Limits of random tree-like discrete structures

Exchangeability and continuum limits of discrete random structures

Exchangeable representations of complex random structures are useful in several ways, in particular providing a moderately general way to derive continuum limits of discrete random structures. I shall describe an old example (continuum random trees) and a more recent example (dense graph limits). Thinking this way about road routes suggests challenging new problems in the plane. Mathematics Sub...

Topics in discrete random structures

This thesis presents four papers on problems in discrete probability. A common theme of the articles is to take some class of discrete structures, impose some randomness, and then consider what happens asymptotically as the size of the structure tends to infinity. Paper I concerns first–passage percolation on Cartesian graph powers G of some fixed base graph G as d → ∞. We propose a natural asy...

Splicing on tree-like structures

In this paper, we provide a method to accelerate the power of splicing systems. We introduce the splicing systems on trees to be built as partially annealed single strands, that is a quite similar notion and a natural extension of splicing systems on strings. Trees are a common and useful data structure in computer science and have a biological counterpart such as molecular sequences with secon...

Ramsey properties of random discrete structures

We study thresholds for Ramsey properties of random discrete structures. In particular, we determine the threshold for Rado’s theorem for solutions of partition regular systems of equations in random subsets of the integers and we prove the 1-statement of the conjectured threshold for Ramsey’s theorem for random hypergraphs. Those results were conjectured by Rödl and Ruciński and similar result...

Stochastic Analysis of Tree-like Data Structures