Mokhtar H. Konsowa

Commute times and the effective resistances of random trees

In this article we study the commute and hitting times of simple random walks on spherically symmetric random trees in which every vertex of level n has outdegree 1 with probability 1−qn and outdegree 2 with probability qn. Our argument relies on the link between the commute times and the effective resistances of the associated electric networks when 1 unit of resistance is assigned to each edge of the tree.

Dimensions of random trees

In this paper we show, for Galton-Watson tree T of resistance R, that R-Rn decays exponentially in n where Rn denotes the resistance of the portion of T between the root and level n. We also determine a formula for the resistance dimension of spherically symmetric random trees and prove that it is equal to the fractal dimension. We emphasize the relationship between these dimensions and the type, of being transient or recurrent, of the simple random walks on such trees.

Fractal dimensions and random walks on random trees

Abstract We determine the fractal dimension df of infinite spherically symmetric random trees (all vertices at distance n from the root have the same degree dn where {dn} are independent random variables). If dn takes the value 3 or 2 with probabilities qn and 1−qn, then d f =( log 2) lim n nq n +1 a.s. We show how df is closely related to the type of the simple random walks (SRW) on trees. We prove that the SRW is a.s. transient if df>2 a.s. and a.s. recurrent if df d f = lim n nq n +1 a.s.

A NOTE ON THE CONDUCTIVITY OF RANDOM TREES

We prove that no stochastic domination exists between the effective resistance of a spherically symmetric random tree and that of a branching process in a varying environments tree if they grow according to the same law of distribution.

Random walks on trees and the law of iterated logarithm

In this paper we give an alternative proof for the main result of Konsowa and Mitro (J. Theor. Probab. 4 (3) (1991) 535), Konsowa and Mitro found that the simple random walk (SRW) on infinite trees is transient or recurrent. In part of their work, they considered the case of an -tree in which all the vertices of the same distance n from the root have the same degree which is 3 with probability qn and 2 with probability 1-qn. They proved that the SRW is transient if liminf nqn>1/log 2 and recurrent if limsup nqn

The speed of random walks on trees and electric networks

The speed of the random walk on a tree is the rate of escaping its starting point. It depends on the way that the branching occurs in the sense that if the average number of branching is large, the speed is more likely to be positive. The speed on some models of random trees is calculated via calculating the hitting times of the consecutive levels of the tree.

CONDUCTIVITY OF RANDOM TREES

We prove that the effective resistances of spherically symmetric random trees dominate in mean the effective resistances of random trees corresponding branching processes in varying environments and having the same growth law of spherically symmetric trees. We conjecture that the statement does not necessarily hold true in the case of stochastic domination and give an idea of constructing a counterexample.

Commute times of random walks on trees

In this paper we provide exact formula for the commute times of random walks on spherically symmetric random trees. Using this formula we sharpen some of the results presented in Al-Awadhi et al. to the form of equalities rather than inequalities.

Fractal and resistance dimensions of random trees

In this article we determine a formula for fractal and resistance dimensions of two models of uniformly bounded random trees. The type (transient or recurrent) of the random walk on such trees is ascribed, to some extent, to these dimensions. The results presented in this article generalize some of the results of [6] and [7].