alternative proof for the localization of sinai s walk we give an alternative proof of the localization of sinai s random walk in random environment under weaker hypothesis than the ones used by sinai moreover we give estimates that are stronger than the one of sinai on the localization neighborhood and on the probability for the random walk to stay inside this neighborhood

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

note to the asymptotic behaviour of solutions of damped pendulum equations under forcing although the mathematical or the physical pendulum seems to be one of the most popular objects in modelling see the nice exposition lo and therefore no wonder that for the related autonomous equations studied by means of the planar geometrical methods a vast literature exists see the monographs 2 3 151 or more recently the beautiful modern article 4 there are still several white places of its study in the case of forcing recently the big activity is devoted for proving the existence of harmonics to its equation for the survey of results in this field see s lo or chaotic behaviour of the dynamics see e g 6 or the different approaches in 12 is nevertheless we have found only one reference 16 on the asymptotic behaviour of solutions of the pendulum type equations under forcing which is already 30 years old hence our aim is to extend the results in this field for the equation

quenched annealed and functional large deviations for one dimensional random walk in random environment suppose that the integers are assigned random variables i taking values in the unit interval which serve as an environment this environment defines a random walk x n called a rwre which when at i moves one step to the right with probability i and one step to the left with probability 1 i when the i sequence is i i d greven and den hollander 1994 proved a large deviation principle for x n n conditional upon the environment with deterministic rate function we consider in this paper large deviations both conditioned on the environment quenched and averaged on the environment annealed for the rwre in the ergodic environment case the annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy we also give a detailed qualitative description of the resulting rate functions our techniques differ from those of greven and den hollander and allow us to present also a trajectorial quenched large deviation principle

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

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

the limits of sinai s simple random walk in random environment we study the sample path asymptotics of a class of recurrent diffusion processes with random potentials including examples of sinai s simple random walk in random environment and brox s diffusion process with brownian potential the main results consist of several integral criteria which completely characterize all the possible levy classes therefore providing a very precise image of the almost sure asymptotic behaviors of these processes

on the irrelevant disorder regime of pinning models recent results have lead to substantial progress in understanding the role of disorder in the de localization transition of polymer pinning models notably there is an understanding of the crucial issue of disorder relevance and irrelevance that albeit still partial is now rigorous in this work we exploit interpolation and replica coupling methods to get sharper results on the irrelevant disorder regime of pinning models in particular we compute in this regime the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal ex pansion obtained by field theory methods we also show that the quenched and the quenched averaged correlation length exponents coincide while in general they are ex pected to be different interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals 2000 mathematics subject classification 60k35 60k05 82b41 82b44

the saito kurokawa lifting and functoriality certain nontempered liftings from pgl 2 pgl 2 to pgsp 4 are constructed using the theory of local and global theta lifts the resulting representations on pgsp 4 are the saito kurokawa representations the lifting is shown to be functorial under certain reasonable assumptions on the local langlands correspondence for pgsp 4

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

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

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

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

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

zero white noise limit through dirichlet forms with application to diffusions in a random medium we study the zero white noise limit for diffusions in a continuous multidimensional medium given a continuous function on n w we consider diffusions whose drift term is the gradient ofw and whose diffusion coefficient is constant equal to e we describe the asymptotics of the exit time from a domain and of the law of the process when e tends to zero by applying these results to a random self similar mediumw we prove limit theorems for a diffusion in a random medium our theorems agree with results usually proved through the large deviation principle although in our setup this last tool is not available we extend to the multidimensional case properties of diffusions in a random medium already known in one dimension

on slowdown and speedup of transient random walks in random environment we consider one dimensional random walks in random environment which are transient to the right our main interest is in the study of the sub ballistic regime where at time n the particle is typically at a distance of order o n from the origin kappa in 0 1 we investigate the probabilities of moderate deviations from this behaviour specifically we are interested in quenched and annealed probabilities of slowdown at time n the particle is at a distance of order o n nu 0 from the origin nu 0 in 0 kappa and speedup at time n the particle is at a distance of order n nu 1 from the origin nu 1 in kappa 1 for the current location of the particle and for the hitting times also we study probabilities of backtracking at time n the particle is located around n thus making an unusual excursion to the left for the slowdown our results are valid in the ballistic case as well

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

large deviations for a random walk in random environment let px x z be an i i d collection of 0 1 valued random variables given let x n n 0 be the markov chain on z defined by x 0 0 and x n 1 x n 1 resp x n n 1 with probability px n resp 1 px n it is shown that x n n satisfies a large deviation principle with a continuous rate function that is lim n 1 n log p x n n n i a s for n 1 1 first we derive a representation of the rate function i in terms of a variational problem second we solve the latter explicitly in terms of random continued fractions this leads to a classification and qualitative description of the shape of i in the recurrent case i is nonanalytic at 0 in the transient case i is nonanalytic at c 0 c for some c 0 with linear pieces in between

local and global maass relations we characterize the irreducible admissible spherical representations of mathrm gsp 4 f where f is a p adic field that occur in certain cap representations in terms of relations satisfied by their spherical vector in a special bessel model these local relations are analogous to the maass relations satisfied by the fourier coefficients of siegel modular forms of degree 2 in the image of the saito kurokawa lifting we show how the classical maass relations can be deduced from the local relations in a representation theoretic way without recourse to the construction of saito kurokawa lifts in terms of fourier coefficients of half integral weight modular forms or jacobi forms as an additional application of our methods we give a new characterization of saito kurokawa lifts involving a certain average of fourier coefficients

