Anish Sarkar

Nonuniversality and Continuity of the Critical Covered Volume Fraction in Continuum Percolation

We establish, using mathematically rigorous methods, that the critical covered volume fraction (CVF) for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls. In addition, we show that the critical CVF is a continuous function of the distribution of the radius random variable, in the sense that if a sequence of random variables converges weakly to some random variable, then the critical CVF based on these random variables converges to the critical CVF of the limiting random variable.

On the coverage of space by random sets

Let ξ1, ξ2,… be a Poisson point process of density λ on (0,∞) d , d ≥ 1, and let ρ, ρ1, ρ2,… be i.i.d. positive random variables independent of the point process. Let C := ⋃ i≥1 {ξ i + [0,ρ i ] d }. If, for some t > 0, (0,∞) d ⊆ C, then we say that (0,∞) d is eventually covered by C. We show that the eventual coverage of (0,∞) d depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of ℝ d by such Poisson Boolean models. In addition, we consider the set ⋃{i≥1:X i =1} [i,i+ρ i ], where X 1, X 2,… is a {0,1}-valued Markov chain and ρ1, ρ2,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.

The random connection model in high dimensions

Consider a continuum percolation model in which each pair of points of a d-dimensional Poisson process of intensity [lambda] is connected with a probability which is a function g of the distance between them. We show that under a mild regularity condition on g, the critical value of [lambda], above which an infinite cluster exists a.s., is asymptotic to ([integral operator]Rd g(x)dx)-1 as d --> [infinity].

Rigorous results on the threshold network model

We analyse the threshold network model in which a pair of vertices with random weights are connected by an edge when the summation of the weights exceeds a threshold. We prove some convergence theorems and central limit theorems on the vertex degree, degree correlation and the number of prescribed subgraphs. We also generalize some results in the spatially extended cases.

Asymptotic properties of sums of upper records

Arnold and Villaseñor (1999) raised several questions for upper records, including characterizing all limit distributions of normalized partial sums of upper records. We provide some answers in the case when the distribution from which the samples are drawn is bounded above. When the distribution is not bounded above, we give sufficient conditions on the distribution for the properly normalized partial sums to converge to a standard normal distribution. We show that our conditions are general enough so that the examples provided by Arnold and Villaseñor (1999) are covered by our results.

Random directed forest and the Brownian web

Consider the d dimensional lattice Z d where each vertex is open or closed with probability p or 1 − p respectively. An open vertex u := (u(1),u(2),... ,u(d)) is connected by an edge to another open vertex which has the minimum L1 distance among all the open vertices with x(d) > u(d). It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of disjoint trees for d ≥ 4. In addition for d = 2, we show that when properly scaled, the family of its paths converge in distribution to the Brownian web.

Waiting time distributions of runs in higher order Markov chains

We consider a {0,1}-valuedm-th order stationary Markov chain. We study the occurrences of runs where two 1’s are separated byat most/exactly/at least k 0’s under the overlapping enumeration scheme wherek≥0 and occurrences of scans (at leastk1 successes in a window of length at mostk, 1≤k1≤k) under both non-overlapping and overlapping enumeration schemes. We derive the generating function of first two types of runs. Under the conditions, (1) strong tendency towards success and (2) strong tendency towards reversing the state, we establish the convergence of waiting times of ther-th occurrence of runs and scans to Poisson type distributions. We establish the central limit theorem and law of the iterated logarithm for the number of runs and scans up to timen.

Hack’s law in a drainage network model: A Brownian web approach

Hack [Studies of longitudinal stream profiles in Virginia and Maryland (1957). Report], while studying the drainage system in the Shenandoah valley and the adjacent mountains of Virginia, observed a power law relation

Backbends in Directed Percolation

l\sim a^{0.6}

High density asymptotics of the Poisson random connection model

between the length