Related papers: Geometry of Permutation Limits
We consider a constrained version of the shortest path problem on the complete graphs whose edges have independent random lengths and costs. We establish the asymptotic value of the minimum length as a function of the cost-budget within a…
The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to…
In this article we continue the study of the quenched distributions of transient, one-dimensional random walks in a random environment. In a previous article we showed that while the quenched distributions of the hitting times do not…
We study scaling limits of random permutations ("permutons") constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In…
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one…
We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a…
We introduce a new oriented evolving graph model inspired by biological networks. A node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes. This leads to a statistical…
For real world systems, nonuniform medium is ubiquitous. Therefore, we investigate the diffusion-limited-aggregation process on a two dimensional directed small-world network instead of regular lattice. The network structure is established…
A sorting network is a shortest path from 12..n to n..21 in the Cayley graph of the symmetric group S(n) generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove…
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…
The paper contains an exposition of recent as well as old enough results on determinantal random point fields. We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of the…
We investigate the random permutation matrices induced by the Chinese restaurant processes with $(\alpha,\theta)$-seating. When $\alpha=0,\theta>0$, the permutations are those following Ewens measures on symmetric groups, and have been…
We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree…
We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly…
We consider random resistor networks with nodes given by a point process on $\mathbb{R}^d$ and with random conductances. The length range of the electrical filaments can be unbounded. We assume that the randomness is stationary and ergodic…
A watermelon is a set of $p$ Bernoulli paths starting and ending at the same ordinate, that do not intersect. In this paper, we show the convergence in distribution of two sorts of watermelons (with or without wall condition) to processes…
We prove some general results about the asymptotics of the distribution of the number of cycles of given length of a random permutation whose distribution is invariant under conjugation. These results were first established to be applied in…
Paths are important structural elements in complex networks because they are finite (unlike walks), related to effective node coverage (minimum spanning trees), and can be understood as being dual to star connectivity. This article…
We consider connectivity properties of certain i.i.d. random environments on $\Z^d$, where at each location some steps may not be available. Site percolation and oriented percolation can be viewed as special cases of the models we consider.…
In this article, we study the pointwise asymptotic behavior of iterated convolutions on the one dimensional lattice Z. We generalize the so-called local limit theorem in probability theory to complex valued sequences. A sharp rate of…