zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean curvature motion we consider the glauber dynamics for the ising model with boundary conditions at zero temperature or at a temperature that goes to zero with the system size hence the quotation marks in the title in dimension d 3 we prove that an initial domain of linear size l of spins disappears within a time which is at most l2 log l c and at least l2 c log l for some c 0 the proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe on large time scales the evolution of the interface between and domains the analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice our result apart from the spurious logarithmic factors is the first rigorous confirmation of the lifshitz law const l2 conjectured on heuristic grounds 8 13 in dimension d 2 can be shown to be of order l2 without logarithmic corrections the upper bound was proven in 6 and here we provide the lower bound for d 2 we also prove that the spectral gap of the generator behaves like for l large as conjectured in 2 2011 wiley periodicals inc

central limit theorems for infinite series of queues and applications to simple exclusion we prove that a tagged particle in asymmetric simple exclusion satisfies a central limit theorem when properly rescaled to obtain this result we derive several results of positive or negative correlation for occupation times of a server in a series of queues which imply various central limit theorems

a phase transition behavior for brownian motions interacting through their ranks consider a time varying collection of n points on the positive real axis modeled as exponentials of n brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time if at each time point we divide the points by their sum under suitable assumptions the rescaled point process converges to a stationary distribution depending on n and the vector of drifts as time goes to infinity this stationary distribution can be exactly computed using a recent result of pal and pitman the model and the rescaled point process are both central objects of study in models of equity markets introduced by banner fernholz and karatzas in this paper we look at the behavior of this point process under the stationary measure as n tends to infinity under a certain continuity at the edge condition on the drifts we show that one of the following must happen either i all points converge to 0 or ii the maximum goes to 1 and the rest go to 0 or iii the processes converge in law to a non trivial poisson dirichlet distribution the underlying idea of the proof is inspired by talagrand s analysis of the low temperature phase of derrida s random energy model of spin glasses the main result establishes a universality property for the bfk models and aids in explicit asymptotic computations using known results about the poisson dirichlet law

diffusive limit of a tagged particle in asymmetric simple exclusion processes invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d 3 the method of proof is by martingale techniques which rely on the fact that symmetric random walks are transient in high dimensions c 2000 john wiley sons inc

current fluctuations of the one dimensional symmetric simple exclusion process with step initial condition for the symmetric simple exclusion process on an infinite line we calculate exactly the fluctuations of the integrated current qt during time t through the origin when in the initial condition the sites are occupied with density a on the negative axis and with density b on the positive axis all the cumulants of qt grow like sqrt t in the range where q t sim sqrt t the decay exp qt3 t of the distribution of qt is non gaussian our results are obtained using the bethe ansatz and several identities derived recently by tracy and widom for exclusion processes on the infinite line

randomized polynuclear growth with a columnar defect we study a variant of poly nuclear growth where the level boundaries perform continuous time discrete space random walks and study how its asymptotic behavior is affected by the presence of a columnar defect on the line we prove that there is a non trivial phase transition in the strength of the perturbation above which the law of large numbers for the height function is modified

hitting times for independent random walks on ℤd we consider a system of asymmetric independent random walks on z d denoted by t t e r stationary under the product poisson measure v p of marginal density p 0 we fix a pattern a an increasing local event and denote by the hitting time of a by using a loss network representation of our system at small density we obtain a coupling between the laws of t conditioned on r t for all times t when d 3 this provides bounds on the rate of convergence of the law of t conditioned on r t toward its limiting probability measure as t tends to infinity we also treat the case where the initial measure is close to v without being product

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

on two types of obstructed random walks abstract two examples of random walks on a lattice which are subject to obstructing conditions are discussed i a random walk on a d dimensional lattice containing a periodic array of traps ii a random walk in a set of colliding random walks in d 1 expressions are derived for the probability distribution of the lattice point where the walk comes to an end and the average total number of distinct lattice points visited in example i and for the probability distribution of a walker s displacement in particular the mean square displacement for long times in example ii

occupation time large deviations of two dimensional symmetric simple exclusion process we prove a large deviations principle for the occupation time of a site in the two dimensional symmetric simple exclusion process the decay probability rate is of order t logt and the rate function is given by 2 sin 1 2 1 sin 1 2 1 2 the proof relies on a large deviations principle for the polar empirical measure which contains an interesting log scale spatial average a contraction principle permits us to deduce the occupation time large deviations from the large deviations for the polar empirical measure