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

localization of random walks in one dimensional random environments we consider a random walk on the one dimensional semi lattice 0 1 2 we prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment the size of this neighbourhood is estimated the limit parameters of the walks are also determined

moderate deviations for diffusions with brownian potentials we present precise moderate deviation probabilities in both quenched and annealed settings for a recurrent diffusion process with a brownian potential our method relies on fine tools in stochastic calculus including kotani s lemma and lamperti s representation for exponential functionals in particular our result for quenched moderate deviations is in agreement with a recent theorem of comets and popov probab theory related fields 126 2003 571 609 who studied the corresponding problem for sinai s random walk in random environment

on classical saito kurokawa liftings there exist two different generalizations of the classical saito kurokawa lifting to modular forms with square free level one lifting produces modular forms with respect to 0 m the other one with respect to the paramodular group m we shall give an alternative and unified construction of both liftings using group theoretic methods the construction shows that a single elliptic modular form may in fact have many saito kurokawa liftings we also obtain precise information about the spin l function of the resulting siegel modular forms

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

annealed tail estimates for a brownian motion in a drifted brownian potential we study brownian motion in a drifted brownian potential kawazu and tanaka j math soc japan 49 1997 189 211 exhibited two speed regimes for this process depending on the drift they supplemented these laws of large numbers by central limit theorems which were recently completed by hu shi and yor trans amer math soc 351 1999 3915 3934 using stochastic calculus we studied large deviations ann probab 29 2001 1173 1204 showing among other results that the rate function in the annealed setting that is after averaging over the potential has a flat piece in the ballistic regime in this paper we focus on this subexponential regime proving that the probability of deviating below the almost sure speed has a polynomial rate of decay and computing the exponent in this power law this provides the continuous time analogue of what dembo peres and zeitouni proved for the transient random walk in random environment comm math phys 181 1996 667 683 our method takes a completely different route making use of lamperti s representation together with an iteration scheme

limit law for transition probabilities and moderate deviations for sinai s random walk in random environment we consider a one dimensional random walk in random environment in the sinai s regime our main result is that logarithms of the transition probabilities after a suitable rescaling converge in distribution as time tends to infinity to some functional of the brownian motion we compute the law of this functional when the initial and final points agree also among other things we estimate the probability of being at time t at distance at least z from the initial position when z is larger than ln2t but still of logarithmic order in time

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

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 limit distribution of sinai s random walk in random environment abstract recently sinai proved that if x n is a one dimensional random walk in random environment which is recurrent then log n 2 x n converges in distribution here we calculate the limit distribution explicitly

metastability in stochastic dynamics of disordered mean field models we study a class of markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding gibbs measures we derive the analog of the wentzell freidlin theory in this case showing that any transition can be decomposed with probability exponentially close to one into a deterministic sequence of admissible transitions for these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor the distributions of the rescaled transition times are shown to converge to the exponential distribution we exemplify our results in the context of the random field curie weiss model

siegel cusp forms of degree 2 are determined by their fundamental fourier coefficients we prove that a siegel cusp form of degree 2 for the full modular group is determined by its set of fourier coefficients a s with 4 det s ranging over odd squarefree integers as a key step to our result we also prove that a classical cusp form of half integral weight and level 4n with n odd and squarefree is determined by its set of fourier coefficients a d with d ranging over odd squarefree integers a result that was previously known only for hecke eigenforms

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

lyapunov functions for random walks and strings in random environment we study two typical examples of countable markov chains in random environment using the lyapunov functions method random walk and random string in random environment in each case we construct an explicit lyapunov function investigating the behavior of this function we get the classification for recurrence transience ergodicity we obtain new results for random strings in random environment though we simply review well known results for random walks using our approach

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

spectral analysis of sinai s walk for small eigenvalues sinai s walk can be thought of as a random walk on z with random potential v with v weakly converging under diffusive rescaling to a two sided brownian motion we consider here the generator l n of sinai s walk on n n o z with dirichlet conditions on n n by means of potential theory for each h 0 we show the relation between the spectral properties of l n for eigenvalues of order o exp h n and the distribution of the h extrema of the rescaled potential v n x v nx n defined on 1 1 information about the h extrema of v n is derived from a result of neveu and pitman concerning the statistics of h extrema of brownian motion as first application of our results we give a proof of a refined version of sinai s localization theorem