Related papers: Effective resistance of random trees

We consider a continuous time random walk on the rooted binary tree of depth $n$ with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by…

We re-examine a lower-tail upper bound for the random variable $$X=\prod_{i=1}^{\infty}\min\left\{\sum_{k=1}^iE_k,1\right\},$$ where $E_1,E_2,\ldots\stackrel{iid}\sim\text{Exp}(1)$. This bound has found use in root-finding and seed-finding…

We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson…

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights…

Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ matrix for which the entry in the $i$-th row, $j$-th column is the level (distance from the root) of the first…

Let $W$ be a nonnegative random variable with expectation $1$. For all $r \geqslant 2$, we consider the total mass $Z_r^\infty$ of the associated Mandelbrot multiplicative cascade in the $r$-ary tree. For all $n \geqslant 1$, we also…

We study the detection error probability associated with a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent detectors. The root of the tree represents a fusion center that makes the…

For each $n \ge 1$, let $\mathrm{d}^n=(d^{n}(i),1 \le i \le n)$ be a sequence of positive integers with even sum $\sum_{i=1}^n d^n(i) \ge 2n$. Let $(G_n,T_n,\Gamma_n)$ be uniformly distributed over the set of simple graphs $G_n$ with degree…

We consider self-repelling elastic manifolds with a domain $[-N,N]^d \cap \mathbb{Z}^d$, that take values in $\mathbb{R}^D$. Our main result states that when the dimension of the domain is $d=2$ and the dimension of the range is $D=1$, the…

The concept of effective resistance, originally introduced in electrical circuit theory, has been extended to the setting of graphs by interpreting each edge as a resistor. In this context, the effective resistance between two vertices…

It is known that electrical networks with resistors are related to the Laplace operator and random walk on weighted graphs. In this paper we consider more general electrical networks with coils, capacitors, and resistors. We give two…

We show that for every $k$, the probability that a randomly selected vertex of a random binary search tree on $n$ nodes is at distance $k-1$ from the closest leaf converges to a rational constant $c_k$ as $n$ goes to infinity.

We consider an Erdos-Renyi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N->infinity one obtains a graph of finite mean degree c. In…

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative…

This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaym\'e-Galton-Watson tree with critical offspring distribution $\xi$,…

The exact expression for the effective resistance between any two vertices of the $N$-cycle graph with four nearest neighbors $C_{N}(1,2)$, is given. It turns out that this expression is written in terms of the effective resistance of the…

We consider solutions to the maximum recursion on weighted branching trees given by$$X\,{\buildrel d\over=}\,\bigvee_{i=1}^{N}{A_iX_i}\vee B,$$where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in\mathbb{N}}$ are random positive…

A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests…

Let $\tau$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study…

We study the general low-energy effective action on long open strings, such as confining strings in pure gauge theories. Using Lorentz invariance, we find that for a string of length R, the leading deviation from the Nambu-Goto energy…