tracer diffusion in lattice gases it has been proved that a tracer particle in a reversible lattice gas converges to brownian motion however only in a few particular cases has a strictly positive self diffusion coefficientd been established here we supply the missing piece and show thatd 0 in general the exceptions are one dimensional lattice gases with nearest neighbor jumps only for whichd 0 the proof establishes a variational formula ford which could be used to obtain realistic bounds

self diffusion in simple models systems with long range jumps we review some exact results for the morion of a tagged particle in simple models then we study the density dependence of the sill diffusion coefficient d n in lattice systems with simple symmetric exclusion in which the particles can jump with equal rates to a set of n neighboring sites we obtain positive upper and lower bounds on f n n 1 d n d n 0 1 for is an element of 0 1 computer simulations for the square triangular and one dimensional lattices suggest that fn becomes effectively independent of n for n greater than or equal to 20

existence of quasi stationary distributions a renewal dynamical approach we consider markov processes on the positive integers for which the origin is an absorbing state quasi stationary distributions qsd s are described as fixed points of a transformation phi in the space of probability measures under the assumption that the absorption time at the origin r of the process starting from state x goes to infinity in probability as x rightarrow infty we show that the existence of a operatorname qsd is equivalent to e xe lambda r infty for some positive lambda and x we also prove that a subsequence of phi n delta x converges to a minimal operatorname qsd for a birth and death process we prove that phi n delta x converges along the full sequence to the minimal operatorname qsd the method is based on the study of the renewal process with interarrival times distributed as the absorption time of the markov process with a given initial measure mu the key tool is the fact that the residual time in that renewal process has as stationary distribution the distribution of the absorption time of phi mu

exclusion processes with degenerate rates convergence to equilibrium and tagged particle stochastic lattice gases with degenerate rates namely conservative particle systems where the exchange rates vanish for some configurations have been introduced as simplified models for glassy dynamics we introduce two particular models and consider them in a finite volume of size l in contact with particle reservoirs at the boundary we prove that as for non degenerate rates the inverse of the spectral gap and the logarithmic sobolev constant grow as l2 it is also shown how one can obtain via a scaling limit from the logarithmic sobolev inequality the exponential decay of a lyapunov functional for a degenerate parabolic differential equation porous media equation we analyze finally the tagged particle displacement for the stationary process in infinite volume in dimension larger than two we prove that in the diffusive scaling limit it converges to a brownian motion with non degenerate diffusion coefficient

improving the asymptotically necessary and sufficient conditions for stability of fuzzy control this paper improves some results in antonio sala carlos arino asymptotically necessary and sufficient conditions for stability and performance in fuzzy control applications of polya s theorem fuzzy sets and systems 158 2007 2671 2686 in the relaxation of the stability results only a smaller number of artificial decision variables are required thus we obtain the equivalent stability results with lower requirement on the computation

hydrodynamics and large deviation for simple exclusion processes we prove the hydrodynamical limit for weakly asymmetric simple exclusion processes a large deviation property with respect to this limit is established for the symmetric case we treat also the situation where a slow reaction creation and annihilation of particles is present

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

exclusion process for particles of arbitrary extension hydrodynamic limit and algebraic properties the behaviour of extended particles with exclusion interaction on a one dimensional lattice is investigated the basic model is called l asep as a generalization of the asymmetric exclusion process asep to particles of arbitrary length l stationary and dynamical properties of the l asep with periodic boundary conditions are derived in the hydrodynamic limit from microscopic properties of the underlying stochastic many body system in particular the hydrodynamic equation for the local density evolution and the time dependent diffusion constant of a tracer particle are calculated as a fundamental algebraic property of the symmetric exclusion process the su 2 symmetry is generalized to the case of extended particles

large deviations for noninteracting infinite particle systems a large deviation property is established for noninteracting infinite particle systems previous large deviation results obtained by the authors involved a singlei function because the cases treated always involved a unique invariant measure for the process in the context of this paper there is an infinite family of invariant measures and a corresponding infinite family ofi functions governing the large deviations

shock fluctuations in the asymmetric simple exclusion process we consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively q p ferrari et al 1991 have shown that if the initial measure isv a product measure with densities and to the left and right of the origin respectively then there exists a microscopic shock for the system a shock is a random positionx t such that the system as seen from this position at timet has asymptotic product distributions with densities and to the left and right of the origin respectively uniformly int we compute the diffusion coefficient of the shockd lim t t 1 e x t 2 ex t 2 and findd p q 1 1 1 as conjectured by spohn 1991 we show that in the scale sqrt t the position ofx t is determined by the initial distribution of particles in a region of length proportional tot we prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities and this is the so called dynamical phase transition under shock initial conditions we show how the density fluctuation fields depend on the initial configuration

on the dirichlet problem for asymmetric zero range process on increasing domains in order to obtain hitting time estimates for the asymmetric zero range process azrp on d in dimensions d 3 we characterize the principal eigenvalue of the generator of the azrp with dirichlet boundary on special domains we obtain a donsker varadhan variational representation and show that the corresponding eigenfunction is unique in a natural class of functions

generalized notions of amenability ii this paper continues the investigation of the first two authors begun in part i it is shown that approximate amenability and approximate contractibility are the same properties as are uniform approximate amenability and amenability bounded approximate contractibility and bounded approximate amenability are characterized by the existence of suitable operator bounded approximate identities for the diagonal ideal results are given on banach sequence algebras lipschitz algebras and beurling algebras and on the crucial role of approximate identities a new proof is given for a result due to n gronbaek on characterizing of amenability for beurling algebras

approximate identities in approximate amenability abstract we answer several open questions in the theory of approximate amenability for banach algebras first we give examples of banach algebras which are boundedly approximately amenable but which do not have bounded approximate identities this answers a question open since the year 2000 when ghahramani and loy founded the notion of approximate amenability we give a nice condition for a c o direct sum of amenable banach algebras to be approximately amenable which gives us a reasonably large and varied class of such examples then we examine our examples in some detail and thereby find answers to other open questions the two notions of bounded approximate amenability and bounded approximate contractibility are not the same the direct sum of two approximately amenable banach algebras does not have to be approximately amenable and a 1 codimensional closed ideal in a boundedly approximately amenable banach algebra need not be approximately amenable

regularity of quasi stationary measures for simple exclusion in dimension d 5 we consider the symmetric simple exclusion process on zz d for d geq 5 and study the regularity of the quasi stationary measures of the dynamics conditioned on not occupying the origin for each rho in 0 1 we establish uniqueness of the density of quasi stationary measures in l 2 d nur where nur is the stationary measure of density rho this in turn permits us to obtain sharp estimates for p nur tau t where tau is the first time the origin is occupied

approximate amenability of tensor products of banach algebras abstract examples constructed by the first author and charles read make it clear that many of the hereditary properties of amenability no longer hold for approximate amenability these and earlier results of the authors also show that the presence of a bounded approximate identity often entails positive results here we show that the tensor product of approximately amenable algebras need not be approximately amenable and investigate conditions under which a and b being approximately amenable implies or is implied by a b or a b being approximately amenable once again the role of having a bounded approximate identity comes to the fore our methods also enable us to prove that if a b is amenable then so are a and b a result proved by barry johnson in 1996 under an additional assumption

correlation inequalities for interacting particle systems with duality we prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other a process which we call here the symmetric inclusion process sip or repel each other a generalized version of the well known symmetric exclusion process as an application new correlation inequalities are obtained for the sip as well as for some interacting diffusions which are used as models of heat conduction the so called brownian momentum process and the brownian energy process these inequalities are counterparts of the inequalities in the opposite direction for the symmetric exclusion process showing that the sip is a natural bosonic analogue of the symmetric exclusion process which is fermionic finally we consider a boundary driven version of the sip for which we prove duality and then obtain correlation inequalities

t e harris contributions to interacting particle systems and percolation this is a bird s eye view of t e harris work on interacting particle systems and percolation and of its impact on later work in probability theory and mathematical physics nal illness and death was a great loss to me personally and professionally as well as to probability theory in southern california and beyond for three decades he and i were the primary probabilists at usc and ucla respectively our approach to mathematics and our mathematical interests were very similar my wife chris and i enjoyed many wonderful social occasions at the harris home in beverly hills with ted and connie they were great hosts the conversation was always stimulating they were interested in so many things ted initiated the southern california probability symposium in about 1970 it is probably the oldest meeting of its type in the us and continues to provide an exciting place for interactions among southern california probabilists to this day in two of the years 1989 and 2007 the meeting was dedicated to ted the rst was on the occasion of his 70th birthday and the second was in his memory even after his retirement ted attended the usc probability seminar regularly always sitting in the front row and asking perceptive questions few mathematicians have had a greater ratio of number of ideas to number of papers ted wrote many fewer papers 31 than most prominent mathematicians but each is a jewel in each of the several areas in which he worked he was among the

large deviations for poisson systems of independent random walks we prove large deviation theorems for occupation time functionals of independent random walks started from a poisson field on zd in dimensions 1 and 2 the large deviation tails are larger than exponential exact asymptotics are derived

