rumors in a network who s the culprit we provide a systematic study of the problem of finding the source of a rumor in a network we model rumor spreading in a network with the popular susceptible infected si model and then construct an estimator for the rumor source this estimator is based upon a novel topological quantity which we term rumor centrality we establish that this is a maximum likelihood ml estimator for a class of graphs we find the following surprising threshold phenomenon on trees which grow faster than a line the estimator always has nontrivial detection probability whereas on trees that grow like a line the detection probability will go to 0 as the network grows simulations performed on synthetic networks such as the popular small world and scale free networks and on real networks such as an internet as network and the u s electric power grid network show that the estimator either finds the source exactly or within a few hops of the true source across different network topologies we compare rumor centrality to another common network centrality notion known as distance centrality we prove that on trees the rumor center and distance center are equivalent but on general networks they may differ indeed simulations show that rumor centrality outperforms distance centrality in finding rumor sources in networks which are not tree like

the universal relation between scaling exponents in first passage percolation it has been conjectured in numerous physics papers that in ordinary rst passage percolation on integer lattices the uctuation exponent and the wandering exponent are related through the universal relation 2 1 irrespective of the dimension this is sometimes called the kpz relation between the two exponents this article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense

variational formula for the time constant of first passage percolation consider first passage percolation with positive stationary ergodic weights on the square lattice in d dimensions let t x be the first passage time from the origin to x in z d the convergence of t nx n to the time constant as n tends to infinity is a consequence of the subadditive ergodic theorem this convergence can be viewed as a problem of homogenization for a discrete hamilton jacobi bellman hjb equation by borrowing several tools from the continuum theory of stochastic homogenization for hjb equations we derive an exact variational formula duality principle for the timeconstant under a symmetry assumption we will use the variational formula to construct an explicit iteration that produces the limit shape

first passage percolation network flows and electrical resistances we show that the first passage times of first passage percolation on 2 are such that p 0n n decay geometrically as n where may represent any of the four usual first passage time processes the former estimate requires no moment condition on the time coordinates but there exists a geometrically decaying estimate for the latter quantity if and only if the time coordinate distribution has finite moment generating function near the origin here is the time constant and 0 we study the line to line first passage times and describe applications to the maximum network flow through a randomly capacitated subsection of 2 and to the asymptotic behaviour of the electrical resistance of a subsection of 2 when the edges of the subsection are wires in an electrical network with random resistances in the latter case we show for example that if each edge resistance equals 1 or ohms with probabilities p and 1 p respectively then the effective resistance rn across opposite faces of an n by n box satisfies the following a if p 1 2 then p rn 1 as n b if p 1 2 then there exists p such that p p 1 underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim inf r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim sup r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant v p 1

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

complexity for extended dynamical systems we consider dynamical systems for which the spatial extension plays an important role for these systems the notions of attractor entropy and topological entropy per unit time and volume have been introduced previously in this paper we use the notion of kolmogorov complexity to introduce for extended dynamical systems a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems we introduce this notion as an almost sure limit on orbits of the system moreover we prove a kind of variational principle for this complexity

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

on the total and strict total positivity of the kernels associated with parabolic initial boundary value problems abstract this paper is concerned with the total and strict total positivity of the integral kernels which relate the solutions of parabolic initial boundary value problems to the initial and boundary data maximum principle arguments in conjunction with a characterisation of total positivity in terms of variation diminishing properties are used to prove total positivity of both the initial value and boundary value kernels under mild assumptions similar methods yield strict total positivity when the coefficients appearing in the equation are analytic in either the time or the space variable in this way we extend results of karlin and mcgregor on the total positivity of the kernel associated with the initial data their proofs used the determinantal definition of total positivity and exploited the work of gohberg and krein on the green s functions of sturm liouville operators

conservative nonlinear difference scheme for the cahn hilliard equation ii abstract numerical solutions for the cahn hilliard equation is considered using the crank nicolson type finite difference method existence of the solution for the difference scheme has been shown by brouwer fixed point theorem stability convergence and error analysis of the scheme are shown we also show that the scheme preserves the discrete mass even though the linearized scheme in 1 is conditionally stable and does not preserve the mass

