Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

Maximal Arithmetic Progressions in Random Subsets

Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0, 1}N . By an application of the Chen-Stein method, we show that U −2 log N/ log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we ...

Arithmetic Progressions of Length Three in Subsets of a Random Set

For integers 1 ≤ M ≤ n, let R(n,M) denote the uniform probability space which consists of all the M -element subsets of [n] = {0, 1, . . . , n − 1}. It is shown that for every α > 0 there exists a constant C such that if M = M(n) ≥ C √ n then, with probability tending to 1 as n → ∞, the random set R ∈ R(n,M) has the property that any subset of R with at least α|R| elements contains a 3-term ari...

Large Deviations for Random Walk in Random

Primes in Arithmetic Progressions to Large Moduli

where <p is Euler's function and n(x) counts all primes up to x. Indeed, (1.1) is known to hold, but only under the proviso that q grow more slowly than some fixed power of logx (Siegel-Walfisz theorem). Yet, for many applications, it is important to weaken this restriction. The assumption of the Generalized Riemann Hypothesis (GRH) yields (1.1) in the much wider range q < X I / 2/ log2+e x . I...

Large Deviations for Random Trees.

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.