encounter distribution of two random walkers on a finite one dimensional interval we analyse the first passage properties of two random walkers confined to a finite one dimensional domain for the case of absorbing boundaries at the endpoints of the interval we derive the probability that the two particles meet before either one of them becomes absorbed at one of the boundaries for the case of reflecting boundaries we obtain the mean first encounter time of the two particles our approach leads to closed form expressions that are more easily tractable than a previously derived solution in terms of the weierstrass elliptic function

a bounded approximation of weakly compact operators abstract we study bounded approximation properties of operators it is shown that for every weakly compact operator t on a banach space x if t is uniformly approximated on each compact subset of x by finite rank operators then t 2 can be approximated in the same topology by finite rank operators with the operator norm less than or equal to t 2

random tree type partitions as a model for acyclic polymerization holtsmark 3 2 stable distribution of the supercritical gel we study the asymptotic behavior in the thermodynamic limit as n of a particular probability distribution on the set of all forests of trees on a set of n elements monomers we establish the existence of three stages of polymerization dependent upon the ration of association and dissociation rates of monomers and we analyse the near critical and supercritical stages in the supercritical stage we discover that the molecular weight of the largest connected component gel has the holtsmark distribution

the weak metric approximation property we introduce and investigate the weak metric approximation property of banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property among others we show that if a banach space has the approximation property and is 1 complemented in its bidual then it has the weak metric approximation property we also study the lifting of the weak metric approximation property from banach spaces to their dual spaces this enables us in particular to show that the subspace of c0 constructed by johnson and schechtman does not have the weak metric approximation property

the asymptotic distributions of the size of the largest length of a reversible markov process of polymerization in this paper the subcritical near critical and supercritical asymptotic behaviour of a reversible markov process as a chemical model for polymerization was studied we establish the existence of three distinct stages subcritical near critical and supercritical stages of polymerization in the thermodynamic limit as n depending on the value of strength of the fragmentation reaction these three stages correspond to the size of the largest length of polymers of size n to be itself of order log n na 1 2 a 1 and n respectively especially a 2 2 1 when the coagulation reaction rate constant rij of the polymers satisfies rij i j with 1 2 3 2

percolation critical exponents in scale free networks we study the behavior of scale free networks having connectivity distribution p k k 2l close to the percolation threshold we show that for networks with 3 l 4 known to undergo a transition at a finite threshold of dilution the critical exponents are different than the expected mean field values of regular percolation in infinite dimensions networks with 2 l 3 possess only a percolative phase nevertheless we show that in this case percolation critical exponents are well defined near the limit of extreme dilution where all sites are removed and that also then the exponents bear a strong l dependence the regular mean field values are recovered only for l 4

equilibrium ring formation in polymer solutions this paper delivers a flexible formalism for handling equilibrium ring formation based on graphical models of polymerization it includes as special cases the flory stockmayer ra sub f model the flory a sub f rb sub g model and gordon s branching process formalism when simple ring formation occurs in equireactive systems ist also includes the jacobson stockmayer ra sub 2 and hoeve ra sub f models the formalism is built form first principles in statistical mechanics and all assumptions are clearly stated all parameters re given in terms of thermodynamic variables with ring weights generalizing the jacobson stockmayer gaussian random walk the formalism yields results for branching ra sub f a sub f rb sub g and ra sub f rb sub g polymer models equireactivity then gives explicit solutions the equireactive ra sub f rb sub g model compares favorability with data from gel point vs dilition experiments with the exception of the spanning tree approximation graphical models of piolymerization suffer from combinations of the following defects equireactivity assumptions restrictions to one type of monomer or bond absence of rings or absence of fused rings this paper provided a promising exact approach to handling all of these problems simultaneously

pre and post gel size distributions in ir reversible polymerisation a class of irreversible coagulation processes can be modelled by smoluchowski s coagulation equation with rate constants kij a b i j cij non negative a b and c for c not 0 a gelation transition occurs the authors obtain explicit solutions for the size distribution ck t with ck 0 delta k1 next they construct and solve the equations for reversible polymerisation by incorporating break up processes in the kinetic equation with a unimolecular fragmentation rate fij lambda ninjkij ni j the degeneracy factors nk obey k 1 nk 1 2 sigma kijninj with i j k and n1 1 and the strength parameter lambda exp g kbt where the binding energy g to infinity for irreversible coagulation explicit results are only given for flory s polymerisation models raf and braf 1 in the vicinity of the gel point the authors verify the scaling hypothesis and calculate critical exponents

