arXiv subjects

Alexandre Stauffer

Publications and source records attributed to Alexandre Stauffer.

36 records · Page 2Linked to original sources

arXiv2018

Percolation of Lipschitz surface and tight bounds on the spread of information among mobile agents

We consider the problem of spread of information among mobile agents on the torus. The agents are initially distributed as a Poisson point process on the torus, and move as independent simple random walks. Two agents can share information whenever they are at the same vertex of the torus. We study the so-called flooding time: the amount of time it takes for information to be known by all agents. We establish a tight upper bound on the flooding time, and introduce a technique which we believe can be applicable to analyze other processes involving mobile agents.

cs.DM↗

arXiv2017

Critical density of activated random walks on transitive graphs

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

math.PR↗

arXiv2017

Random walks in random conductances: decoupling and spread of infection

Let $(G,μ)$ be a uniformly elliptic random conductance graph on $\mathbb{Z}^d$ with a Poisson point process of particles at time $t=0$ that perform independent simple random walks. We show that inside a cube $Q_K$ of side length $K$, if all subcubes of side length $\ell<K$ inside $Q_K$ have sufficiently many particles, the particles return to stationarity after $c\ell^2$ time with a probability close to $1$. We also show this result for percolation clusters on locally finite graphs. Using this mixing result, we show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery, where we show positive speed and that the infection survives indefinitely with positive probability.

math.PR↗

arXiv2016

Polynomial mixing of the edge-flip Markov chain for unbiased dyadic tilings

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] \times [b2^{-t}, (b+1)2^{-t}] for non-negative integers a,b,s,t. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Ω(n^{1.38}), improving upon the previously best lower bound of Ω(n\log n) coming from the diameter of the chain.

math.PR↗

arXiv2015

Space-time percolation and detection by mobile nodes

Consider the model where nodes are initially distributed as a Poisson point process with intensity $λ$ 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 $λ$ 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 $λ$ since, for small enough $λ$, 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.

math.PR↗

arXiv2015

Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $σ$ has weight $λ^{|σ|}$, where $λ$ is a positive real parameter, and $|σ|$ is the total length of the edges in $σ$. Empirically, this model exhibits a "phase transition" at $λ=1$ (corresponding to the uniform distribution): for $λ<1$ distant edges behave essentially independently, while for $λ>1$ very large regions of aligned edges appear. We substantiate this picture as follows. For $λ<1$ sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for $λ>1$ we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.

math.PR↗

arXiv2015

Dynamics of Lattice Triangulations on Thin Rectangles

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $λ^{|σ|}$ where $λ>0$ is a parameter and $|σ|$ denotes the total edge length of the triangulation. When $λ\in(0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp(Ω(n^2))$ for $λ>1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $λ=1$.

math.PR↗

arXiv2015

A Lyapunov function for Glauber dynamics on lattice triangulations

We study random triangulations of the integer points $[0,n]^2 \cap\mathbb{Z}^2$, where each triangulation $σ$ has probability measure $λ^{|σ|}$ with $|σ|$ denoting the sum of the length of the edges in $σ$. Such triangulations are called \emph{lattice triangulations}. We construct a height function on lattice triangulations and prove that, in the whole subcritical regime $λ<1$, 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 $λ<1$. 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 $λ<1$. 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.

math.PR↗

arXiv2014

Intersection and mixing times for reversible chains

Suppose X and Y are two independent irreducible Markov chains on n states. We consider the intersection time, which is the first time their trajectories intersect. We show for reversible and lazy chains that the total variation mixing time is always upper bounded by the expected intersection time taken over the worst starting states. For random walks on trees we show the two quantities are equivalent. We obtain an expression for the expected intersection time in terms of the eigenvalues for reversible and transitive chains. For such chains we also show that it is up to constants the geometric mean of n and E[I], where I is the number of intersections up to the uniform mixing time. Finally for random walks on regular graphs we obtain sharp inequalities that relate the expected intersection time to maximum hitting time and mixing time.

math.PR↗

arXiv2014

Balls into bins via local search: cover time and maximum load

We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m = n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m > n.

math.PR↗

arXiv2013

Phase transition for finite-speed detection among moving particles

Consider the model where particles are initially distributed on $\mathbb{Z}^d, \, d\geq 2$, according to a Poisson point process of intensity $λ>0$, 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 $λ_* > 0$, so that if the initial density of particles $λ< λ_*$, then the target can avoid detection forever.

math.PR↗

arXiv2013

Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times

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 μ 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/μ 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.

math.PR↗

arXiv2012

Balls into Bins via Local Search

We propose a natural process for allocating n balls into n bins that are organized as the vertices of an undirected graph G. Each ball first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. In our main result, we prove that this process yields a maximum load of only Θ(\log \log n) on expander graphs. In addition, we show that for d-dimensional grids the maximum load is Θ\Big(\big(\frac{\log n}{\log \log n}\big)^{\frac{1}{d+1}}\Big). Finally, for almost regular graphs with minimum degree Ω(\log n), we prove that the maximum load is constant and also reveal a fundamental difference between random and arbitrary tie-breaking rules.

math.PR↗

arXiv2012

The Isolation Time of Poisson Brownian Motions

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.

math.PR↗

arXiv2012

Perturbing the hexagonal circle packing: a percolation perspective

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 Π_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 Π_t to contain an infinite component converges to that of continuum percolation (which was shown---based on a Monte Carlo estimate---by Balister, Bollobás 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.

math.PR↗

arXiv2011

Characterizing Optimal Sampling of Binary Contingency Tables via the Configuration Model

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.

math.PR↗

arXiv2010

Mobile Geometric Graphs: Detection, Coverage and Percolation

We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of the graph be a Poisson point process in R^d with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes if their distance is at most r. We study three features in this model: detection (the time until a target point---fixed or moving---is within distance r from some node of the graph), coverage (the time until all points inside a finite box are detected by the graph), and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these features by combining ideas from stochastic geometry, coupling and multi-scale analysis.

math.PR↗

arXiv2010

Mobile Geometric Graphs, and Detection and Communication Problems in Mobile Wireless Networks

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(-Θ(t/\log t)) in two dimensions, and \exp(-Θ(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(-Ω(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.

math.PR↗