Related papers: Permute, Graph, Map, Derange

The planar rigidity problem asks, given a set of m pairwise distances among a set P of n unknown points, whether it is possible to reconstruct P, up to a finite set of possibilities (modulo rigid motions of the plane). The celebrated…

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.

This work is inspired by a recent study of a two-dimensional stochastic fragmentation model. We show that the configurational entropy of this model exhibits log-periodic oscillations as a function of the sample size, by exploiting an exact…

Consider a random permutation of $kn$ objects that permutes $n$ disjoint blocks of size $k$ and then permutes elements within each block. Normalizing its cycle lengths by $kn$ gives a random partition of unity, and we derive the limit law…

We show that if pn >> log n, the binomial random graph G_{n,p} has an approximate Hamilton decomposition. More precisely, we show that in this range G_{n,p} contains a set of edge-disjoint Hamilton cycles covering almost all of its edges.…

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the…

We consider a class of inhomogeneous media known as composite media that is often encountered in experimental sciences and investigate the persistence probability of a random walker in such a system. Analytical and numerical results for the…

We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance…

We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we…

Two models of binary fragmentation are introduced in which a time dependent transition size produces two regions of fragment sizes above and below the transition size. In the models we consider a fixed rate of fragmentation for the largest…

We show that the number of cycles in a random permutation chosen according to generalized Ewens measure is normally distributed and compute asymptotic estimates for the mean and variance.

Let $\mathcal{S}_n$ be the permutation group on $n$ elements, and consider a random walk on $\mathcal{S}_n$ whose step distribution is uniform on $k$-cycles. We prove a well-known conjecture that the mixing time of this process is…

In this note we put forward a conjecture on the average optimal length for bipartite matching with a finite number of elements where the different lengths are independent one from the others and have an exponential distribution.

We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations…

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs between the existence of an absolutely continuous invariant probability measure and infinite measure depending…

We investigate the probability for the largest segment in with total displacement $Q$ in an $N$-step random walk to have length $L$. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the…

Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more…

The major challenge in designing a discriminative learning algorithm for predicting structured data is to address the computational issues arising from the exponential size of the output space. Existing algorithms make different assumptions…

Defant and Zheng introduced a consecutive-pattern-avoiding stack sort map $SC_{\sigma}$, where the stack must avoid a consecutive pattern $\sigma$. Seidel and Sun disproved a conjecture in Defant and Zheng's paper about the maximum…

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central…