thick points for planar brownian motion and the erdős taylor conjecture on random walk and conjectured that the limit exists and equals 1 tr a s the importance of determining the value of this limit is clarified in 1 3 below where this value appears in the power laws governing the local time of the walk the erd6s taylor conjecture was quoted in the book by r v sz 19 w but to the best of our knowledge the bounds in 1 1 were not improved prior to the present paper as it turns out an important step towards our solution of the erd6s taylor conjecture was the formulation by perkins and taylor 17 of an analogous problem on the maximal occupation measure that planar brownian motion run for unit time can

eigenvalue problems with fully discontinuous operators and critical exponents abstract we construct here more examples of multivoque leray lions operators that are strongly monotonic and we introduce a new abstract theorem on critical points for multivalued operators available for unbounded domains

existence and multiplicity of solutions for a discontinuous problems with critical sobolev exponents abstract in this paper we consider the equation p u u p 2 u f x u in r n with discontinuous nonlinearity where 1 p n 0 is a real parameter and p n p n p is the critical sobolev exponent under proper conditions on f applying the nonsmooth critical point theory for locally lipschitz functionals we obtain at least one nontrivial nonnegative solution provided that 0 and for any k n it has k pairs of nontrivial solutions if k where 0 and k are positive numbers in particular we obtain the existence results for f is discontinuous in just one point

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

infinite volume limit of the abelian sandpile model in dimensions d 3 we study the abelian sandpile model on mathbb z d in d 3 we prove existence of the infinite volume addition operator almost surely with respect to the infinite volume limit of the uniform measures on recurrent configurations we prove the existence of a markov process with stationary measure and study ergodic properties of this process the main techniques we use are a connection between the statistics of waves and uniform two component spanning trees and results on the uniform spanning forest measure on mathbb z d

the infinite volume limit of dissipative abelian sandpiles we construct the thermodynamic limit of the stationary measures of the bak tang wiesenfeld sandpile model with a dissipative toppling matrix sand grains may disappear at each toppling we prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic markov process leaving it invariant we show how to extend the dhar formalism of the abelian group of toppling operators to infinite volume in order to obtain a compact abelian group with a unique haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches

non criticality criteria for abelian sandpile models with sources and sinks we prove that the abelian sandpile model on a random binary and binomial tree as introduced in redig ruszel and saada j stat phys 147 653 677 2012 is not critical for all branching probabilities p 1 by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps we obtain some more information about the exponential tail of the avalanche radius next we study the sandpile model on zd with some additional dissipative sites we provide examples and sufficient conditions for non criticality we also make a connection with the parabolic anderson model finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non criticality in the presence of a finite number of sources using a connection with the homogeneous pinning model

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

cover times for brownian motion and random walks in two dimensions lett x denote the rst hitting time of the disc of radius centered at x for brownian motion on the two dimensional torus t 2 we prove that sup x2t2t x j log j 2 2 as 0 the same applies to brownian motion on any smooth compact connected two dimensional riemannian manifold with unit area and no boundary as a consequence we prove a conjecture due to aldous 1989 that the number of steps it takes a simple random walk to cover all points of the lattice torus z 2 is asymptotic to 4n 2 logn 2 determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete so far this structure was only studied non rigorously in the physics literature we also establish a conjecture due to kesten and r ev esz that describes the asymptotics for the number of steps needed by simple random walk in z 2 to cover the disc of radius n

thick points of the gaussian free field let u c be a bounded domain with smooth boundary and let f be an instance of the continuum gaussian free field on u with respect to the dirichlet inner product u f x g x dx the set t a u of a thick points of f consists of those z u such that the average of f on a disk of radius r centered at z has growth a log 1 r as r 0 we show that for each 0 a 2 the hausdorff dimension of t a u is almost surely 2 a that 2 a t a u when 0 2 furthermore we prove that t a u is invariant under conformal transformations in an appropriate sense the notion of a thick point is connected to the liouville quantum gravity measure with parameter y given formally by dz e 2 f z dz considered by duplantier and sheffield

multiple solutions for a class of elliptic hemivariational inequalities the existence of multiple solutions to elliptic hemivariational inequality problems in bounded domains is investigated via a suitable nonsmooth version of a classical technique due to struwe and a recent saddle point theorem for locally lipschitz continuous functionals

some results for long range exclusion processes we examine the long range exclusion process introduced by spitzer and studied by liggett and answer some of the open questions raised by liggett in particular we show the existence of equilibria corresponding to bounded dual harmonic functions and that the process can have right discontinuous paths at strictly positive times we also show that explosions when they occur do so at fixed times determined by the initial configuration finally we give an example for which the configuration with all sites occupied is not stable although the rate at which particles arrive at any given site for that configuration is infinite

