complete convergence theorem for a competition model in this paper we consider a hierarchical competition model durrett and swindle have given sufficient conditions for the existence of a nontrivial stationary distribution here we show that under a slightly stronger condition the complete convergence theorem holds and hence there is a unique nontrivial stationary distribution

metastable behavior of stochastic dynamics a pathwise approach in this paper a new approach to metastability for stochastic dynamics is proposed the basic idea is to study the statistics of each path performing time averages along the evolution metastability would be characterized by the fact that the process of these time averages converges under a suitable rescaling to a measure valued markov jump process here this convergence is shown for the curie weiss mean field dynamics and also for a model with spatial structure harris contact process

percolation in half spaces equality of critical densities and continuity of the percolation probability renormalization arguments are developed and applied to independent nearest neighbor percolation on various subsets of d d 2 yielding equality of the critical densities p c for a half space quarter space etc and ford 2 equality with the limit of slab critical densities continuity of the phase transition for the half space quarter space etc i e vanishing of the percolation probability p atp p c

a two stage contact process on scale free networks we study a two stage contact process on scale free networks as a model for the spread of epidemics we show that any virus starting from a single vertex with arbitrarily small infection rates can last for a super polynomial time with positive probability if the power law exponent 2 this is in sharp contrast with the mean field analysis the estimation of the metastable density is also provided

contact processes on random graphs with power law degree distributions have critical value 0 if we consider the contact process with infection rate on a random graph on n vertices with power law degree distributions mean field calculations suggest that the critical value c of the infection rate is positive if the power 3 physicists seem to regard this as an established fact since the result has recently been generalized to bipartite graphs by gomez gardenes et al proc natl acad sci usa 105 2008 1399 1404 here we show that the critical value c is zero for any value of 3 and the contact process starting from all vertices infected with a probability tending to 1 as n maintains a positive density of infected sites for time at least exp n 1 for any 0 using the last result together with the contact process duality we can establish the existence of a quasi stationary distribution in which a randomly chosen vertex is occupied with probability it is expected that c as 0 here we show that 1 2 3 and so 2 for 3 thus even though the graph is locally tree like does not take the mean field critical value 1

the box crossing property for critical two dimensional oriented percolation we consider critical oriented bernoulli percolation on the square lattice mathbb z 2 we prove a russo seymour welsh type result which allows us to derive several new results concerning the critical behavior the sub linear polynomial fluctuations contrast with the supercritical regime where w n is known to behave linearly in n it is also different from the critical picture obtained for non oriented bernoulli percolation in which the scaling limit is non degenerate in both directions all our results extend to the graphical representation of the one dimensional contact process

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

existence of a constant for finite system extinction we show the existence of a constant 0 such that if n is the extinction time for a supercritical contact process on 1 n d starting from full occupancy then log e n nd tend to as n tends to infinity

different clustering regimes in systems of hierarchically interacting diffusions we study a system of interacting diffusions dx t a x t x t dt g x t dw t indexed by the hierarchical group as a genealogical two genotype model where x t denotes the frequency of say type a with hierarchically determined degrees of relationship between colonies in the case of short interaction range it is known that the system clusters that is locally one genotype dies out we focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel a via its recurrent potential kernel one of these regimes will be further investigated by mean field methods for general interaction range we shall also relate the behavior of large finite systems indexed by finite subsets n of to that of the infinite system on the way we will establish relations between hitting times of random walks and their potentials

on the long term behavior of some finite particle systems we consider the problem of comparing large finite and infinite systems with locally interacting components and present a general comparison scheme for the case when the infinite system is nonergodic we show that this scheme holds for some specific models one of these is critical branching random walk onz d let t denote this system and let denote a finite version of t defined on the torus n n d z d ford 3 we prove that for stationary shift ergodic initial measures with density that ift n andt n 2n 1 d s 0 asn then v 0 is the set of extremal invariant measures for the infinite system t andq s is the transition function of feller s branching diffusion we prove several extensions and refinements of this result the other systems we consider are the voter model and the contact process

symmetries of statistics on lattice paths between two boundaries abstract we prove that on the set of lattice paths with steps n 0 1 and e 1 0 that lie between two fixed boundaries t and b which are themselves lattice paths the statistics number of e steps shared with b and number of e steps shared with t have a symmetric joint distribution to do so we give an involution that switches these statistics preserves additional parameters and generalizes to paths that contain steps s 0 1 at prescribed x coordinates we also show that a similar equidistribution result for path statistics follows from the fact that the tutte polynomial of a matroid is independent of the order of its ground set we extend the two theorems to k tuples of paths between two boundaries and we give some applications to dyck paths generalizing a result of deutsch to watermelon configurations to pattern avoiding permutations and to the generalized tamari lattice finally we prove a conjecture of nicolas about the distribution of degrees of k consecutive vertices in k triangulations of a convex n gon to achieve this goal we provide a new statistic preserving bijection between certain k tuples of non crossing paths and k flagged semistandard young tableaux which is based on local moves reminiscent of jeu de taquin

