For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary finite-state reversible Markov chains. Under weak conditions, the covering time for such processes is at leastc times the covering time for the corresponding i.i.d. process.

Lower bounds for covering times for reversible Markov chains and random walks on graphs

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Reversible Markov Chains and Random Walks on Graphs

12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related

Asymptotic enumeration methods

Let (B t (s), 0 ≤ s < ∞) be reflecting inhomogeneous Brownian motion with drift t - s at time s, started with B t (0) = 0. Consider the random graph script G sign(n, n -1 + tn -4/3 ), whose largest components have size of order n 2/3 . Normalizing by n -2/3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (x i ), and clusters with sizes x i and x j merge at rate x i x j . The multiplicative coalescent is shown to be a Feller process on l 2 . The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time -∞; the existence of such a process is not obvious.

/pdf/brownian-excursions-critical-random-graphs-and-the-3algyvo84e.pdf

Brownian excursions, critical random graphs and the multiplicative coalescent

https://statweb.stanford.edu/~cgates/PERSI/papers/strong87.pdf

Strong uniform times and finite random walks

This paper is concerned with a random walk process {l , n1i, . , ri} in which 772n = 70=0 and -i 0 for i = 1, 2, -., 2n. This process is called a Bernoulli excursion. The main object is to find the distribution, the moments, and the asymptotic distribution of the random variable ow, defined by 2nwo = + 1r? + S" + 72n. The results derived have various applications in the theory of probability, including random graphs, tournaments and order statistics. BROWNIAN EXCURSION; RANDOM GRAPHS; ORDER STATISTICS; TOURNAMENTS

A bernoulli excursion and its various applications

This paper gives a survey of the historical development of the solutions of various fluctuation problems in the theory of queues from the point of view of the mathematical methods used. These methods include Markov chains, Markov processes, integral equations, complex functions, generating functions, Laplace transforms, factorization of functions, operator calculus, Banach algebras and some particular methods, such as calculus of finite differences and combinatorics. In addition, the paper contains several recent results of the author for semi-Markov queuing processes.

On fluctuation problems in the theory of queues

Let K be a finite graph with vertex set V= {x0, x,, ., x_1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent or cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by v, (n = 1, 2, - - -) the location of the traveler at the end of the nth flight, and by vo the initial location. It is assumed that the transition probabilities P{v, = xi n-= xi}, xi V, xi E V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial

Random flights on regular graphs

Throughout this paper we shall be concerned with a sequence of mutually independent and identically distributed random variables ξ 1 ξ 2 , · · ·, ξ n , · · · taking on real values. We shall use the notation ζ n = ξ 1 + · · · + ξ n for n = 1, 2, · · · and ζ 0 = 0.

On the distribution of the maximum of sums of mutually independent and identically distributed random variables

This paper is a survey of the solutions of various first-passage time problems in the theory of queues.

Lajos Takács

Papers

A bernoulli excursion and its various applications

On fluctuation problems in the theory of queues

Random flights on regular graphs

On the distribution of the maximum of sums of mutually independent and identically distributed random variables

Combinatorial and analytic methods in the theory of queues