Related papers: A density version for H\"aggstr\"om's theorem

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which…

Let d \geq d_0 be a sufficiently large constant. A (n,d,c \sqrt{d}) graph G is a d-regular graph over n vertices whose second largest (in absolute value) eigenvalue is at most c \sqrt{d}. For any 0 < p < 1, G_p is the graph induced by…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…

Consider the continuous greedy paths model: given a $d$-dimensional Poisson point process with positive marks interpreted as masses, let $\mathrm P(\ell)$ denote the maximum mass gathered by a path of length $\ell$ starting from the origin.…

We determine the probability that a random polynomial of degree $n$ over $\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it is given by a rational function of $p$ that is invariant under replacing $p$ by $1/p$.

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an…

Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite…

We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both…

We consider the problem of decomposing some $t$-uniform hypergraph $G$ into copies of another, say $H$, with nonnegative rational weights. For fixed $H$ on $k$ vertices, we show that this is always possible for all $G$ having sufficiently…

It is shown that in the complex trajectory representation of quantum mechanics, the Born's Psi^{\star}\Psi probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this…

We consider Bernoulli first-passage percolation on the $d$-dimensional hypercubic lattice with $d \geq 2$. The passage time of edge $e$ is $0$ with probability $p$ and $1$ with probability $1-p$, independently of each other. Let $p_c$ be…

We consider independent edge percolation models on Z, with edge occupation probabilities p_<x,y> = p if |x-y| = 1, 1 - exp{- beta / |x-y|^2} otherwise. We prove that oriented percolation occurs when beta > 1 provided p is chosen…

In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…

In this paper we study the Spanning Tree Congestion problem, where we are given a graph $G=(V,E)$ and are asked to find a spanning tree $T$ of minimum maximum congestion. Here, the congestion of an edge $e\in T$ is the number of edges…

First-passage percolation is a random growth model which has a metric structure. An infinite geodesic is an infinite sequence whose all sub-sequences are shortest paths. One of the important quantity is the number of infinite geodesics…

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has…

We prove that any involution-invariant probability measure on the space of trees with maximum degrees at most d arises as the local limit of a convergent large girth graph sequence. This answers a question of Bollobas and Riordan.

We extend a result of Lyons (2016) from fractional tiling of finite graphs to a version for infinite random graphs. The most general result is as follows. Let $\bf P$ be a unimodular probability measure on rooted networks $(G, o)$ with…

We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property…

We study a problem on edge percolation on product graphs $G\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\{0,1\}$ connected by an edge. Every edge in $G\times K_2$ is present with probability $p$ independent…