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((ξi,Xi), i ≥ 1). Here(ξi) and g(·) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(·) is essentially a “maximum ” or “minimum” function. We draw attention to the theoretical question of endogeny: inthe associated recursive tree process X i,aretheX i measurable functions of the innovations process (ξ i)? 1. Introduction. Write

Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean p ≤ 1/2. How many of these Bernoullis one must look at in order to find a path of length n from the root which maximizes, up to a factor of 1 −ε, the sum of the Bernoullis along the path? In the case p = 1/2 (the critical value for nontriviality), it is shown to take Θ(ε −1 n) steps. In the case p < 1/2, the number of steps is shown to be at least n · exp(constε −1/2). This last result matches the known upper bound from [Algorithmica 22 (1998) 388–412] in a certain family of subcases.

Let S(v) be a function defined on the vertices v of the infinite binary tree. One algorithm to seek large positive values of S is the Metropolis-type Markov chain (Xn ) defined by P (Xn+1 = wjXn = v) = 1 3 e b(S(w)\GammaS(v)) 1 + e b(S(w)\GammaS(v)) for each neighbor w of v, where b is a parameter ("1=temperature") which the user can choose. We introduce and motivate study of this algorithm under a probability model for the objective function S, in which S is "tree-indexed simple random walk", that is the increments ¸(e) = S(w) \Gamma S(v) along parent-child edges e = (v; w) are independent and P (¸ = 1) = p; P (¸ = \Gamma1) = 1 \Gamma p. This algorithm has a "speed" r(p; b) = lim n n \Gamma1 ES(Xn ). We study the speed via a mixture of rigorous arguments, non-rigorous arguments and Monte Carlo simulations, and compare with a deterministic greedy algorithm which permits rigorous analysis. Formalizing the non-rigorous arguments presents a challenging problem. Mathematically, th...

Consider the complete n-graph with independent exponential (mean n) edge-weights. Let M (c; n) be the maximal size of subtree for which the average edge-weight is at most c. It is shown that M (c; n) transitions from o(n) to \Omega\Gamma n) around some critical value c(0), which can be specified in terms of a fixed point of a mapping on probability distributions. Research supported by N.S.F. Grant DMS96-22859 1 1 Introduction To each edge e of the complete graph on f1; 2; : : : ; ng attach a weight w e , where the (w e ) are independent exponential (mean n) r.v.'s. Call this randomly-weighted graph W n . For each subtree of W n , that is each tree t whose vertex-set is a subset of f1; 2; : : : ; ng, write jtj for the number of edges of t and w(t) = P e2t w e for the weight of the tree. So w(t)=jtj is the average edge-weight of the tree t. Consider the maximum size of tree with average edge-weight at most c: M(c; n) = maxfjtj : w(t)=jtj cg: (1) It is natural to guess that th...

We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite [26, 31]. In this paper, we confirm a conjecture of Aldous [3, 4] that E [Z] < ∞ while E [Z log Z] = ∞. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks. 1

In this work we prove the bivariate uniqueness property for two “maxtype” recursive distributional equations which then lead to the proof of endogeny for the associated recursive tree processes. Thus providing two concrete instances of the general theory developed by Aldous and Bandyopadhyay [3]. The first example discussed here deals with the construction of a frozen percolation process on a infinite regular binary tree. For this we prove that the construction do not involve any external randomness. It is also shown that same is true for any r-regular tree and more interestingly for any infinite regular Galton-Watson branching process trees with mild moment condition on the progeny distribution. The second example is proving the endogeny for the Logistic recursive distributional equation which appears for studying the asymptotic limit of the random assignment problem using local-weak convergence method. The two examples are quite unrelated and hence illustrate a broad range of applicability of the general methods of [3].

This paper considers a problem in extreme value theory from a computational complexity viewpoint. Suppose that {Snk: n ≥ 1, k ≤ K(n)} are random variables, with K(n) growing perhaps quite rapidly. Let Mn: = max k≤K(n) Snk. A prototypical classical extreme value theorem takes the form fn(Mn) → Z, where convergence is to a constant or a distribution. When K(n) grows rapidly with