Related papers: The Clairvoyant Demon Has a Hard Task

Equip each point $x$ of a homogeneous Poisson process $\mathcal{P}$ on $\mathbb{R}$ with $D_x$ edge stubs, where the $D_x$ are i.i.d. positive integer-valued random variables with distribution given by $\mu$. Following the stable…

Quantum percolation describes the problem of a quantum particle moving through a disordered system. While certain similarities to classical percolation exist, the quantum case has additional complexity due to the possibility of Anderson…

We consider a model of a discrete time "interacting particle system" on the integer line where infinitely many changes are allowed at each instance of time. We describe the model using chameleons of two different colours, {\it viz}., red…

A permutation may be represented by a collection of paths in the plane. We consider a natural class of such representations, which we call tangles, in which the paths consist of straight segments at 45 degree angles, and the permutation is…

In this paper we consider a discrete or continuous landscape with correlations and we consider a source of light (a sun) at infinity emitting parallel rays of light making a slope l with the horizontal plane. Depending on the value of l…

We show that a large class of site percolation processes on any planar graph contains either zero or infinitely many infinite connected components. The assumptions that we require are: tail triviality, positive association (FKG) and that…

We consider $m$-divisible non-crossing partitions of $\{1,2,\ldots,mn\}$ with the property that for some $t\leq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such…

Let A be the annulus in R^2 centered at the origin with inner and outer radii r(1-\epsilon) and r, respectively. Place points {x_i} in R^2 according to a Poisson process with intensity 1 and let G_A be the random graph with vertex set {x_i}…

In the bond percolation model on a lattice, we colour vertices with $n_c$ colours independently at random according to Bernoulli distributions. A vertex can receive multiple colours and each of these colours is individually observable. The…

Let $0<a<b<\infty$, and for each edge $e$ of $Z^d$ let $\omega_e=a$ or $\omega_e=b$, each with probability 1/2, independently. This induces a random metric $\dist_\omega$ on the vertices of $Z^d$, called first passage percolation. We prove…

A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…

Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of…

The problem of linear and circular permutations of n identical objects in m boxes, where a limit l is imposed on the number of objects in a box, is considered. In the linear case, where the boxes are arranged as a row, two methods of…

Let $0<a<b<\infty$ be fixed scalars. Assign independently to each edge in the lattice $\mathbb{Z}^2$ the value $a$ with probability $p$ or the value $b$ with probability $1-p$. For all $u,v\in\mathbb{Z}^2$, let $T(u,v)$ denote the first…

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond…

We show that the problem of directed percolation on an arbitrary lattice is equivalent to the problem of m directed random walkers with rather general attractive interactions, when suitably continued to m=0. In 1+1 dimensions, this is dual…

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…

In $r$-neighbor bootstrap percolation on the vertex set of a graph $G$, a set $A$ of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least $r$ previously infected neighbors. When the…

This is the second in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let $G$ be a graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored, where $L$ is a…

We investigate percolation on a randomly directed lattice, an intermediate between standard percolation and directed percolation, focusing on the isotropic case in which bonds on opposite directions occur with the same probability. We…