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

on the solution of moment problems by reproducing kernel methods abstract a general method is presented for finding necessary and sufficient conditions in order that a complex valued function be represented as an integral of a given integrand with respect to some bounded nonnegative measure on the real axis extensions of the method to hilbert space representations and multivariant representations are noted

extinction time for the contact process on general graphs we consider the contact process on finite and connected graphs and study the behavior of the extinction time that is the amount of time that it takes for the infection to disappear in the process started from full occupancy we prove without any restriction on the graph g that if the infection rate lambda is larger than the critical rate of the one dimensional process then the extinction time grows faster than exp g log g kappa for any constant kappa 1 where g denotes the number of vertices of g also for general graphs we show that the extinction time divided by its expectation converges in distribution as the number of vertices tends to infinity to the exponential distribution with parameter 1 these results complement earlier work of mountford mourrat valesin and yao in which only graphs of bounded degrees were considered and the extinction time was shown to grow exponentially in n here we also provide a simpler proof of this fact

a generalization of diagonal flips in a convex polygon a diagonal flip is an operation that converts one triangulation of a convex polygon into another in this paper we generalize this operation to prove some results including the next theorem theorem let e 1 and e 2 be any two sets of edges in a convex polygon such that each e has no t 1 pairwise crossing edges and e i u e contains t 1 pairwise crossing edges for any edge e e i then e 1 can be transformed into e 2 by a sequence of diagonal flips such that each flip changes one edge to another preserving the above property

hypergeometric series in a series expansion of the directed bond percolation probability on the square lattice the asymmetric directed bond percolation adbp problem with an asymmetry parameterk is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice it is shown that the first correction term dn 1 k is expressed by gauss hypergeometric series with a variablek since the adbp includes the ordinary directed bond percolation as a special case withk 1 our results give another proof for the baxter guttmann s conjecture thatdn 1 1 is given by the catalan number which was recently proved by bousquet melou direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the adbp on the square lattice the analysis bydlog pade approximations suggests that the critical value depends onk while asymmetry does not change the critical exponent of percolation probability

the contact process on finite homogeneous trees let d be a homogeneous tree in which every vertex has d 1 neighbours where d 2 the contact process on such a tree is known to have three distinct phases we consider the process on a finite subtree namely the rooted tree of depth h and branching factor d and relate the behaviour of the process on the infinite tree to its behaviour on the finite tree for large h in the phase of strong survival we show that with probability independent of h the process on the subtree starting from a single infection survives for a time which is doubly exponential in h and almost exponential in the number of vertices of the finite tree in the phase of weak survival on the infinite tree the survival time on the finite tree is approximately linear in h in the phase of no survival the survival time on the finite tree is linear in h if one starts with all vertices initially infected and bounded by a random variable independent of h with an exponential tail if one starts from a single infection

on directed compact percolation near a damp wall a percolation probability for directed compact percolation near a damp wall which interpolates between the previously examined cases is derived exactly we find that the critical exponent 2 in common with the dry wall rather than the value previously found in the wet wall and bulk cases the solution is found via a mapping to a particular model of directed walks we evaluate the exact generating function for this walk model which is also related to the asep model of traffic flow we compare the underlying mathematical structure of the various cases previously considered and this one by reviewing the common framework of solution via the mapping to different directed walk models

the contact process on a finite set ii in this paper we complete the work started in part i we show that if sigma n is the time that the contact process on 1 ldots n first hits the empty set then for lambda lambda c the critical value for the process on z there is a positive constant gamma lambda so that log sigma n n rightarrow gamma lambda in probability as n rightarrow infty we also give a new simple proof that sigma n e sigma n converges to a mean one exponential the keys to the proof of the first result are a planar graph duality for the contact process and an observation of j chayes and l chayes that exponential decay rates for connections in strips approach the decay rates in the plane as the width of the strip goes to infty

the asymmetric contact process at its second critical value the asymmetric contact process onz has two distinct critical values 1 2 at least with sufficient asymmetry one can consider the process on 0 n and analyze the time which we call n till complete vacany starting from complete occupation its behavior has already been resolved for all regions of except for 2 for this value schinazi proved that limn log n logn 2 in probability and conjectured that n n2 converges in distribution it is that result that we prove in this paper we rely heavily on the brownian motion behavior of the edge particle which comes from galves and presutti and kuczek

nilpotent ideals in having a fixed class of nilpotence combinatorics and enumeration we study the combinatorics of ad nilpotent ideals of a borel subalgebra of sl n 1 c we provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence we study the relationships between these results and the combinatorics of dyck paths based upon a remarkable bijection between ad nilpotent ideals and dyck paths finally we propose a q t analogue of the catalan number c n these q t catalan numbers count on the one hand ad nilpotent ideals with respect to dimension and class of nilpotence and on the other hand admit interpretations in terms of natural statistics on dyck paths

return polynomials for non intersecting paths above a surface on the directed square lattice we enumerate sets of n non intersecting t step paths on the directed square lattice which are excluded from the region below the surface y 0t o which they are initially attached in particular we obtain a product formula for the number of star configurations in which the paths have arbitrary fixed endpoints we also consider the return polynomial r w y m0 r w y m m

the asymmetric contact process on a finite set the contact process onz has one phase transition let c be the critical value at which the transition occurs let n be the extinction time of the contact process on 0 n durrett and liu 1988 durrett and schonmann 1988 and durrett schonmann and tanaka 1989 have respectively proved that the subcritical supercritical and critical phases can be characterized using a large finite system instead ofz in the following way there are constants 1 and 2 such that if c limn log n n 2 if c limn n n and limn n n4 0 in probability in this paper we consider the asymmetric contact process onz when it has two distinct critical values c1 c2 we show that for c1 c2 limn n n 1 where i is an edge speed and for c2 limn log n logn 2 in probability

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

scaling inequalities for oriented percolation we look at seven critical exponents associated with two dimensional oriented percolation scaling theory implies that these quantities satisfy four equalities we prove five related inequalitites

correlation lengths for oriented percolation oriented percolation has two correlation lengths one in the space and one in the time direction in this paper we define these quantities for the two dimensional model in terms of the exponential decay of suitably chosen quantities and study the relationship between the various definitions the definitions are used in a companion paper to prove inequalities between critical exponents