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

clustering in the one dimensional three color cyclic cellular automaton cyclic cellular automaton the author has previously shown that this process fluctuates meaning that each lattice site changes color infinitely often so that there is no final state for the system the focus of the current work is on the clustering properties of this system this paper demonstrates that the one dimensional three color cyclic cellular automaton clusters and the mean cluster size as a function of time t is asymptotic to ct1 2 where c is an explicitly calculable constant the method of proof also allows us to compute asymptotic estimates of the mean interparticle distance for a one dimensional system of particles which undergo deterministic motion and which annihilate upon collision no clustering results are known about the four color process but evidence is presented to suggest that the mean cluster size of such systems grows at a rate different from t1 2

large deviations of interacting particle systems in infinite volume we prove a large deviation theorem from the hydrodynamical limit for the empirical measure of ginzburg landau and zero range processes in infinite volume starting from deterministic initial configurations in the ginzburg landau case the main tool is the study of the evolution of the h 1 norm and in the zero range case the attractiveness which allows couplings

multi species simple exclusion processes abstract a motility mechanism based on a simple exclusion process where the movement of discrete agents on a lattice is either unbiased symmetric or biased asymmetric is considered estimates of diffusivities from tracking data do not describe the population level response of the system this mismatch between the individual level and population level behaviour can be resolved by averaging the individual level mechanism in terms of an expected site occupancy new insight into simple exclusion processes is obtained by representing the system as a series of interacting subpopulations this formalism leads to a system of nonlinear advection diffusion equations which can be interpreted in terms of the agent fluxes these interactions have consequences for both agent based modelling and continuum modelling in cell biology such as tracking subpopulations of cells within a total cell population

equilibrium fluctuation of the atlas model we study the fluctuation of the atlas model where a unit drift is assigned to the lowest ranked particle among a semi infinite z z indexed system of otherwise independent brownian particles initiated according to a poisson point process on r r in this context we show that the joint law of ranked particles after being centered and scaled by t 14t 14 converges as t t to the gaussian field corresponding to the solution of the additive stochastic heat equation ashe on r r with the neumann boundary condition at zero this allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional brownian motion fbm in particular we prove a conjecture of pal and pitman ann appl probab 18 2008 2179 2207 about the asymptotic gaussian fluctuation of the ranked particles

power law polya s urn and fractional brownian motion we introduce a natural family of random walks s n on mathbb z that scale to fractional brownian motion the increments x n s n s n 1 in pm 1 have the property that given x k k n the conditional law of x n is that of x n k n where k n is sampled independently from a fixed law mu on the positive integers when mu has a roughly power law decay precisely when mu lies in the domain of attraction of an alpha stable subordinator for 0 alpha 1 2 the walks scale to fractional brownian motion with hurst parameter alpha 1 2 the walks are easy to simulate and their increments satisfy an fkg inequality in a sense we describe they are the natural fractional analogues of simple random walk on mathbb z

approximate amenability is not bounded approximate amenability abstract we obtain a new criterion sufficient for approximate amenability of banach algebras and an associated criterion sufficient for approximate amenability of c 0 direct sums of approximately amenable banach algebras we use these criteria to give examples of banach algebras which are approximately amenable but not boundedly approximately amenable thus we answer a question which was open since the year 2004 this is so to speak the complementary result to one of our earlier ones in the paper f ghahramani c j read approximate identities in approximate amenability j funct anal 262 9 2012 3929 3945 where we gave examples of boundedly approximately amenable banach algebras that are not boundedly approximately contractible

current fluctuations of a system of one dimensional random walks in random environment we study the current of particles that move independently in a common static random environment on the one dimensional integer lattice a two level fluctuation picture appears on the central limit scale the quenched mean of the current process converges to a brownian motion on a smaller scale the current process centered at its quenched mean converges to a mixture of gaussian processes these gaussian processes are similar to those arising from classical random walks but the environment makes itself felt through an additional brownian random shift in the spatial argument of the limiting current process

scaling limits of a tagged particle in the exclusion process with variable diffusion coefficient we prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in z with variable diffusion coefficient the scaling limits are obtained from a similar result for the current through 1 2 for a zero range process with bond disorder for the clt we prove convergence to a fractional brownian motion of hurst exponent 1 4