local discontinuous galerkin methods with implicit explicit multistep time marching for solving the nonlinear cahn hilliard equation abstract in this paper we develop the fully discrete local discontinuous galerkin ldg methods coupled with the implicit explicit imex multistep time marching for solving the nonlinear cahn hilliard equation to be more specific we rewrite the cahn hilliard equation in a novel form by adding and subtracting a linear term then we discretize the spatial derivatives with the ldg methods and the temporal derivative with the imex multistep method finally a series of numerical experiments are given to verify the accuracy efficiency and validness of proposed method

on some growth models with a small parameter we consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models when a certain parameter becomes small the basic example treated is the variant of richardson s growth model on d in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors at rate 1 if it has exactly 1 occupied neighbor and of course at rate 0 if it has no occupied neighbor occupied sites remain occupied forever starting from a single occupied site this model has asymptotic speeds of growth in each direction as time goes to infinity and these speeds determine an asymptotic shape in the usual sense it is proven that as tends to 0 the asymptotic speeds scale as 1 d and the asymptotic shape when renormalized by dividing it by 1 d converges to a cube other similar models which are partially oriented are also studied

multi particle diffusion limited aggregation we consider a stochastic aggregation model on mathbb z d start with particles distributed according to the product bernoulli measure with parameter mu in addition start with an aggregate at the origin non aggregated particles move as continuous time simple random walks obeying the exclusion rule whereas aggregated particles do not move the aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it this evolution is called multi particle diffusion limited aggregation our main result states that if on d 1 the initial density of particles is large enough then with positive probability the aggregate has linearly growing arms that is there exists a constant c 0 so that at time t the aggregate contains a point of distance at least ct from the origin for all t the key conceptual element of our analysis is the introduction and study of a new growth process consider a first passage percolation process called type 1 starting from the origin whenever type 1 is about to occupy a new vertex with positive probability instead of doing it it gives rise to another first passage percolation process called type 2 which starts to spread from that vertex each vertex gets occupied only by the process that arrives to it first this process may have three phases extinction type 1 gets eventually surrounded by type 2 coexistence infinite clusters of both types emerge and strong survival type 1 produces an infinite cluster which entraps all type 2 clusters understanding the various phases of this process is of mathematical interest on its own right we establish the existence of a strong survival phase and use this to show our main result

crabgrass measles and gypsy moths an introduction to modern probability on etudie cinq modeles qui permettent d expliquer des processus lies aux systemes de particules en interaction ainsi que d autres processus

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

sublinear variance in first passage percolation for general distributions we prove that the variance of the passage time from the origin to a point x in first passage percolation on mathbb z d is sublinear in the distance to x when d ge 2 obeying the bound c vert x vert log vert x vert under minimal assumptions on the edge weight distribution the proof applies equally to absolutely continuous discrete and singular continuous distributions and mixtures thereof and requires only 2 log moments the main result extends work of benjamini kalai schramm ann prob 31 2003 and benaim rossignol ann inst henri poincare prob stat 3 2008

multifractal analysis for lyapunov exponents on nonconformal repellers for nonconformal repellers satisfying a certain cone condition we establish a version of multifractal analysis for the topological entropy of the level sets of the lyapunov exponents due to the nonconformality the lyapunov exponents are averages of nonadditive sequences of potentials and thus one cannot use birkhoff s ergodic theorem nor the classical thermodynamic formalism we use instead a nonadditive topological pressure to characterize the topological entropy of each level set this prevents us from estimating the complexity of the level sets using the classical gibbs measures which are often one of the main ingredients of multifractal analysis instead we avoid even equilibrium measures and thus in particular g measures by constructing explicitly ergodic measures although not necessarily invariant which play the corresponding role in our work

limiting shape for directed percolation models we consider directed first passage and last passage percolation on the nonnegative lattice z d d 2 with i i d weights at the vertices under certain moment conditions on the common distribution of the weights the limits g x lim n n 1 t nx exist and are constant a s for x r d where t z is the passage time from the origin to the vertex z z d we show that this shape function g is continuous on r d in particular at the boundaries in two dimensions we give more precise asymptotics for the behavior of g near the boundaries these asymptotics depend on the common weight distribution only through its mean and variance in addition we discuss growth models which are naturally associated to the percolation processes giving a shape theorem and illustrating various possible types of behavior with output from simulations

