<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e19" altimg="si13.svg"><mml:mi>k</mml:mi></mml:math>-free lattice points in random walks

Visible Lattice Points in Random Walks

We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability α and 1 − α, respectively) and starting from the origin. We show that, almost surely, the asymptotic proportion of strings of k consecutive visible lattice points visited by such an α-random walk is a certain constant ck(α), which is actually an (explicitly calculabl...

Random Walks on Nite Convex Sets of Lattice Points

Lattice paths and random walks

Lattice paths are ubiquitous in combinatorics and algorithmics, where they are either studied per se, or as a convenient encoding of other structures. Logically, they play an important role in Philippe’s papers. For instance, they are central in his combinatorial theory of continued fractions, to which Chapter 3 of this volume is devoted. In this chapter, we present a collection of seven papers...

Self-avoiding random walks on lattice strips.

A self-avoiding walk on an infinitely long lattice strip of finite width will asymptotically exhibit an end-to-end separation proportional to the number of steps. A proof of this proposition is presented together with comments concerning an earlier attempt to deal with the matter. In addition, some unproved, yet "obvious," conjectures concerning self-avoiding walks are cited as basic propositio...

Random Points and Lattice Points in Convex Bodies

Assume K ⊂ Rd is a convex body and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong to X? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? We are interested in these questions mainly in two cases. The first is when X is a random sample of n uniform, independent ...