For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...

Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of “word of mouth ” in the promotion of new products. Recently, motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target? We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here, and we provide the first provable approximation guarantees for efficient algorithms. Using an analysis framework based on submodular functions, we show that a natural greedy strategy obtains a solution that is provably within 63 % of optimal for several classes of models; our framework suggests a general approach for reasoning about the performance guarantees of algorithms for these types of influence problems in social networks. We also provide computational experiments on large collaboration networks, showing that in addition to their provable guarantees, our approximation algorithms significantly out-perform nodeselection heuristics based on the well-studied notions of degree centrality and distance centrality from the field of social networks.

We describe a simple one-person card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of Baik-Deift-Johansson which yields limiting probability laws via hard analysis of Toeplitz determinants. 1991 Mathematics Subject Classifications: Primary 60C05, 05E10, 15A52, 60F05. Research supported by N.S.F. Grant MCS 96-22859 1 Introduction This survey paper treats two themes in parallel. One theme is a purely mathematical question: describe the asymptotic law (probability distribution) of the length of the longest increasing subsequence of a random permutation. This question has been studied by a variety of increasingly technically sophisticated methods over the last 30 years. We outline three, apparently quite unrelated, methods in sections 2 - 4. The other theme is a card game, patience sorting. This gam...

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ‘‘phantom traffic jams’ ’ even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ‘‘freeze by heating’’? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well. CONTENTS

random-cluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the random-cluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey random-cluster measures from the probabilist’s point of view, giving clear statements of some of the many open problems. Secondly, we present new results for such measures, as follows. We discuss the relationship between weak limits of random-cluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of random-cluster measures for all but (at most) countably many values of the parameter p. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way, in part of these arguments. In the second part of this paper is constructed a Markov process whose level-sets are reversible Markov processes with random-cluster measures as unique equilibrium measures. This construction enables a coupling of random-cluster measures for all values of p. Furthermore it leads to a proof of the semicontinuity of the percolation probability, and provides a heuristic probabilistic justification for the widely held belief that there is a first-order phase transition if and only if the cluster-weighting factor q is sufficiently large. 1.

For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution # on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate normalizing constant. The same difficulty arises in computer science problems where one seeks to sample randomly from a large finite distributive lattice whose precise size cannot be ascertained in any reasonable amount of time. The Markov chain Monte Carlo (MCMC) approximate sampling approach to such a problem is to construct and run "for a long time" a Markov chain with long-run distribution #. But determining how long is long enough to get a good approximation can be both analytically and empirically difficult. Recently, Jim Propp and David Wilson have devised an ingenious and efficient algorithm to use the same Markov chains to produce perfect (i.e., exact) samples from #. However, the running t...

In a famous paper [8] Hammersley investigated the length Ln of the longest increasing subsequence of a random n-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly "soft" arguments that limn \Gamma1=2 ELn = 2. This is a known result, but previous proofs (Vershik - Kerov [14]; Logan - Shepp [11]) relied on hard analysis of combinatorial asymptotics. Mathematics subject classification. 60C05, 60K35. Running title. Hammersley's process. Research supported by N.S.F. Grant MCS 92-24857 and the Miller Institute for Basic Research in Science y Research supported by N.S.F. Grant DMS92-04864 1 Introduction An increasing subsequence i 1 ; i 2 ; : : : ; i k of a permutation i ! ß(i) is a subsequence such that i 1 ! i 2 ! : : : ! i k ; ß(i 1 ) ! ß(i 2 ) ! : : : ß(i k ): For instance, the permutation 7 2 8 1 3 4 10 6 9 5 (1) (for which ß(1) = 4; ß(2) =...

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ion. 60K35, 82C22, 82C24, 82C41. 1. Introduction. The nearest neighbor one dimensional asymmetric simple exclusion process is the Markov process t 2 f0; 1g ZZ that corresponds to the following description. At most one particle is allowed at each site. If there is a particle at site x, it jumps at rate p to site x + 1 if there is no particle in x + 1. Analogously if site x 1 is empty, the particle at site x jumps to x 1 at rate q. This process was introduced by Spitzer (1970) and has received a great deal of attention. Its ergodic properties are well understood (Liggett (1976, 1985)). The set of invariant measures is the set of convex combinations of the product measures and blocking measures. These measures concentrate on a denumerable set of con gurations and for p > q have asymptotic density 0 and 1 to the left and right of the origin, respectively. The hydrodynamical limit was studied by Andjel and Vares (1987) and extended by Benassi et al (1991) for monotone initial den

We consider a series of n single-server queues, each with unlimited waiting space and the first-in first-out service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n and/or k . There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavy-traffic limit theorem implies that D(k, n) satisfies an invariance principle as n , converging after normalization to a functional of k-dimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n as n . The case of k n = xn corresponds to a hydrodyna...