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

occupation time large deviations of the voter model this paper is a sequel to 5 and 6 we continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so called voter model t see e g 11 or 8 in keeping with our previous results we show that the large deviations are classical in high dimensions d 5 for t but fat in low dimensions d 4 interaction distinguishes the voter model from the independent particle systems of 5 and 6 and consequently exact computations no longer seem feasible instead we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails

approximate and pseudo amenability of various classes of banach algebras we continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with r j loy it is shown that every boundedly approximately contractible banach algebra has a bounded approximate identity and that the fourier algebra of the free group on two generators is not operator approximately amenable further examples are obtained of l1 semigroup algebras which are approximately amenable but not amenable using these we show that bounded approximate contractibility need not imply sequential approximate amenability results are also given for segal algebras on locally compact groups and algebras of p pseudo functions on discrete groups

quasi stationary measures for conservative dynamics in the infinite lattice we study quasi stationary measures for conservative particle systems in the in finite lattice existence of quasi stationary measures is established for a fairly general class of reversible systems for the special cases of a system of independent random walks and the symmetric simple exclusion process it is shown that qualitative features of quasi stationary measures change drastically with dimension

quasi stationary distributions and diffusion models in population dynamics in this paper we study quasi stationarity for a large class of kolmogorov diffusions the main novelty here is that we allow the drift to go to infty at the origin and the diffusion to have an entrance boundary at infty these diffusions arise as images by a deterministic map of generalized feller diffusions which themselves are obtained as limits of rescaled birth death processes generalized feller diffusions take nonnegative values and are absorbed at zero in finite time with probability 1 an important example is the logistic feller diffusion we give sufficient conditions on the drift near 0 and near infty for the existence of quasi stationary distributions as well as rate of convergence in the yaglom limit and existence of the q process we also show that under these conditions there is exactly one quasi stationary distribution and that this distribution attracts all initial distributions under the conditional evolution if and only if infty is an entrance boundary in particular this gives a sufficient condition for the uniqueness of quasi stationary distributions in the proofs spectral theory plays an important role on l 2 of the reference measure for the killed process

pointwise ergodic theorems for the symmetric exclusion process almost sure convergence theorems are proved for cesaro averages of continous functions in the case of the symmetric exclsion processes in dimension d 3 for the occupation time of a single site the same result is proved in all dimensions

asymptotically necessary and sufficient conditions for stability and performance in fuzzy control applications of polya s theorem stability and performance requirements in fuzzy control of takagi sugeno systems are usually stated as fuzzy summations i e sums of terms related to lyapunov functions which are weighted by membership functions or products of them this paper presents an application to fuzzy control of polya s theorems on positive forms on the standard simplex the result provides a set of progressively less conservative sufficient conditions for proving positivity of fuzzy summations such conditions are less and less conservative as a complexity parameter n increases particular cases of such conditions are those in c h fang y s liu s w kau l hong c h lee a new lmi based approach to relaxed quadratic stabilization of t s fuzzy control systems ieee trans fuzzy systems 14 2006 286 397 x liu q zhang new approaches to h controller designs based on fuzzy observers for t s fuzzy systems via lmi automatica 39 9 2003 1571 1582 with n 2 and 3 respectively the proposed conditions are asymptotically exact i e necessary and sufficient when n tends to infinity or equivalently when a tolerance parameter tends to zero

driven tracer particle in one dimensional symmetric simple exclusion consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction we investigate the behavior of a tagged particle under the action of an external constant driving force we prove that the diffusively rescaled position of the test particle ex e 2t t 0 converges in probability as e 0 to a deterministic function v t the function v depends on the initial distribution of the random environment through a non linear parabolic equation this law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin an einstein relation is satisfied asymptotically when the external force is small

tracer diffusion in stochastic lattice gases we report on rigorous results for self diffusion in stochastic lattice gases 1 scaling limit and convergence to brownian motion ii a variational formula for the tracer diffusion coefficient d iii strict positivity of d except for lattice gases in one dimension with nearest neighbor jumps only iv long time tails

hitting times for special patterns in the symmetric exclusion process on z d we consider the symmetric exclusion process eta t t 0 on 0 1 z d we fix a pattern a eta sum lambda eta i ge k where lambda is a finite subset of z d and k is an integer and we consider the problem of establishing sharp estimates for tau the hitting time of a we present a novel argument based on monotonicity which helps in some cases to obtain sharp tail asymptotics for tau in a simple way also we characterize the trajectories eta s s le t conditioned on tau t