clustering in coagulation fragmentation processes random combinatorial structures and additive number systems asymptotic formulae and limiting laws we develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper in the case when the parametric function of the models is regularly varying with positive exponent the approach is based on khintchine s probabilistic method that grew out of the darwin fowler method in statistical physics our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters components as the total size of a structure number of particles goes to infinity we discover that n1 l 1 is the threshold for the limiting distribution of the largest cluster as a byproduct of our study we prove the independence of the numbers of groups of fixed sizes as n this is in accordance with the general principle of asymptotic independence of sites in mean field models the latter principle is commonly accepted in statistical physics but not rigorously proved

the structure of random partitions of large integers random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability the focus is on limit theorems as the number being partitioned approaches the limiting probability distribution of the appropriately normalized number of parts of some small size is exponential the large parts are described by a particular markov chain a central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes the major tool is a simple construction of random partitions that treats the number being partitioned as a random variable the same technique is useful when some restriction is placed on partitions such as the requirement that all parts must be distinct

on edge exchangeable random graphs we study a recent model for edge exchangeable random graphs introduced by crane and dempsey in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges we study a number of examples and show that the model can produce dense sparse and extremely sparse random graphs one example yields a power law degree distribution we give some examples where the random graph is dense and converges a s in the sense of graph limit theory but also an example where a s every graph limit is the limit of some subsequence another example is sparse and yields convergence to a non integrable generalized graphon defined on 0 infty

new criterion of the approximation property this paper is concerned with the approximation property which is an important property in banach space theory we show that a banach space x has the approximation property if and only if for every banach space y the set of finite rank operators from x to y is dense in the corresponding space of compact operators in the usual topology of uniform convergence on compact sets

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

structure of the condensed phase in the inclusion process we establish a complete picture of condensation in the inclusion process in the thermodynamic limit with vanishing diffusion covering all scaling regimes of the diffusion parameter and including large deviation results for the maximum occupation number we make use of size biased sampling to study the structure of the condensed phase which can extend over more than one lattice site and exhibit an interesting hierarchical structure characterized by the poisson dirichlet distribution while this approach is established in other areas including population genetics or random permutations we show that it also provides a powerful tool to analyse homogeneous condensation in stochastic particle systems with stationary product distributions we discuss the main mechanisms beyond inclusion processes that lead to the interesting structure of the condensed phase and the connection to other generic particle systems our results are exact and we present monte carlo simulation data and recursive numerics for partition functions to illustrate the main points

sharp vanishing thresholds for cohomology of random flag complexes for every k 1 the k th cohomology group h k x q of the random ag complex x x n p passes through two phase transitions one where it appears and one where it vanishes we describe the vanishing threshold and show that it is sharp using the same spectral methods we also nd a sharp threshold for the fundamental group 1 x to have kazhdan s property t combining with earlier results we obtain as a corollary that for every k 3 there is a regime in which the random ag complex is rationally homotopy equivalent to a bouquet of k dimensional spheres

the structure of a random graph at the point of the phase transition consider the random graph models g n edges m and g n prob edge p with m m n 1 n 1 3 n 2 and p p n 1 an 1 3 n for 1 1 define an i component of a random graph as a component which has exactly i more edges than vertices call an i component with i 1 a complex component for both models we show that when a is constant the expected number of complex components is bounded almost surely a s each of these components if any exist has size of order n2 3 and the maximum value of i is bounded in probability we prove that a s the largest suspended tree in each complex component has size of order n2 3 and deletion of all suspended trees results in a smoothed graph of size of order n 3 with the maximum vertex degree 3 the total number of branching vertices i e of degree 3 is bounded in probability thus each complex component is almost surely topologically equivalent to a 3 regular multigraph of a uniformly bounded size lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3 we find a relatively simple integral formula for the limit distribution of the numbers of complex components which implies in particular that all values of the complexity spectrum have positive limiting probabilities we also answer questions raised by erdos and renyi back in 1960 it is proven that there exists p a the limiting planarity probability with 0 p a 1 p oo 1 p oo 0 in particular g n m g n p resp is almost surely nonplanar iff m n 2 n 213 00 np i n 1 3 oo resp

independent process approximations for random combinatorial structures abstract many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables conditioned on the value of a weighted sum of the variables it is interesting to compare the combinatorial structure directly to the independent discrete process without renormalizing the quality of approximation can often be conveniently quantified in terms of total variation distance for functionals which observe part but not all of the combinatorial and independent processes among the examples are combinatorial assemblies e g permutations random mapping functions and partitions of a set multisets e g polynomials over a finite field mapping patterns and partitions of an integer and selections e g partitions of an integer into distinct parts and square free polynomials over finite fields we consider issues common to all the above examples including equalities and upper bounds for total variation distances existence of limiting processes heuristics for good approximations the relation to standard generating functions moment formulas and recursions for computing densities refinement to the process which counts the number of parts of each possible type the effect of further conditioning on events of moderate probability large deviation theory and nonuniform measures on combinatorial objects and the possibility of getting useful results by overpowering the conditioning

random walks on deterministic scale free networks exact results we study the random walk problem on a class of deterministic scale free networks displaying a degree sequence for hubs scaling as a power law with an exponent gamma log 3 log 2 we find exact results concerning different first passage phenomena and in particular we calculate the probability of first return to the main hub these results allow to derive the exact analytic expression for the mean time to first reach the main hub whose leading behavior is given by tau approximately v1 1 gamma where v denotes the size of the structure and the mean is over a set of starting points distributed uniformly over all the other sites of the graph interestingly the process turns out to be particularly efficient we also discuss the thermodynamic limit of the structure and some local topological properties

finite size effects in dynamics of zero range processes the finite size effects prominent in zero range processes exhibiting a condensation transition are studied by using continuous time monte carlo simulations we observe that well above the thermodynamic critical point both static and dynamic properties display fluidlike behavior up to a density c l which is the finite size counterpart of the critical density c c l we determine this density from the crossover behavior of the average size of the largest cluster we then show that several dynamical characteristics undergo a qualitative change at this density in particular the size distribution of the largest cluster at the moment of relocation the persistence properties of the largest cluster and the correlations in its motion are studied

cyclization in bipartite random graphs in this paper the time evolution of a finite bipartite graph initially comprising two sorts of isolated vertices is considered the graph is assumed to evolve by adding edges one at a time each new edge connects either two linked components and forms a new component of a larger order coalescence of graphs or increases by one the number of edges in a given linked component cycling any state of the graph is thus characterized by the set of occupation numbers the numbers of linked components comprising a given numbers of vertexes of the both sorts and a given number of edges once the rate of appearance of an extra edge in the graph being known the master equation governing the time evolution of the probability to find the random graph in a given state is reformulated in terms of the functional generating the probability to find the evolving graph in a given state the exact solution of the evolution equation for the generating functional applies for analyzing the average population numbers of linked components in the limit of large order of the graph the distribution factorizes into two multipliers one of which is just the spectrum of linked components in the infinite bipartite graph the second multiplier includes the dependence on the total size of the graph both these multipliers contain information on the emergence of the giant component that forms at a critical time

polymers and random graphs asymptotic equivalence to branching processes in 1974 falk and thomas did a computer simulation of flory s equireactive raf polymer model rings forbidden and rings allowed asymptotically the rings forbidden model tended to stockmayer s raf distribution in which the sol distribution sticks after gelation while the rings allowed model tended to the flory version of the raf distribution in 1965 whittle introduced the tree and pseudomultigraph models we show that these random graphs generalize the falk and thomas models by incorporating first shell substitution effects moreover asymptotically the tree model displays postgelation sticking hence this phenomenon results from the absence of rings and occurs independently of equireactivity we also show that the pseudomultigraph model is asymptotically identical to the branching process model introduced by gordon in 1962 this provides a possible basis for the branching process model in standard statistical mechanics

condensation in perturbed zero range processes condensation phenomena in zero range processes have attracted significant attention recently here we consider zero range processes in which n sites are occupied by m particles that jump between sites according to a jump rate function given by g let z z1 z2 zn denote the steady state of a zero range process and let z n be the size of the maximum cluster a non complete condensation is defined as an event in which z n n converges to a positive constant less than 1 as n tends to infinity in this work we provide evidence that condensation depends on the detailed form of the jump rates by showing that non complete condensation occurs when g is given by where mk are bounded constants satisfying suitable conditions note that this choice of g is a perturbation of which exhibits perfect condensation i e n z n converges to 0 as n tends to infinity

the convex approximation property of banach spaces abstract we study the convex approximation property of banach spaces to provide a unified approach to various approximation properties including besides the classical ones e g the positive approximation property of banach lattices and the approximation property for pairs of banach spaces our main results concern lifting of metric and weak metric approximation properties from banach spaces to their dual spaces as an easy application it follows that if x or x has the radon nikod m property then the approximation property of x defined by a convex subset of conjugate compact operators containing 0 in particular the positive approximation property of x is metric

statistical mechanics of topological phase transitions in networks we provide a phenomenological theory for topological transitions in restructuring networks in this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges the associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge dual graph of a fully connected network in our studies based on an exact enumeration method monte carlo simulations and theoretical considerations we find a rich variety of topological phase transitions when the temperature is varied these transitions signal singular changes in the essential features of the global structure of the network depending on the energy function chosen the observed transitions can be best monitored using the order parameters f s5s max m i e the size of the largest connected component divided by the number of edges or f k5k max m the largest degree in the network divided by the number of edges if for example the energy is chosen to be e52s max the observed transition is analogous to the percolation phase transition of random graphs for this choice of the energy the phase diagram in the k t plane is constructed single vertex energies of the form e5 i f k i where k i is the degree of vertex i are also studied depending on the form of f ki first order and continuous phase transitions can be observed in case of f k i 52 k i1a ln ki the transition is continuous and at the critical temperature scale free graphs can be recovered finally by abruptly decreasing the temperature nonequilibrium processes e g nucleation and growth of particular topological phases can also be interpreted by the present approach

condensation of non reversible zero range processes in this article we investigate the condensation phenomena for a class of non reversible zero range processes on a fixed finite set by establishing a novel inequality bounding the capacity between two sets and by developing a robust framework to perform quantitative analysis on the metastability of non reversible processes we prove that the condensed site of the corresponding zero range processes approximately behaves as a markov chain on the underlying graph whose jump rate is proportional to the capacity with respect to the underlying random walk the results presented in the current paper complete the generalization of the work of beltran and landim probab theory relat fields 152 781 807 2012 on reversible zero range processes and that of landim commun math phys 330 1 32 2014 on totally asymmetric zero range processes on a one dimensional discrete torus

critical behavior of the heterogeneous random coagulation fragmentation processes in this paper we study the asymptotic distributions of the heterogeneous random coagulation fragmentation processes hcfp which model the coagulation fragmentation and diffusion of clusters of particles on a lattice based on a closed form of the stationary distribution for the hcfp we prove that the mutually dependent clusters with a finite size finite particles in the sub critical stage will become independent in the critical and super critical stages and the asymptotic distributions of the number of clusters may converge to gaussian or poisson distribution according to its size to be small or large in the critical and super critical stages

the random cluster model on the complete graph the random cluster model of fortuin and kasteleyn contains as special cases the percolation ising and potts models of statistical physics when the underlying graph is the complete graph onn vertices then the associated processes are called mean field in this study of the mean field random cluster model with parametersp n andq we show that its properties for any value ofq 0 may be derived from those of an erd s renyi random graph in this way we calculate the critical point c q of the model and show that the associated phase transition is continuous if and only ifq 2 exact formulae are given for c q the density of the largest component the density of edges of the model and the free energy this work generalizes earlier results valid for the potts model whereq is an integer satisfyingq 2 equivalent results are obtained for a fixed edge number random cluster model as a consequence of the results of this paper one obtains large deviation theorems for the number of components in the classical random graph models whereq 1

random subgraphs of finite graphs ii the lace expansion and the triangle condition in a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph in this paper we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph the latter is readily verified for several graphs with vertex set 0 1 r 1 n including the hamming cube on an alphabet of r letters the n cube for r 2 the n dimensional torus with nearest neighbor bonds and n sufficiently large and the n dimensional torus with n 6 and sufficiently spread out long range bonds the conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples

synchronous vs asynchronous dynamics of diffusion controlled reactions abstract an analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice at each time step either both walkers diffuse simultaneously with probability p synchronous event or one of them diffuses while the other remains immobile with complementary probability asynchronous event reaction takes place through same site occupation or position exchange we study the influence of the degree of synchronicity p of the walkers and the lattice size n on the global reaction s efficiency for odd n the purely synchronous case p 1 is always the most effective one while for even n the encounter time is minimized by a combination of synchronous and asynchronous events this new parity effect is fully confirmed by monte carlo simulations on 1d lattices as well as for 2d and 3d lattices in contrast the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of p the relevance of the model for several research areas is briefly discussed

determining mean first passage time on a class of treelike regular fractals relatively general techniques for computing mean first passage time mfpt of random walks on networks with a specific property are very useful since a universal method for calculating mfpt on general graphs is not available because of their complexity and diversity in this paper we present techniques for explicitly determining the partial mean first passage time pmfpt i e the average of mfpts to a given target averaged over all possible starting positions and the entire mean first passage time emfpt which is the average of mfpts over all pairs of nodes on regular treelike fractals we describe the processes with a family of regular fractals with treelike structure the proposed fractals include the t fractal and the peano basin fractal as their special cases we provide a formula for mfpt between two directly connected nodes in general trees on the basis of which we derive an exact expression for pmfpt to the central node in the fractals moreover we give a technique for calculating emfpt which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials making use of the proposed methods we obtain analytically the closed form solutions to pmfpt and emfpt on the fractals and show how they scale with the number of nodes in addition to exhibit the generality of our methods we also apply them to the vicsek fractals and the iterative scale free fractal tree and recover the results previously obtained

the strong approximation property we introduce and investigate the strong approximation property of banach spaces which is strictly stronger than the approximation property and at least formally weaker than the weak bounded approximation property among others we show that the weak bounded approximation property is equivalent to a quantitative strengthening of the strong approximation property some recent results on the approximation property of banach spaces and their dual spaces are improved

a martingale problem for an absorbed diffusion the nucleation phase of condensing zero range processes we prove uniqueness of a martingale problem with boundary conditions on a simplex associated to a differential operator with an unbounded drift we show that the solution of the martingale problem remains absorbed at the boundary once it attains it and that after hitting the boundary it performs a diffusion on a lower dimensional simplex similar to the original one we also prove that in the diffusive time scale condensing zero range processes evolve as this absorbed diffusion

size of the largest cluster under zero range invariant measures we study the finite zero range process with occupancy dependent rate function g under the invariant measure which can be written explicitly in terms of g particles are distributed over sites and we regard all particles at a fixed site as a cluster in the density one case that is equal numbers of particles and sites we determine asymptotically the size of the largest cluster as the number of particles tends to infinity and determine its dependence on the rate function

exact solution for the time evolution of network rewiring models we consider the rewiring of a bipartite graph using a mixture of random and preferential attachment the full mean field equations for the degree distribution and its generating function are given the exact solution of these equations for all finite parameter values at any time is found in terms of standard functions it is demonstrated that these solutions are an excellent fit to numerical simulations of the model we discuss the relationship between our model and several others in the literature including examples of urn backgammon and balls in boxes models the watts and strogatz rewiring problem and some models of zero range processes our model is also equivalent to those used in various applications including cultural transmission family name and gene frequencies glasses and wealth distributions finally some voter models and an example of a minority game also show features described by our model

lifting bounded approximation properties from banach spaces to their dual spaces based on a new reformulation of the bounded approximation property we develop a unified approach to the lifting of bounded approximation properties from a banach space x to its dual x this encompasses cases when x has the unique extension property or x is extendably locally reflexive in particular it is shown that the unique extension property of x permits to lift the metric a approximation property from x to x for any operator ideal a and that there exists a banach space x such that x x are extendably locally reflexive but x x are not

on a random graph evolving by degrees abstract we consider a random multi graph growth process g m on a vertex set n which is a special case of a more general process proposed by laci lovasz in 2002 g 0 is empty and g m 1 is obtained from g m by inserting a new edge e at random specifically the conditional probability that e joins two currently disjoint vertices i and j is proportional to d i d j where d i d j are the degrees of i j in g m and 0 is a fixed parameter the limiting case is the erd s renyi graph process we show that whp g m contains a unique giant component iff c 2 m n c 1 and the size of this giant is asymptotic to n 1 c c where c c is the root of c c 2 c c 2 a phase transition window is proved to be contained essentially in c a n 1 3 c b n 1 4 and we conjecture that 1 4 may be replaced with 1 3 for the multigraph version mg m we show that mg m is connected whp iff m m n n 1 1 we conjecture that for 1 m n is the threshold for connectedness of g m itself

