In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X =^d g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(\cdot) is essentially a ``maximum'' or ``minimum'' function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process X_i, are the X_i measurable functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

ABSTRACT:In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494 · · · ) n -many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximatelyn , for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well.As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree COUNTING WITHOUT SAMPLING 453graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.

Abstract. In this work we introduce a new type of urn model with infinite but countable many colors indexed by an appropriate infinite set. We mainly consider the indexing set of colors to be the d-dimensional integer lattice and consider balanced replacement schemes associated with bounded increment random walks on it. We prove central and local limit theorems for the random color of the n-th selected ball and show that irrespective of the null recurrent or transient behavior of the underlying random walks, the asymptotic distribution is Gaussian after appropriate centering and scaling. We show that the order of any nonzero centering is always O (log n) and the scaling is O √ log n . The work also provides similar results for urn models with infinitely many colors indexed by more general lattices in R d . We introduce a novel technique of representing the random color of the n-th selected ball as a suitably sampled point on the path of the underlying random walk. This helps us to derive the central and local limit theorems.

a b s t r a c tIn this work we consider the infinite color urn model associated with a bounded increment random walk on Z d . This model was first introduced in Bandyopadhyay and Thacker (2013). We prove that the rate of convergence of the expected configuration of the urn at time n with appropriate centering and scaling is of the order O  (log n) −1/2  . Moreover we derive bounds similar to the classical Berry-Esseen bound. Further we show that for the expected configuration a large deviation principle (LDP) holds with a good rate function and speed log n.

We show that for any Cayley graph, the probability (at any p) that the cluster of the origin has size n decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph.

In this article we prove the bivariate uniqueness property for a particular "max-type" recursive distributional equation (RDE). Using the general theory developed in [5] we then show that the corresponding recursive tree process (RTP) has no external randomness, more preciously, the RTP is endogenous. The RDE we consider is so called the Logistic RDE, which appears in the proof of the ζ(2)-limit of the random assignment problem [4] using the local weak convergence method. Thus this work provides a non-trivial application of the general theory developed in [5].

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Entries in microblogging sites such as Twitter are very short: a "tweet "can contain at most 140 characters. Given a user query, retrieving relevant tweets is particularly challenging since their extreme brevity exacerbates the well-known vocabulary mismatch problem. In this preliminary study, we explore standard query expansion approaches as a way to address this problem. Since the tweets are short, we use external corpora as a source for query expansion terms. Specifically, we used the Google Search API (GSA) to retrieve pages from the Web, and used the titles to expand queries. Initial results on the TREC 2011 Microblog test data are very promising. Since many of the TREC topics were oriented towards the news genre, we also tried restricting the GSA to a news site (BBC) in the hope that it would be a cleaner, less noisy source for expansion terms. This turned out to be counter-productive. Some analysis of these results is also included.