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

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

how large a disc is covered by a random walk in n steps we show that the largest disc covered by a simple random walk srw on z 2 after n steps has radius n 1 4 o 1 thus resolving an open problem of revesz random walk in random and non random environments 1990 world scientific teaneck nj for any fixed t the largest disc completely covered at least l times by the srw also has radius n 1 4 o 1 however the largest disc completely covered by each of l independent simple random walks on z 2 after n steps is only of radius n 1 2 2 l o 1 we complement this by showing that the radius of the largest disc completely covered at least a fixed fraction a of the maximum number of visits to any site during the first n steps of the srw on z 2 is n 1 4 o 1 we also show that almost surely for infinitely many values of n it takes about n 1 2 o 1 steps after step n for the srw to reach the first previously unvisited site and the exponent 1 2 is sharp this resolves a problem raised by revesz ann probab 21 1993 318 328

the escape rate of favorite sites of simple random walk and brownian motion consider a simple symmetric random walk on the integer lattice zb for each n let v n denote a favorite site or most visited site of the random walk in the first n steps a somewhat surprising theorem of bass and griffin z wahrsch verw gebiete 70 1985 417 436 says that v is almost surely transient thus disproving a previous conjecture of erd s and revesz mathematical structures computational mathematics mathematical modeling 2 1984 152 157 more precisely bass and griffin proved that almost surely lim infn v n n1 2 logn equals 0 if 11 eleven the present paper studies the rate of escape of v n we show that almost surely the lim inf expression in question is 0 if 1 and is infinity otherwise the corresponding problem for brownian motion is also studied

the abelian sandpile model on an infinite tree we consider the standard abelian sandpile process on the bethe lattice we show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume markov process exhibiting features of self organized criticality

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

uniform measure results for the image of subsets under brownian motion the paper obtains bounds on the hausdorff and packing measures of the imagex e of a borel sete by a transient strictly stable processx t which a s hold for alle and for every measure function h beta gamma s s beta left log s right gamma star in some cases examples are constructed to show that the bounds are sharp

on a class of semilinear schrodinger equations involving critical growth and discontinuous nonlinearities abstract in this paper we deal with a class of schrodinger equation in r n involving critical sobolev exponent and jump discontinuities the basic tool employed here is an approximation technique with periodic functions and variational arguments based on a linking theorem for locally lipschitz functionals

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

approaching criticality via the zero dissipation limit in the abelian avalanche model the discrete height abelian sandpile model was introduced by bak tang wiesenfeld and dhar as an example for the concept of self organized criticality when the model is modified to allow grains to disappear on each toppling it is called bulk dissipative we provide a detailed study of a continuous height version of the abelian sandpile model called the abelian avalanche model which allows an arbitrarily small amount of dissipation to take place on every toppling we prove that for non zero dissipation the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure we show that in the whole non zero dissipation regime the model is not critical i e spatial covariances of local observables decay exponentially we then study the zero dissipation limit and prove that the self organized critical model is recovered both for the stationary measure and for the dynamics we obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality we rigorously establish the mean field value of this exponent for d 4

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

a new approach to strong embeddings we revisit strong approximation theory from a new perspective culminating in a proof of the komlos major tusnady embedding theorem for the simple random walk the proof is almost entirely based on a series of soft arguments and easy inequalities the new technique inspired by stein s method of normal approximation is applicable to any setting where stein s method works in particular one can hope to take it beyond sums of independent random variables

on the maximum satisfiability of random formulas say that a k cnf a formula is p satisfiable if there exists a truth assignment satisfying a fraction 1 2 k pp 2 k of its clauses note that every k cnf formula is 0 satisfiable let fk n m denote a random k cnf formula on n variables with m clauses for every k 2 and every r 0 we determine p and e k eo k2 k 2 such that with probability tending to 1 as n a random k cnf formula fk n rn is p satisfiable but not pp satisfiable

hausdorff dimension of the contours of symmetric additive levy processes let x 1 x n denote n independent symmetric levy processes on r d the corresponding additive levy process is defined as the following n parameter random field on r d mathfrak x t x 1 t 1 cdots x n t n quad t in bf r n khoshnevisan and xiao ann probab 30 1 62 100 2002 have found a necessary and sufficient condition for the zero set mathfrak x 1 0 of mathfrak x to be non trivial with positive probability they also provide bounds for the hausdorff dimension of mathfrak x 1 0 which hold with positive probability in the case that mathfrak x 1 0 can be non void here we prove that the hausdorff dimension of mathfrak x 1 0 is a constant almost surely on the event mathfrak x 1 0 neq varnothing moreover we derive a formula for the said constant this portion of our work extends the well known formulas of horowitz israel j math 6 176 182 1968 and hawkes j lond math soc 8 517 525 1974 both of which hold for one parameter levy processes more generally we prove that for every nonrandom borel set f in 0 n the hausdorff dimension of mathfrak x 1 0 cap f is a constant almost surely on the event mathfrak x 1 0 cap f neq varnothing this constant is computed explicitly in many cases