busemann functions and infinite geodesics in two dimensional first passage percolation we study first passage percolation on mathbb z 2 where the edge weights are given by a translation ergodic distribution addressing questions related to existence and coalescence of infinite geodesics some of these were studied in the late 1990s by c newman and collaborators under strong assumptions on the limiting shape and weight distribution in this paper we develop a framework for working with distributional limits of busemann functions and use it to prove forms of newman s results under minimal assumptions for instance we show a form of coalescence of long finite geodesics in any deterministic direction we also introduce a purely directional condition which replaces newman s global curvature condition and whose assumption we show implies the existence of directional geodesics without this condition we prove existence of infinite geodesics which are directed in sectors last we analyze distributional limits of geodesic graphs proving almost sure coalescence and nonexistence of infinite backward paths this result relates to the conjecture of nonexistence of bigeodesics

a shape theorem for the spread of an infection in ksb we studied the following model for the spread of a rumor or infection there is a gas of so called a particles each of which performs a continuous time simple random walk on z d with jump rate da we assume that just before the start the number of a particles at x na x 0 has a mean a poisson distribution and that the na x 0 x z d are independent in addition there are b particles which perform continuous time simple random walks with jump rate db we start with a finite number of b particles in the system at time 0 the positions of these initial b particles are arbitrary but they are nonrandom the b particles move independently of each other the only interaction occurs when a b particle and an a particle coincide the latter instantaneously turns into a b particle ksb gave some basic estimates for the growth of the set b t x z d a b particle visits x during 0 t

multiple phase transitions in long range first passage percolation on square lattices we consider a model of long range first passage percolation on the d dimensional square lattice in which any two distinct vertices are connected by an edge having exponentially distributed passage time with mean where 0 is a fixed parameter and is the l1 norm on we analyze the asymptotic growth rate of the set t which consists of all such that the first passage time between the origin 0 and x is at most t as t we show that depending on the values of there are four growth regimes i instantaneous growth for 2d 1 like the nearest neighbor first passage percolation model corresponding to 2015 wiley periodicals inc

singularity points for first passage percolation let 0 a b be fixed scalars assign independently to each edge in the lattice z 2 the value a with probability p or the value b with probability 1 p for all u v z 2 let t u denote the first passage time between u and v we show that there are points x r 2 such that the time constant in the direction of x namely lim n n 1 e p t 0 nx is not a three times differentiable function of p

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

asymptotic shapes for stationary first passage percolation this paper deals with first passage percolation where the usual i i d condition is weakened to stationarity and ergodicity the well known asymptotic shape result is known to extend to this case it is easy to give necessary conditions for a compact set b subset mathbb r d to arise as the asymptotic shape for some stationary measure on the passage times our main result says that these conditions are also sufficient

large deviations for the chemical distance in supercritical bernoulli percolation the chemical distance d x y is the length of the shortest open path between two points x and y in an infinite bernoulli percolation cluster in this work we study the asymptotic behavior of this random metric and we prove that for an appropriate norm depending on the dimension and the percolation parameter the probability of the event 0 x d 0 x x 1 e 1 e exponentially decreases when x 1 tends to infinity from this bound we also derive a large deviation inequality for the corresponding asymptotic shape result

central limit theorem for first passage percolation time across thin cylinders we prove that first passage percolation times across thin cylinders of the form 0 n hn hn d 1 obey gaussian central limit theorems as long as hn grows slower than n1 d 1 it is an open question as to what is the fastest that hn can grow so that a gaussian clt still holds under the natural but unproven assumption about existence of fluctuation and transversal exponents and strict convexity of the limiting shape in the direction of 1 0 0 we prove that in dimensions 2 and 3 the clt holds all the way up to the height of the unrestricted geodesic we also provide some numerical evidence in support of the conjecture in dimension 2

internal diffusion limited aggregation on discrete groups having exponential growth the internal diffusion limited aggregation dla has been introduced by diaconis and fulton rend sem mat univ pol torino 49 95 119 1991 it is a growth model defined on an infinite set and associated to a markov chain on this set we focus here on sets which are finitely generated groups with exponential growth we prove a shape theorem for the internal dla on such groups associated to symmetric random walks for that purpose we introduce a new distance associated to the green function which happens to have some interesting properties in the case of homogeneous trees we also get the right order for the fluctuations of that model around its limiting shape