zero constant formula for first passage observables in bounded domains in this letter we develop an analytical approach which provides an explicit determination of mean first passage times mfpts for random walks in bounded domains for a wide class of transport processes in particular we derive for the first time explicit expressions of mfpts for emblematic models of transport in complex media such as diffusion on deterministic and random fractals this approach relies on a scale invariance hypothesis and a large volume expansion of the mfpt which actually proves to be very accurate even for small system sizes as shown by numerical simulations this explicit determination of mfpts can be straightforwardly generalized to further useful first passage observables such as occupation times and splitting probabilities

a random graph with a subcritical number of edges a random graph gn prob edge p p c n 0 ci 2j as n oo the random graph does not have a cycle of length 1 another result is that with probability approaching 1 the size of the vth largest component either equals an integer closest to alog bn v log5 2 n a a c b b c or is one less than this integer provided that v oo and v o n log5 2 n

efficiency of encounter controlled reaction between diffusing reactants in a finite lattice topology and boundary effects abstract the role of dimensionality euclidean vs fractal spatial extent boundary effects and system topology on the efficiency of diffusion reaction processes involving two simultaneously diffusing reactants is analyzed we present numerically exact values for the mean time to reaction as gauged by the mean walklength before reactive encounter obtained via application of the theory of finite markov processes and via monte carlo simulation as a general rule we conclude that for sufficiently large systems the efficiency of diffusion reaction processes involving two synchronously diffusing reactants two walker case relative to processes in which one reactant of a pair is anchored at some point in the reaction space one walker plus trap case is higher and is enhanced the lower the dimensionality of the system this differential efficiency becomes larger with increasing system size and for periodic systems its asymptotic value may depend on the parity of the lattice imposing confining boundaries on the system enhances the differential efficiency relative to the periodic case while decreasing the absolute efficiencies of both two walker and one walker plus trap processes analytic arguments are presented to provide a rationale for the results obtained the insights afforded by the analysis to the design of heterogeneous catalyst systems is also discussed

global first passage times of fractal lattices the global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time t we find simple expressions for the mean global first passage time for five fractals the d dimensional sierpinski gasket t fractal hierarchical percolation model mandelbrot given curve and a deterministic tree we also find an exact expression for the second moment and show that the variance of the first passage time var t scales with the number of nodes within the fractal n such that var t similar to n 4 d where d is the spectral dimension

inverted regions induced by geometric constraints on a classical encounter controlled binary reaction abstract the efficiency of an encounter controlled reaction between two independently mobile reactants on a lattice is characterized by the mean number n of steps to reaction the two reactants are distinguished by their mass with the light walker performing a jump to a nearest neighbor site in each time step while the heavy walker hops only with a probability p we associate p with the temperature of the system to account for geometric exclusion effects in the reactive event two reaction channels are specified for the walkers irreversible reaction occurs either in a nearest neighbor collision or when the two reactants attempt to occupy the same site lattices subject to periodic and to confining boundary conditions are considered for periodic lattices depending on the initial state the reaction time either falls off monotonically with p or displays a local minimum with respect to p occurrence of the latter signals a regime where the efficiency of the reaction effectively decreases with increasing temperature such behavior can also occur when one averages over all initial conditions but it disappears if the jump probability of the light walker falls below a characteristic threshold value even more robust behavior can occur on lattices subject to confining boundary conditions depending on the initial conditions the reaction time as a function of p may increase monotonically decrease monotonically display a single maximum or even a maximum and minimum in the latter case one can identify distinct regimes where the above noted inversion in reaction efficiency can occur we document both numerically and theoretically that these inversion regions are a consequence of a strictly classical interplay between excluded volume effects implicit in the specification of the two reaction channels and the system s dimensionality and spatial extent our results highlight situations where the description of an encounter controlled reactive event cannot be described by a single effective diffusion coefficient we also distinguish between the inversion region identified here and the marcus inverted region which arises in electron transfer reactions

scaling anomalies in the sol gel transition coagulating systems are characterized by the fact that aggregates of masses g and l react irreversibly to form a larger aggregate of mass the reaction rates for this process are assumed to depend only on the particle masses this model is described by the corresponding kinetic equations if the are assumed to grow fast enough it happens that at some finite time tc a fraction of the mass goes into a cluster of infinite size near and at the gel time the mass spectrum cg t can be described in scaling terms that is there exists a divergent typical mass in terms of which the spectrum can be expressed as well as two exponents describing the g dependence of the mass spectrum these exponents have been extensively studied both numerically and in terms of the case in which which is exactly solvable so far the exact model strongly suggested the validity of certain general expressions for these exponents in terms of the homogeneity degree of the these however were not confirmed numerically in the following we study the exact model in the case in which the initial conditions display an algebraic tail and show that in this case the scaling exponents can be calculated analytically and do not conform to the expressions initially suggested

