Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability

This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity? The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence (k_n)_{n\geq 1}. The fundamental breakthrough of this paper is to generalise the previous results to any (reasonable) sequence of integers (k_n)_{n\geq 1}, which enables us, in particular, to solve the above satisfiability question. We also analyse the effect of introducing a natural equivalence relation on the set of Boolean expressions. This new "quotient" model happens to exhibit a very interesting threshold (or saturation) phenomenon at k_n = n/ln n.Comment: Long version of arXiv:1304.561

Smoothing equations for large P\'olya urns

Consider a balanced non triangular two-color P\'olya-Eggenberger urn process, assumed to be large which means that the ratio sigma of the replacement matrix eigenvalues satisfies 1/2<sigma <1. The composition vector of both discrete time and continuous time models admits a drift which is carried by the principal direction of the replacement matrix. In the second principal direction, this random vector admits also an almost sure asymptotics and a real-valued limit random variable arises, named WDT in discrete time and WCT in continous time. The paper deals with the distributions of both W. Appearing as martingale limits, known to be nonnormal, these laws remain up to now rather mysterious. Exploiting the underlying tree structure of the urn process, we show that WDT and WCT are the unique solutions of two distributional systems in some suitable spaces of integrable probability measures. These systems are natural extensions of distributional equations that already appeared in famous algorithmical problems like Quicksort analysis. Existence and unicity of the solutions of the systems are obtained by means of contracting smoothing transforms. Via the equation systems, we find upperbounds for the moments of WDT and WCT and we show that the laws of WDT and WCT are moment-determined. We also prove that WDT is supported by the whole real line and admits a continuous density (WCT was already known to have a density, infinitely differentiable on R\{0} and not bounded at the origin)

The relation between tree size complexity and probability for Boolean functions generated by uniform random trees

We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the tree size tends to infinity and derive a relation between the tree size complexity and the probability of a function. This is done by first expressing trees representing a particular function as expansions of minimal trees representing this function and then computing the probabilities by means of combinatorial counting arguments relying on generating functions and singularity analysis

Parameterised branching processes:A functional version of Kesten &amp; Stigum theorem

Let (Zn,n≥0) be a supercritical Galton–Watson process whose offspring distribution μ has mean λ&gt;1 and is such that ∫xlog+(x)dμ(x)&lt;+∞. According to the famous Kesten &amp; Stigum theorem, (Zn/λn) converges almost surely, as n→+∞. The limiting random variable has mean 1, and its distribution is characterised as the solution of a fixed point equation. In this paper, we consider a family of Galton–Watson processes (Zn(λ),n≥0) defined for λ ranging in an interval I⊂(1,∞), and where we interpret λ as the time (when n is the generation). The number of children of an individual at time λ is given by X(λ), where (X(λ))λ∈I is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that E(X(λ))=λ&gt;1 for all λ∈I. This allows us to define Zn(λ) the number of elements in the nth generation at time λ. Set Wn(λ)=Zn(λ)/λn for all n≥0 and λ∈I. We prove that, under some moment conditions on the process X, the sequence of processes (Wn(λ),λ∈I)n≥0 converges in probability as n tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.</p

Large deviations principle for a stochastic process with random reinforced relocations

Stochastic processes with random reinforced relocations have been introduced in the physics literature to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at random in its whole past according to some ``memory kernel'', and jumps to its value at that random time. We prove a quenched large deviations principle for the value of the process at large times. The difficulty in proving this result comes from the fact that the process is not Markov because of the relocations. Furthermore, the random inter-relocation times act as a random environment

Voronoi cells in random split trees

We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $n\exp(-\mathrm{const}\sqrt{\log n})$. This discrepancy persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different "influence" parameters (called "speeds" in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex

Voronoi cells in random split trees

We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $n\exp(-\mathrm{const}\sqrt{\log n})$. This discrepancy persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different "influence" parameters (called "speeds" in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex