continuous we study the existence of solutions for the nonlinear two point boundary value problems on time scalex t f t x t x t email protected 0 1 x 0 0 x s 1 0 we do not need any growth restrictions on f

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

multiplicity results for some second order four point boundary value problems abstract in this paper by using a fixed point theorem of functional type in a cone some new results for the multiplicity of positive solutions of second order four point boundary value problems are obtained the emphasis is put on the nonlinear term involved with the first order derivative

a distance exponent for liouville quantum gravity let gamma in 0 2 and let h be the random distribution on mathbb c which describes a gamma liouville quantum gravity lqg cone also let kappa 16 gamma 2 4 and let eta be a whole plane space filling sle kappa curve sampled independent from h and parametrized by gamma quantum mass with respect to h we study a family mathcal g epsilon epsilon 0 of planar maps associated with h eta called the lqg structure graphs a k a mated crt maps which we conjecture converge in probability in the scaling limit with respect to the gromov hausdorff topology to a random metric space associated with gamma lqg in particular mathcal g epsilon is the graph whose vertex set is epsilon mathbb z with two such vertices x 1 x 2 in epsilon mathbb z connected by an edge if and only if the corresponding curve segments eta x 1 epsilon x 1 and eta x 2 epsilon x 2 share a non trivial boundary arc due to the peanosphere description of sle decorated lqg due to duplantier et al liouville quantum gravity as a mating of trees 2014 the graph mathcal g epsilon can equivalently be expressed as an explicit functional of a correlated two dimensional brownian motion so can be studied without any reference to sle or lqg we prove non trivial upper and lower bounds for the cardinality of a graph distance ball of radius n in mathcal g epsilon which are consistent with the prediction of watabiki prog theor phys suppl 114 1 17 1993 for the hausdorff dimension of lqg using subadditivity arguments we also prove that there is an exponent chi 0 for which the expected graph distance between generic points in the subgraph of mathcal g epsilon corresponding to the segment eta 0 1 is of order epsilon chi o epsilon 1 and this distance is extremely unlikely to be larger than epsilon chi o epsilon 1

exponential tail bounds for loop erased random walk in two dimensions let m n be the number of steps of the loop erasure of a simple random walk on 2 from the origin to the circle of radius n we relate the moments of m n to es n the probability that a random walk and an independent loop erased random walk both started at the origin do not intersect up to leaving the ball of radius n this allows us to show that there exists c such that for all n and all k 1 2 e m k n and hence to establish exponential moment bounds for m n this implies that there exists c 0 such that for all n and all 0 p m n e m n 2e c using similar techniques we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any 0 such that for all n and 0 p m n 1 e m n ce c

existence of three positive solutions to a second order boundary value problem on a measure chain growth conditions are imposed on f such that the boundary value problem x t f x t t t1 t2 x t1 x t1 0 and x t2 x t2 0 where t1 t2 from a measure chain t has at least three positive solutions by way of the five functionals fixed point theorem

on the convergence of a distributed augmented lagrangian method for nonconvex optimization in this paper we propose a distributed algorithm for optimization problems that involve a separable possibly nonconvex objective function subject to convex local constraints and linear coupling constraints the method is based on the accelerated distributed augmented lagrangians adal algorithm that was recently developed by the authors to address convex problems here we extend this line of work in two ways first we establish convergence of the method to a local minimum of the problem using assumptions that are common in the analysis of nonconvex optimization methods to the best of our knowledge this is the first work that shows convergence to local minima specifically for a distributed augmented lagrangian al method applied to nonconvex optimization problems distributed al methods are known to perform very well when used to solve convex problems second we propose a more general and decentralized rule to select the stepsizes of the method this improves on the authors original adal method where the stepsize selection used global information at initialization numerical results are included to verify the correctness and efficiency of the proposed distributed 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

first order algorithm with mathcal o rm ln 1 epsilon convergence for epsilon equilibrium in two person zero sum games we propose an iterated version of nesterov s first order smoothing method for the two person zero sum game equilibrium problem min x in q 1 max y in q 2 x rm t ay max y in q 2 min x in q 1 x rm t ay this formulation applies to matrix games as well as sequential games our new algorithmic scheme computes an epsilon equilibrium to this min max problem in mathcal o left frac a delta a rm ln 1 epsilon right first order iterations where a is a certain condition measure of the matrix a this improves upon the previous first order methods which required mathcal o 1 epsilon iterations and it matches the iteration complexity bound of interior point methods in terms of the algorithm s dependence on epsilon unlike interior point methods that are inapplicable to large games due to their memory requirements our algorithm retains the small memory requirements of prior first order methods our scheme supplements nesterov s method with an outer loop that lowers the target epsilon between iterations this target affects the amount of smoothing in the inner loop computational experiments both in matrix games and sequential games show that a significant speed improvement is obtained in practice as well and the relative speed improvement increases with the desired accuracy as suggested by the complexity bounds

primal dual subgradient methods for convex problems in this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure our methods are primal dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem besides other advantages this useful feature provides the methods with a reliable stopping criterion the proposed schemes differ from the classical approaches divergent series methods mirror descent methods by presence of two control sequences the first sequence is responsible for aggregating the support functions in the dual space and the second one establishes a dynamically updated scale between the primal and dual spaces this additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set however we always assume the uniform boundedness of subgradients we present the variants of subgradient schemes for nonsmooth convex minimization minimax problems saddle point problems variational inequalities and stochastic optimization in all situations our methods are proved to be optimal from the view point of worst case black box lower complexity bounds

upper and lower solution method for a fourth order four point boundary value problem on time scales in this work we consider a fourth order four point boundary value problem on time scales we establish criteria for the existence of a solution by developing the upper and lower solution method and the monotone iterative technique

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

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

a distributed algorithm for convex constrained optimization under noise we present a novel distributed algorithm for convex constrained optimization problems that are subject to noise corruption and uncertainties the proposed scheme can be classified as a distributed stochastic approximation method where a unique feature here is that we allow for multiple noise terms to appear in both the computation and communication stages of the distributed iterative process specifically we consider problems that involve multiple agents optimizing a separable convex objective function subject to convex local constraints and linear coupling constraints this is a richer class of problems compared to those that can be handled by existing distributed stochastic approximation methods which consider only consensus constraints and fewer sources of noise the proposed algorithm utilizes the augmented lagrangian al framework which has been widely used recently to solve deterministic optimization problems in a distributed way we show that the proposed method generates sequences of primal and dual variables that converge to their respective optimal sets almost surely

twin solutions of boundary value problems for differential equations on measure chains a new twin fixed point theorem is applied to obtain the existence of at least two positive solutions for the right focal boundary value problem on a measure chain t y t f y t 0 t 0 1 t y 0 y 1 0

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

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

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

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

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

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

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

triple positive solutions for some second order boundary value problem on a measure chain in this paper we study the existence of three positive solutions for the second order two point boundary value problem on a measure chain x d d t p t f t x s t x d t 0 email protected t 1 t 2 a 1x t 1 a 2x d t 1 0 a 3x s t 2 a 4x d s t 2 0 where f t 1 s t 2 x 0 xr 0 is continuous and p t 1 s t 2 0 a nonnegative function that is allowed to vanish on some subintervals of t 1 s t 2 of the measure chain the method involves applications of a new fixed point theorem due to bai and ge z b bai w g ge existence of three positive solutions for some second order boundary value problems comput math appl 48 2004 699 707 the emphasis is put on the nonlinear term f involved with the first order delta derivative x d t

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