相关论文: Soft edge results for longest increasing paths on …
We consider a regular $n$-ary tree of height $h$, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we…
We study the properties of least time trajectories for particles moving on a two dimensional surface which consists of piecewise homogeneous regions. The particles are assumed to move with different constant speeds on different regions and…
Edge expansion is a parameter indicating how well-connected a graph is. It is useful for designing robust networks, analysing random walks or information flow through a network and is an important notion in theoretical computer science.…
We provide a framework for proving convergence to the directed landscape, the central object in the Kardar-Parisi-Zhang universality class. For last passage models, we show that compact convergence to the Airy line ensemble implies…
Consider a line segment placed on a two-dimensional grid of rectangular tiles. This paper addresses the relationship between the length of the segment and the number of tiles it visits (i.e. has intersection with). The square grid is also…
This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…
In this paper we study maximum size and minimum weight planar matchings of inhomogenous random bipartite graphs. Our motivation for this study comes from efficient usage of cross edges in relay networks for overall improvement in network…
This research concerns coefficient conditions for linear differential equations in the unit disc of the complex plane. In the higher order case the separation of zeros (of maximal multiplicity) of solutions is considered, while in the…
The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this…
Considered typical processes of rod bending under strong longitudinal compression. The dynamic equation of bending correspponds to a perturbation of the two- dimensional Laplace equation. It is established that, for these processes,…
The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution at each point of the real line in terms of decorated planar…
A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected $n$-vertex graph has a path cover with at most $\lceil n/2 \rceil$ paths. We prove Gallai's conjecture for…
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 \le j \le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard "sprinkling"…
The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise…
A system of non-intersecting squared Bessel processes is considered which all start from one point and they all return to another point. Under the scaling of the starting and ending points when the macroscopic boundary of the paths touches…
We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a~subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph…
Consider an infinite graph with nodes initially labeled by independent Bernoulli random variables of parameter p. We address the density classification problem, that is, we want to design a (probabilistic or deterministic) cellular…
Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain an ${\mathsf M}_3$-sublattice). We call the interval $I = [o, i]$ of $L$ \emph{rectangular}, if there are complementary $a, b \in I$ such that $a$ is to the…
We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the…