Abstract:
Consider the model where nodes are initially distributed as a Poisson point process with intensity $\lambda$ over $\mathbb{R}^d$ and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance $r$ of their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of $\mathbb{R}^d$, is detected by at least one node. We consider the case where the target particle can move according to any continuous function and can adapt its motion based on the location of the nodes. We show that there exists a sufficiently large value of $\lambda$ so that the target will eventually be detected almost surely. This means that the target cannot evade detection even if it has full information about the past, present and future locations of the nodes. Also, this establishes a phase transition for $\lambda$ since, for small enough $\lambda$, with positive probability the target can avoid detection forever. A key ingredient of our proof is to use fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time.

Abstract:
We study random triangulations of the integer points $[0,n]^2 \cap\mathbb{Z}^2$, where each triangulation $\sigma$ has probability measure $\lambda^{|\sigma|}$ with $|\sigma|$ denoting the sum of the length of the edges in $\sigma$. Such triangulations are called \emph{lattice triangulations}. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime $\lambda<1$, the function behaves as a \emph{Lyapunov function} with respect to Glauber dynamics; that is, the function is a supermartingale. We show the applicability of the above result by establishing several features of lattice triangulations, such as tightness of local measures, exponential tail of edge lengths, crossings of small triangles, and decay of correlations in thin rectangles. These are the first results on lattice triangulations that are valid in the whole subcritical regime $\lambda<1$. In a very recent work with Caputo, Martinelli and Sinclair, we apply this Lyapunov function to establish tight bounds on the mixing time of Glauber dynamics in thin rectangles that hold for all $\lambda<1$. The Lyapunov function result here holds in great generality; it holds for triangulations of general lattice polygons (instead of the $[0,n]^2$ square) and also in the presence of arbitrary constraint edges.

Abstract:
We consider the activated random walk model on general vertex-transitive graphs. A central question for this model is whether the critical density $\mu_c$ for sustained activity is strictly between $0$ and $1$. It was known that $\mu_c>0$ on $\mathbb{Z}^d$, $d \geq 1$, and that $\mu_c < 1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $\mu_c \to 0$ as $\lambda \to 0$ in all transient graphs, implying that $\mu_c < 1$ for small enough sleeping rate. We also show that $\mu_c < 1$ for any sleeping rate in any graph in which simple random walk has positive speed. Furthermore, we prove that $\mu_c >0$ in any amenable graph, and that $\mu_c \in (0,1)$ for any sleeping rate on regular trees.

Abstract:
We consider the hexagonal circle packing with radius 1/2 and perturb it by letting the circles move as independent Brownian motions for time t. It is shown that, for large enough t, if \Pi_t is the point process given by the center of the circles at time t, then, as t\to\infty, the critical radius for circles centered at \Pi_t to contain an infinite component converges to that of continuum percolation (which was shown---based on a Monte Carlo estimate---by Balister, Bollob\'as and Walters to be strictly bigger than 1/2). On the other hand, for small enough t, we show (using a Monte Carlo estimate for a fixed but high dimensional integral) that the union of the circles contains an infinite connected component. We discuss some extensions and open problems.

Abstract:
Consider the model where particles are initially distributed on $\mathbb{Z}^d, \, d\geq 2$, according to a Poisson point process of intensity $\lambda>0$, and are moving in continuous time as independent simple symmetric random walks. We study the escape versus detection problem, in which the target, initially placed at the origin of $\mathbb{Z}^d, \, d\geq 2$, and changing its location on the lattice in time according to some rule, is said to be detected if at some finite time its position coincides with the position of a particle. We consider the case where the target can move with speed at most 1, according to any continuous function and can adapt its motion based on the location of the particles. We show that there exists sufficiently small $\lambda_* > 0$, so that if the initial density of particles $\lambda < \lambda_*$, then the target can avoid detection forever.

Abstract:
Static wireless networks are by now quite well understood mathematically through the random geometric graph model. By contrast, there are relatively few rigorous results on the practically important case of mobile networks, in which the nodes move over time; moreover, these results often make unrealistic assumptions about node mobility such as the ability to make very large jumps. In this paper we consider a realistic model for mobile wireless networks which we call mobile geometric graphs, and which is a natural extension of the random geometric graph model. We study two fundamental questions in this model: detection (the time until a given "target" point - which may be either fixed or moving - is detected by the network), and percolation (the time until a given node is able to communicate with the giant component of the network). For detection, we show that the probability that the detection time exceeds t is \exp(-\Theta(t/\log t)) in two dimensions, and \exp(-\Theta(t)) in three or more dimensions, under reasonable assumptions about the motion of the target. For percolation, we show that the probability that the percolation time exceeds t is \exp(-\Omega(t^\frac{d}{d+2})) in all dimensions d\geq 2. We also give a sample application of this result by showing that the time required to broadcast a message through a mobile network with n nodes above the threshold density for existence of a giant component is O(\log^{1+2/d} n) with high probability.

Abstract:
A binary contingency table is an m x n array of binary entries with prescribed row sums r=(r_1,...,r_m) and column sums c=(c_1,...,c_n). The configuration model for uniformly sampling binary contingency tables proceeds as follows. First, label N=\sum_{i=1}^{m} r_i tokens of type 1, arrange them in m cells, and let the i-th cell contain r_i tokens. Next, label another set of tokens of type 2 containing N=\sum_{j=1}^{n}c_j elements arranged in n cells, and let the j-th cell contain c_j tokens. Finally, pair the type-1 tokens with the type-2 tokens by generating a random permutation until the total pairing corresponds to a binary contingency table. Generating one random permutation takes O(N) time, which is optimal up to constant factors. A fundamental question is whether a constant number of permutations is sufficient to obtain a binary contingency table. In the current paper, we solve this problem by showing a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as N goes to \infty. Our finding shows surprising differences from recent results for binary symmetric contingency tables.

Abstract:
Let the nodes of a Poisson point process move independently in $\R^d$ according to Brownian motions. We study the isolation time for a target particle that is placed at the origin, namely how long it takes until there is no node of the Poisson point process within distance $r$ of it. In the case when the target particle does not move, we obtain asymptotics for the tail {probability} which are tight up to constants in the exponent in dimension $d\geq 3$ and tight up to logarithmic factors in the exponent for dimensions $d=1,2$. In the case when the target particle is allowed to move independently of the Poisson point process, we show that the best strategy for the target to avoid isolation is to stay put.

Abstract:
We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate \mu\ while at the same time a random walker moves on G at rate 1 but only along edges which are open. On the d-dimensional torus with side length n, we prove that in the subcritical regime, the mixing times for both the full system and the random walker are n^2/\mu\ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice Z^d holds for this model as well.

Abstract:
We consider the problem of distributing a vaccine for immunizing a scale-free network against a given virus or worm. We introduce a new method, based on vaccine dissemination, that seems to reflect more accurately what is expected to occur in real-world networks. Also, since the dissemination is performed using only local information, the method can be easily employed in practice. Using a random-graph framework, we analyze our method both mathematically and by means of simulations. We demonstrate its efficacy regarding the trade-off between the expected number of nodes that receive the vaccine and the network's resulting vulnerability to develop an epidemic as the virus or worm attempts to infect one of its nodes. For some scenarios, the new method is seen to render the network practically invulnerable to attacks while requiring only a small fraction of the nodes to receive the vaccine.