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

metastability in a condensing zero range process in the thermodynamic limit zero range processes with decreasing jump rates are known to exhibit condensation where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value we study such a process on a one dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed super critical particle density we show that the process exhibits metastability with respect to the condensate location i e the suitably accelerated process of the rescaled location converges to a limiting markov process on the unit torus this process has stationary independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice our result extends previous work for fixed lattices and diverging density in beltran and landim probab theory relat fields 152 3 4 781 807 2012 and we follow the martingale approach developed there and in subsequent publications besides additional technical difficulties in estimating error bounds for transition rates the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells corresponding to a typical set of configurations with a particular condensate location the total exit rates from individual wells turn out to diverge in the limit which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells which is of a general nature and can be adapted to other models

a graph valued markov process as rings allowed polymerization model subcritical behavior in previous papers random tree type partitions of a finite set as a stochastic model for acyclic polymerization were studied in this paper the model is extended one important step further by allowing rings formation the main result is that in the subcritical domain the asymptotic behavior of the process remains essentially the same as in the acyclic case despite the fact that cycles are not excluded

exact mean first passage time on the t graph we consider a simple random walk on the t fractal and we calculate the exact mean time taug to first reach the central node i0 the mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph except i0 by means of analytic techniques based on decimation procedures we find the explicit expression for taug as a function of the generation g and of the volume v of the underlying fractal our results agree with the asymptotic ones already known for diffusion on the t fractal and more generally they are consistent with the standard laws describing diffusion on low dimensional structures

thermodynamic limit for the invariant measures in supercritical zero range processes we prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles it is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system we show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density this improves the result of groskinsky schutz and spohn where convergence is obtained for the finite dimensional marginals we obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk

condensation for a fixed number of independent random variables a family of m independent identically distributed random variables indexed by a chemical potential 0 represents piles of particles as increases to the mean number of particles per site converges to a maximal density c the distribution of particles conditioned on the total number of particles equal to n does not depend on canonical ensemble for fixed m as n goes to infinity the canonical ensemble measure behave as follows removing the site with the maximal number of particles the distribution of particles in the remaining sites converges to the grand canonical measure with density c the remaining particles concentrate condensate on a single site

localization for a random walk in slowly decreasing random potential we consider a continuous time random walk x in a random environment on such that its potential can be approximated by the function v given by v x sigma w x frac b 1 alpha x 1 alpha where w a brownian motion with diffusion coefficient 0 and parameters b are such that b 0 and 0 1 2 we show that p a s where p is the averaged law lim t to infty frac x t c ln ln t 1 ln t frac 1 alpha 1 with c frac 2 alpha b sigma 2 1 2 alpha in fact we prove that by showing that there is a trap located around c ln ln t 1 ln t frac 1 alpha with corrections of smaller order where the particle typically stays up to time t this is in sharp contrast to what happens in the pure sinai s regime where the location of this trap is random on the scale ln2t

metastability of reversible condensed zero range processes on a finite set let r s x s r be the jump rates of an irreducible random walk on a finite set s reversible with respect to some probability measure m for alpha 1 let g n r be given by g 0 0 g 1 1 g k k k 1 alpha k 2 consider a zero range process on s in which a particle jumps from a site x occupied by k particles to a site y at rate g k r x y let n stand for the total number of particles in the stationary state as n up arrow infinity all particles but a finite number accumulate on one single site we show in this article that in the time scale n1 alpha the site which concentrates almost all particles evolves as a random walk on s whose transition rates are proportional to the capacities of the underlying random walk