an efron stein inequality for nonsymmetric statistics on etablit l analogue d un resultat d efron et stein 1981 que l on demontre a l aide d une technique d espace de hilbert introduite par vitale 1984

the weil petersson metric and volumes of 3 dimensional hyperbolic convex cores we present a coarse interpretation of the weil petersson distance dwp x y between two finite area hyperbolic riemann surfaces x and y using a graph of pants decompositions introduced by hatcher and thurston the combinatorics of the pants graph reveal a connection between riemann surfaces and hyperbolic 3 manifolds conjectured by thurston the volume of the convex core of the quasi fuchsian manifold q x y with x and y in its conformal boundary is comparable to the weil petersson distance dwp x y in applications we relate the weil petersson distance to the hausdorff dimen sion of the limit set and the lowest eigenvalue of the laplacian for q x y and give a new finiteness criterion for geometric limits mathematics department university of chicago 5734 s university ave chicago il 60637 e mail address brock math uchicago edu

shape theorem lyapounov exponents and large deviations for brownian motion in a poissonian potential we derive a large deviation principle governing the position of a d dimensional brownian motion moving in a poissonian potential the derivation of this large deviation principle and the form of the rate function rely on a result similar to the shape theorem of first passage percolation this result produces certain constants which play in this multidimensional situation a similar role as the lyapounov exponents in the one dimensional case the large deviation principle enables us to investigate the transition of regime which occurs between the small h and the large h case for brownian motion with a constant drift h moving in the same potential 1994 john wiley sons inc

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

rate of convergence of the mean for sub additive ergodic sequences abstract for sub additive ergodic processes x m n with weak dependence we analyze the rate of convergence of e x 0 n n to its limit g we define an exponent given roughly by e x 0 n n g n and assuming existence of a fluctuation exponent that gives var x 0 n n 2 we provide a lower bound for of the form the main requirement is that 1 2 in the case 1 2 and under the assumption var x 0 n o n log n for some 0 we prove c for a dependent constant c these results show in particular that non diffusive fluctuations are associated to non trivial various models including first passage percolation directed polymers the minimum of a branching random walk and bin packing fall into our general framework and the results apply assuming exists in the case of first passage percolation in z d we provide a version of 1 2 without assuming existence of

tight geodesics in the curve complex the curve graph mathcal g associated to a compact surface is the 1 skeleton of the curve complex defined by harvey masur and minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein we prove some finiteness results for such geodesics for example we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of we deduce some consequences for the action of the mapping class group on mathcal g in particular we show that it satisfies an acylindricity condition and that the stable lengths of pseudoanosov elements are rational with bounded denominator

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

geodesics and spanning trees for euclidean first passage percolation the metric d q q on the set q of particle locations of a homogeneous poisson process on r d defined as the infimum of i q i q i 1 1 over sequences in q starting with q and ending with q where denotes euclidean distance has nontrivial geodesics when a 1 the cases 1 are the euclidean first passage percolation fpp models introduced earlier by the authors while the geodesics in the case are exactly the paths from the euclidean minimal spanning trees forests of aldous and steele we compare and contrast results and conjectures for these two situations new results for 1 and any d include inequalities on the fluctuation exponents for the metric x 1 2 and for the geodesies 3 4 in strong enough versions to yield conclusions not yet obtained for lattice fpp almost surely every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic from every q for d 2 and 2 further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces

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

geodesics of random riemannian metrics we analyze the disordered riemannian geometry resulting from random perturbations of the euclidean metric we focus on geodesics the paths traced out by a particle traveling in this quenched random environment by taking the point of the view of the particle we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric and we provide an explicit form for its radon nikodym derivative we use this result to prove a local markov property along an unbounded geodesic demonstrating that it eventually encounters any type of geometric phenomenon we also develop in this paper some general results on conditional gaussian measures our main theorem states that a geodesic chosen with random initial conditions chosen independently of the metric is almost surely not minimizing to demonstrate this we show that a minimizing geodesic is guaranteed to eventually pass over a certain bump surface which locally has constant positive curvature by using jacobi fields we show that this is sufficient to destabilize the minimizing property

dimension and rank for mapping class groups we study the large scale geometry of the mapping class group mcg s our main result is that for any asymptotic cone of mcg s the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of mcg s an application is a proof of brock farb s rank conjecture which asserts that mcg s has quasi flats of dimension n if and only if it has a rank n free abelian subgroup hamenstadt has also given a proof of this conjecture using different methods we also compute the maximum dimension of quasi flats in teichmuller space with the weil petersson metric

exact limiting shape for a simplified model of first passage percolation on the plane we derive the limiting shape for the following model of first passage bond percolation on the two dimensional integer lattice the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions the passage times of horizontal bonds are bernoulli distributed while the passage times of vertical bonds are all equal to a deterministic constant to analyze the percolation model we couple it with a one dimensional interacting particle system this particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far away particles we prove a law of large numbers for a tagged particle in this process and the shape result for the percolation is obtained as a corollary

boundary control for the heat equation with non linear boundary condition soit d un domaine borne de r n avec n 1 a frontiere lisse d et soit une normale a d pointant vers l exterieur soit u x t une fonction de x dans d et t l evolution de u est decrite par le probleme aux valeurs limites et initiales suivant u t u dans d 0 t u g u f dans d 0 t u 0 u 0 dans d sous certaines conditions on demontre l existence d une solution faible unique

improved upper bounds on the asymptotic growth velocity of eden clusters we consider the asymptotic shape of clusters in the eden model on a d dimensional hypercubical lattice we discuss two improvements for the well known upper bound to the growth velocity in different directions by that of the independent branching process ibp in the ibp each cell gives rise to a daughter cell at a neighboring site at a constant rate in the first improvement we do not allow such births along the bond connecting the cell to its mother cell in the second we iteratively evolve the system by a growth as ibp for a duration delta followed by culling process in which if any cell produced a descendant within this interval who occupies the same site as the cell itself then the descendant is removed we study the improvement on the upper bound on the velocity for different dimensions d the bounds are asymptotically exact in the large d limit but in d 2 the improvement over the ibp approximation is only a few percent

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

first passage percolation the stationary case if the passage time of the edges of the d lattice are stationary ergodic and have finite moment of orderp d then a s the set of vertices that can be reached within timet has an asymptotic shape ast

energy ssp imex runge kutta methods for the cahn hilliard equation we consider the cahn hilliard equation which describes phase separation phenomenon the equation is discretized by using a fourth order compact difference scheme in space and strong stability preserving ssp implicit explicit imex runge kutta methods in time the new methods have two distinct features 1 the large time steps can be used in the numerical simulation because the energy is stable and 2 the energy functional decreases by time unconditional energy stability of first order and second order methods are proved numerical experiments are given to demonstrate the performance of the proposed methods

asymptotic entropy and green speed for random walks on countable groups we study asymptotic properties of the green metric associated with transient random walks on countable groups we prove that the rate of escape of the random walk computed in the green metric equals its asymptotic entropy the proof relies on integral representations of both quantities with the extended martin kernel in the case of finitely generated groups where this result is known benjamini and peres probab theory related fields 98 1994 91 112 we give an alternative proof relying on a version of the so called fundamental inequality relating the rate of escape the entropy and the logarithmic volume growth extended to random walks with unbounded support

second order implicit explicit total variation diminishing schemes for the euler system in the low mach regime abstract in this work we consider the development of implicit explicit total variation diminishing tvd methods also termed ssp strong stability preserving for the compressible isentropic euler system in the low mach number regime the proposed scheme is asymptotically stable with a cfl condition independent of the mach number in addition it degenerates in the low mach number regime to a consistent discretization of the incompressible system since it has been proved that implicit schemes of order higher than one cannot be tvd ssp 30 we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing tvd methods in space for this particular class of schemes the tvd property is first proved on a linear model advection equation and then extended to the isentropic euler case the result is a method which interpolates from the first to the second order both in space and time it preserves the monotonicity of the solution and is highly accurate for all choices of the mach number moreover the time step is only restricted by the non stiff part of the system we finally show thanks to one and two dimensional test cases that the method indeed possesses the claimed properties