brownian intersection local times upper tail asymptotics and thick points we equip the intersection of p independent brownian paths in r d d 2 with the natural measure l defined by projecting the intersection local time measure via one of the brownian motions onto the set of intersection points given a bounded domain u c r d we show that as a the probability of the event l u a decays with an exponential rate of a 1 p where is described in terms of a variational problem in the important special case when u is the unit ball in r 3 and p 2 we characterize in terms of an ordinary differential equation we apply our results to the problem of finding the hausdorff dimension spectrum for the thick points of the intersection of two independent brownian paths in r 3

mean field avalanche size exponent for sandpiles on galton watson trees we show that in abelian sandpiles on infinite galton watson trees the probability that the total avalanche has more than t topplings decays as t 1 2 we prove both quenched and annealed bounds under suitable moment conditions our proofs are based on an analysis of the conductance martingale of morris probab theory relat fields 125 259 265 2003 that was previously used by lyons et al electron j probab 13 58 1702 1725 2008 to study uniform spanning forests on mathbb z d d ge 3 and other transient graphs

the dual variational principle and elliptic problems with discontinuous nonlinearities on etudie des problemes aux valeurs limites elliptiques du type u f u p x x u 0 x ou r n est un domaine borne et f peut avoir des discontinuites on cherche des solutions en utilisant le principe d action duale de clarke

asymmetric abeiian sandpile models in the abelian sandpile models introduced by dhar long time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system dhar proposed a simple procedure theburning algorithm as a possible test of whether a configuration is recurrent and later with majumdar verified the correctness of this test when the toppling rules of the sandpile are symmetric we observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles we also obtain necessary and sufficient conditions for the validity of the original test the results are applied to a family of deterministic one dimensional sandpile models originally studied by lee liang and tzeng

the most visited site of brownian motion and simple random walk let l t x be the local time at x for brownian motion and for each t let bar v t inf x underline underline 0 l t x vee l t x mathop sup limits y l t y the absolute value of the most visited site for brownian motion up to time t in this paper we prove that v t is transient and obtain upper and lower bounds for the rate of growth of v t the main tools used are the ray knight theorems and william s path decomposition of a diffusion an invariance principle is used to get analogous results for simple random walks we also obtain a law of the iterated logarithm for v t

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

thick points for intersections of planar sample paths let l x n x denote the number of visits to x z 2 of the simple planar random walk x up to step n let x be another simple planar random walk independent of x we show that for any 0 b 1 2 there are n 1 2 b o 1 points x e z 2 for which l x n x l x n x b 2 logn 4 this is the discrete counterpart of our main result that for any a 1 the hausdorff dimension of the set of thick intersection points x for which limsup r 0 i x r r 2 log r 4 a 2 is almost surely 2 2a here i x r is the projected intersection local time measure of the disc of radius r centered at x for two independent planar brownian motions run until time 1 the proofs rely on a multi scale refinement of the second moment method in addition we also consider analogous problems where we replace one of the brownian motions by a transient stable process or replace the disc of radius r centered at x by x rk for general sets k

the abelian sandpile model on a random binary tree we study the abelian sandpile model on a random binary tree using a transfer matrix approach introduced by dhar and majumdar we prove exponential decay of correlations and in a small supercritical region i e where the branching process survives with positive probability exponential decay of avalanche sizes this shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes our main technical tools are 1 a recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well defined stochastic solution 2 quenched and annealed estimates of the eigenvalues of a product of n random transfer matrices

approach to criticality in sandpiles a popular theory of self organized criticality predicts that the stationary density of the abelian sandpile model equals the threshold density of the corresponding fixed energy sandpile we recently announced that this density conjecture is false when the underlying graph is any of z2 the complete graph kn the cayley tree the ladder graph the bracelet graph or the flower graph in this paper we substantiate this claim by rigorous proof and extensive simulations we show that driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink nevertheless we do find and prove a relationship between the two models the threshold density of the fixed energy sandpile is the point at which the driven dissipative sandpile begins to lose a macroscopic amount of sand to the sink

abelian sandpile model on the bethe lattice the authors study bak tang and wiesenfeld s abelian sandpile model 1987 of self organised criticality on the bethe lattice exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained they also determine the probability distribution of the number of distinct sites that topple at least once the number of toplings at the origin and the total number of toplings in an avalanche the probability that an avalanche consists of more than n toplings varies as n 1 2 for large n the probability that its duration exceeds t decreases as 1 t for large t these exponents are the same as for the critical percolation clusters in mean field theory

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

non differentiable functionals and applications to elliptic problems with discontinuous nonlinearities abstract multiple critical points theorems for non differentiable functionals are established applications both to elliptic variational hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

thick points of super brownian motion we determine for a super brownian motion xt t 0 in d d 3 the precise gauge function such that almost surely on survival up to time t open image in new window improving a result of barlow evans and perkins about the most visited sites of super brownian motion we also determine upper and lower bounds for the hausdorff dimension spectrum of thick points refining the multifractal analysis of super brownian motion by taylor and perkins the upper bound conjectured to be sharp involves a constant which can be characterized in terms of the upper tails of the associated equilibrium palm distribution