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

efficiency of encounter controlled reaction between diffusing reactants in a finite lattice non nearest neighbor effects abstract the influence of non nearest neighbor displacements on the efficiency of diffusion reaction processes involving one and two mobile diffusing reactants is studied an exact analytic result is given for dimension d 1 from which for large lattices one can recover the asymptotic estimate reported 30 years ago by lakatos lindenberg and shuler for dimensions d 2 3 we present numerically exact values for the mean time to reaction as gauged by the mean walklength before reactive encounter obtained via the theory of finite markov processes and supported by monte carlo simulations qualitatively different results are found between processes occurring on d 1 versus d 1 lattices and between results obtained assuming nearest neighbor only versus non nearest neighbor displacements

phase transition for perfect condensation and instability under the perturbations on jump rates of the zero range process let z z1 z2 zn represent the steady state of a zero range process in which n sites are occupied by m particles with a jump rate between sites given by g if m n a particle density of 1 and z n is the maximum cluster size perfect condensation occurs if n z n converges to 0 in probability as n tends to infinity in this paper we improve the description of the conditions for perfect condensation first introduced by jeon et al 2000 ann prob 28 1162 and jeon and march 2000 stochastic models proc int conf on stochastic models in honor of professor donald a dawson ottawa canada 10 13 june 1998 p 233 applying the results to a few special cases we demonstrate the existence of an interesting phase transition and conclude that the maximum cluster size in a zero range process is unstable with respect to fluctuations in the jump rate g

the strong approximation property and the weak bounded approximation property abstract we show that the strong approximation property strong ap respectively strong cap and the weak bounded approximation property respectively weak bcap are equivalent for every banach space this gives a negative answer to oja s conjecture as a consequence we show that each of the spaces c 0 and l 1 has a subspace which has the ap but fails to have the strong ap

effects of confinement on the statistics of encounter times exact analytical results for random walks in a partitioned lattice we study the effects of temporarily and permanently confining domains on the statistics of first passage times in finite lattices in one and two dimensions we present exact results for the mean and variance of the first passage time between arbitrary sites in the following 1 a finite one dimensional lattice partitioned into temporarily confining domains and 2 a finite two dimensional lattice with reflecting boundaries for a single random walker and an immobile target in the one dimensional case we also present the full first passage time distribution via numerical inversion of laplace transforms

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

condensation in stochastic particle systems with stationary product measures we study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities we review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit this leads to strengthened results on weak convergence for subcritical systems and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions extending previous results which focused on attractive and finite systems we use relative entropy techniques which provide simple proofs making use of general versions of local limit theorems for independent random variables

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed

time evolution of a random graph an initially empty no edges graph of the order of m is assumed to evolve by adding one edge at a time this edge can connect either two linked components and form a new component of a larger order coalescence of graphs or increase by one the number of edges in a given linked component the evolution equation for the generating functional of the probability to find in the system a given set of occupation numbers the numbers of graphs of the order of g having exactly edges at time t is formulated and solved exactly the expression for the graph composition spectrum is derived and analysed in the limit of large m

analytic expression for the mean time to absorption for a random walker on the sierpinski gasket the exact analytic expression for the mean time to absorption or mean walk length for a particle performing a random walk on a finite sierpinski gasket with a trap at one vertex is found to be t n 3 n 5 n 1 4 5 n ensuremath 3 n 3 n 1 1 where n denotes the generation index of the gasket and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket in terms of the number n n of sites on the gasket and the spectral dimension stackrel ifmmode tilde else fi d of the gasket the precise asymptotic behavior for large n n is t n ensuremath rightarrow 1 3 2n n 2 stackrel ifmmode tilde else fi d ensuremath sim n 1 464 this serves as a partial check on our result as it is a intermediate between the known results t ensuremath sim n 2 d 1 and t ensuremath sim n mathrm ln n d 2 for random walks on d dimensional euclidean lattices and b consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice

phase transitions on markovian bipartite graphs an application of the zero range process we analyze the existence and the size of the giant component in the stationary state of a markovian model for bipartite multigraphs in which the movement of the edge ends on one set of vertices of the bipartite graph is a zero range process the degrees being static on the other set the analysis is based on approximations by independent variables and on the results of molloy and reed for graphs with prescribed degree sequences the possible types of phase diagrams are identified by studying the behavior below the zero range condensation point as a specific example we consider the so called evans interaction in particular we examine the values of a critical exponent describing the growth of the giant component as the value of the dilution parameter controlling the connectivity is increased above the critical threshold rigorous analysis spans a large portion of the parameter space of the model exactly at the point of zero range condensation these results supplemented with conjectures supported by monte carlo simulations suggest that the phenomenological landau theory for percolation on graphs is not broken by the fluctuations

corrigendum and addendum to the strong approximation property and the weak bounded approximation property j funct anal 266 2014 5439 5447 abstract we correct a mistake in the proof of 2 theorem 1 1 and a proof of the main result in 2 is presented without depending on 2 theorem 1 1