first passage percolation on mathbb z 2 a simulation study first passage percolation on mathbb z 2 is a model for describing the spread of an infection on the sites of the square lattice the infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges in this paper the speed of the growth and the shape of the infected set is studied by aid of large scale computer simulations with focus on continuous passage time distributions it is found that the most important quantity for determining the value of the time constant which indicates the inverse asymptotic speed of the growth is mathbf e min tau 1 ldots tau 4 where tau 1 ldots tau 4 are i i d passage time variables the relation is linear for a large class of passage time distributions furthermore the directional time constants are seen to be increasing when moving from the axis towards the diagonal so that the limiting shape is contained in a circle with radius defined by the speed along the axes the shape comes closer to the circle for distributions with larger variability

differentiability at the edge of the percolation cone and related results in first passage percolation we study first passage percolation in two dimensions using measures on passage times with b inf supp 0 and mu b p geq vec p c the threshold for oriented percolation we first show that for each such the boundary of the limit shape for is differentiable at the endpoints of flat edges in the so called percolation cone we then conclude that the limit shape must be non polygonal for all of these measures furthermore the associated richardson type growth model admits infinite coexistence and if is not purely atomic the graph of infection has infinitely many ends we go on to show that lower bounds for fluctuations of the passage time given by newman piza extend to these measures we establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for this result confirms a prediction of newman and piza ann probab 23 977 1005 1995 and zhang ann probab 36 331 362 2008 under the assumption of finite radius of curvature for the limit shape in these directions we obtain a power law lower bound for the variance and an inequality between the exponents and

boundary control problems with nonlinear state equation first order necessary conditions of optimality are obtained for boundary control problems governed by parabolic equations with nonlinear boundary value conditions

contact processes in several dimensions this paper deals primarily with the basic contact processes introduced and studied by harris 12t4 these are random evolutions on the state space s a l l subsets of z a zd the d dimensional integer lattice with extremely simple local dynamics namely if we think of the contact process as representing the spread of an infection t es denoting the set of infected sites at time t then x e it becomes healthy at exponential rate 1 while x z a t is infected at a rate proport ional to the number of sites neighboring x where infection is present the proportionali ty constant 2 is called the infection parameter contact processes are perhaps the simplest s valued markov processes which exhibit a critical phenomenon infection emanating from a single site dies out with probabili ty one for small positive 2 but has positive probability of surviving for all time when is large there is a critical value 2 d where the transition occurs principal objectives of analysis are the precise formulation of the critical phenomenon and detailed description of the ergodic behavior both below and above a c our recent survey articles 10 and i 6 provide introductions to contact processes and to the general field of interacting s valued systems respectively we will make constant use of notation and techniques from the surveys assuming that the reader is familiar with thern the article 10 describes in some detail the current state of knowledge concerning one dimensional contact processes the theory for d 1 is now fairly complete with two major exceptions 1 is not even determined to one decimal place 1 18 2 e there is an

the poisson formula for groups with hyperbolic properties the poisson boundary of a group g with a probability measure is the space of ergodic components of the time shift in the path space of the associated random walk via a generalization of the classical poisson formula it gives an integral representation of bounded harmonic functions on g in this paper we develop a new method of identifying the poisson boundary based on entropy estimates for conditional random walks it leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the poisson boundary with natural topological boundaries for several classes of groups word hyperbolic groups and discontinuous groups of isometries of gromov hyperbolic spaces groups with inflnitely many ends cocompact lattices in cartan hadamard manifolds discrete subgroups of semisimple lie groups

variational formulas and cocycle solutions for directed polymer and percolation models we discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium namely the limiting time constant for last passage percolation and the limiting free energy for directed polymers the results are valid for models in arbitrary dimension steps of the admissible paths can be general the environment process is ergodic under spatial translations and the potential accumulated along a path can depend on the environment and the next step of the path the variational formulas come in two types one minimizes over gradient like cocycles and another one maximizes over invariant measures on the space of environments and paths minimizing cocycles can be obtained from busemann functions when these can be proved to exist the results are illustrated through 1 1 dimensional exactly solvable examples periodic examples and polymers in weak disorder