diffusion of finite sized hard core interacting particles in a one dimensional box tagged particle dynamics we solve a nonequilibrium statistical mechanics problem exactly namely the single file dynamics of n hard core interacting particles the particles cannot pass each other of size delta diffusing in a one dimensional system of finite length l with reflecting boundaries at the ends we obtain an exact expression for the conditional probability density function rho t yt t vertical bar yt 0 that a tagged particle t t 1 n is at position yt at time t given that it at time t 0 was at position yt 0 using a bethe ansatz we obtain the n particle probability density function and by integrating out the coordinates and averaging over initial positions of all particles but particle t we arrive at an exact expression for rho t yt t vertical bar yt 0 in terms of jacobi polynomials or hypergeometric functions going beyond previous studies we consider the asymptotic limit of large n maintaining l finite using a nonstandard asymptotic technique we derive an exact expression for rho t yt t vertical bar yt 0 for a tagged particle located roughly in the middle of the system from which we find that there are three time regimes of interest for finite sized systems a for times much smaller than the collision time t tau coll 1 rho d 2 where rho n l is the particle concentration and d is the diffusion constant for each particle the tagged particle undergoes a normal diffusion b for times much larger than the collision time t tau coll but times smaller than the equilibrium time t tau eq l 2 d we find a single file regime where rho t yt t vertical bar yt 0 is a gaussian with a mean square displacement scaling as t 1 2 and c for times longer than the equilibrium time t tau eq rho t yt t vertical bar yt 0 approaches a polynomial type equilibrium probability density function notably only regimes a and b are found in the previously considered infinite systems

negative dependence and the geometry of polynomials we introduce the class of strongly rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter this class covers important models such as determinantal measures e g product measures and uniform random spanning tree measures and distributions for symmetric exclusion processes we show that strongly rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to liggett pemantle and wagner respectively moreover we extend lyons recent results on determinantal measures and we construct counterexamples to several conjectures of pemantle and wagner on negative dependence and ultra log concave rank sequences

some properties of k step exclusion processes we introduce k step exclusion processes as generalizations of the simple exclusion process we state their main equilibrium properties when the underlying stochastic matrix corresponds to a random walk or is positive recurrent and reversible finally we prove laws of large numbers for tagged and second class particles

meteor process on mathbb z d the meteor process is a model for mass redistribution on a graph the case of finite graphs was analyzed in billey et al on meteors earthworms and wimps ann appl probab 2014 this paper is devoted to the meteor process on mathbb z d the process is constructed and a stationary distribution is found convergence to this stationary distribution is proved for a large family of initial distributions the first two moments of the mass distribution at a vertex are computed for the stationary distribution for the one dimensional lattice mathbb z the net flow of mass between adjacent vertices is shown to have bounded variance as time goes to infinity an alternative representation of the process on mathbb z as a collection of non crossing paths is presented the distributions of a tracer particle in this system of non crossing paths are shown to be tight as time goes to infinity

levy flights in nonhomogeneous media distributed order fractional equation approach a jumping process defined in terms of the l evi distributed jumping size and the poissonian position dependent waiting time with the algebraic jumping rate is discussed on the assumption that parameters of both distributions are themselves random variables which are determined from given probability distributions the fractional equation for the distributed l evy order parameter ensuremath mu is derived and solved the solution is of the form of a combination of the fox functions and simple scaling is lacking the problem of accelerated diffusion is also discussed the case of the distributed waiting time parameter ensuremath theta is similarly solved and the solution offers a possibility to manage processes which are characterized by more general forms of the jumping rate not only algebraic moreover we mention a possibility that the parameters ensuremath mu and ensuremath theta are mutually dependent

a triangulation approach to asymptotically exact conditions for fuzzy summations many takagi sugeno t s fuzzy control synthesis problems in the literature are expressed as the problem of finding decision variables in a double convex sum fuzzy summation of positive definite matrices matricespsila coefficients in the summation take values in the standard simplex this paper presents a triangulation approach to the problem of generating simplicial partitions of the standard simplex in order to set up a family of sufficient conditions and in parallel another family of necessary ones for fuzzy summations the conditions proposed in this paper are asymptotically exact as the size of the involved simplices decreases its conservativeness vanishes for a sufficiently fine partition sufficiently dense mesh of vertex points the set of conditions is in the form of linear matrix inequalities lmis for which efficient software is available