Related papers: Critical random graphs: Diameter and mixing time
The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced…
Suppose that $\CG$ is a finite, connected graph and $X$ is a lazy random walk on $\CG$. The lamplighter chain $X^\diamond$ associated with $X$ is the random walk on the wreath product $\CG^\diamond = \Z_2 \wr \CG$, the graph whose vertices…
We derive an exact closed-form analytical expression for the distribution of the cover time for a random walk over an arbitrary graph. In special case, we derive simplified exact expressions for the distributions of cover time for a…
The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We present an efficient…
It is well known that many random graphs with infinite variance degrees are ultrasmall. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least $k$ is approximately…
We show that the measure on markings of $\mathbf {Z}_n^d$, $d\geq3$, with elements of ${0,1}$ given by i.i.d. fair coin flips on the range $\mathcal {R}$ of a random walk $X$ run until time $T$ and 0 otherwise becomes indistinguishable from…
We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size…
We study the contact process on random graphs with low infection rate $\lambda$. For random $d$-regular graphs, it is known that the survival time is $O(\log n)$ below the critical $\lambda_c$. By contrast, on the Erd\H{o}s-R\'enyi random…
We study the joint components in a random `double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices, which contains both a red and a blue spanning…
The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as \emph{mixing}, finds widespread applications in several areas of quantum information and computation. Of particular interest in most…
Periodic orbits in chaotic systems form clusters, whose elements traverse approximately the same points of the phase space. The distribution of cluster sizes depends on the length n of orbits and the parameter p which controls closeness of…
We study monotone paths in Erd\H{o}s-R\'enyi random graphs on numbered vertices. Benjamini & Tzalik established a phase transition at $p = \frac{\log n}{n}$ for this model. We refine the critical value to $p = \frac{\log n - \log \log n…
Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order q^(1/k). The same also holds when the generating set…
We consider supercritical bond percolation on a family of high-girth $d$-regular expanders. Alon, Benjamini and Stacey (2004) established that its critical probability for the appearance of a linear-sized ("giant'') component is…
Consider $n$ points distributed uniformly in $[0,1]^d$. Form a graph by connecting two points if their mutual distance is no greater than $r(n)$. This gives a random geometric graph, $\gnrn$, which is connected for appropriate $r(n)$. We…
We show that in three different critical regimes, the masses of the connected components of rank-2 multiplicative random graph converge to lengths of excursions of a thinned L\'{e}vy process, perhaps with random coefficients. The three…
We investigate the average hitting times of simple random walks on the $k$-th power graph $C_N^k$ of the cycle graph $C_N$. First, we show that the average hitting times are characterized by a difference equation corresponding to the graph…
Consider a system of coalescing random walks where each individual performs random walk over a finite graph G, or (more generally) evolves according to some reversible Markov chain generator Q. Let C be the first time at which all walkers…
We study the time that the simple exclusion process on the complete graph needs to reach equilibrium in terms of total variation distance. For the graph with n vertices and 1<<k<n/2 particles we show that the mixing time is of order…
Determining the total variation mixing time of Kac's random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing…