No Arabic abstract
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $mathbb Z^d$ and, more generally, for infinite graphs. We then apply these notions to critical percolation clusters, where the various dimensions have different values.
We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the dust). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial Holder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.
We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others.
We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattic
We consider Gibbs distributions on permutations of a locally finite infinite set $Xsubsetmathbb{R}$, where a permutation $sigma$ of $X$ is assigned (formal) energy $sum_{xin X}V(sigma(x)-x)$. This is motivated by Feynmans path representation of the quantum Bose gas; the choice $X:=mathbb{Z}$ and $V(x):=alpha x^2$ is of principal interest. Under suitable regularity conditions on the set $X$ and the potential $V$, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.
In a batch of synchronized queues, customers can only be serviced all at once or not at all, implying that service remains idle if at least one queue is empty. We propose that a batch of $n$ synchronized queues in a discrete-time setting is quasi-stable for $n in {2,3}$ and unstable for $n geq 4$. A correspondence between such systems and a random-walk-like discrete-time Markov chain (DTMC), which operates on a quotient space of the original state-space, is derived. Using this relation, we prove the proposition by showing that the DTMC is transient for $n geq 4$ and null-recurrent (hence quasi-stability) for $n in {2,3}$ via evaluating infinite power sums over skewed binomial coefficients. Ignoring the special structure of the quotient space, the proposition can be interpreted as a result of Polyas theorem on random walks, since the dimension of said space is $d-1$.