arXiv subjects

Rick Durrett

Publications and source records attributed to Rick Durrett.

At least 37 records · Page 2Linked to original sources

arXiv2019

The Contact Process on Random Graphs and Galton-Watson Trees

The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $λ_2=0$ and (ii) when it is geometric($p$) we have $λ_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $λ_c(n)$ for "prolonged persistence" on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $λ_c(n) \to 0$ we give estimates on the rate of convergence. Physicists tell us that $λ_c(n) \sim 1/Λ(n)$ where $Λ(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.

math.PR↗

arXiv2019

Coexistence in chase-escape

We study a competitive stochastic growth model called chase-escape in which red particles spread to adjacent uncolored sites and blue only to adjacent red sites. Red particles are killed when blue occupies the same site. If blue has rate-1 passage times and red rate-$λ$, a phase transition occurs for the probability red escapes to infinity on $\mathbb Z^d$, $d$-ary trees, and the ladder graph $\mathbb Z \times \{0,1\}$. The result on the tree was known, but we provide a new, simpler calculation of the critical value, and observe that it is a lower bound for a variety of graphs. We conclude by showing that red can be stochastically slower than blue, but still escape with positive probability for large enough $d$ on oriented $\mathbb Z^d$ with passage times that resemble Bernoulli bond percolation.

math.PR↗

arXiv2019

Asymptotic behavior of the Brownian frog model

We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix $r>0$ and place a particle at each point $x$ of a unit intensity Poisson point process $\mathcal P \subseteq \mathbb R^d - \mathbb B(0,r)$. Around each point in $\mathcal{P}$, put a ball of radius $r$. A particle at the origin performs Brownian motion. When it hits the ball around $x$ for some $x \in \mathcal P$, new particles begin independent Brownian motions from the centers of the balls in the cluster containing $x$. Subsequent visits to the cluster do nothing. This waking process continues indefinitely. For $r$ smaller than the critical threshold of continuum percolation, we show that the set of activated points in $\mathcal P$ approximates a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process.

math.PR↗

arXiv2019

The Zealot Voter Model

Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0's are ordinary voters and 1's are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontrivial stationary distribution, space is represented by a tree. The dynamics are a variant of the biased voter: if $x$ has degree $d(x)$ then at rate $d(x)p_k$ the individual at $x$ consults $k\ge 1$ neighbors. If at least one neighbor is 1, they adopt state 1, otherwise they become 0. In addition at rate $p_0$ individuals with opinion 1 change to 0. As in the contact process on trees, we are interested in determining when the zealots survive and when they will survive locally.

math.PR↗

arXiv2019

SIR epidemics on evolving graphs

We consider evoSIR, a variant of the SIR model, on Erd\H os-Renyi random graphs in which susceptibles with an infected neighbor break that connection at rate $ρ$ and rewire to a randomly chosen individual. We compute the critical infection rate $λ_c$ and the probability of a large epidemic by showing that they are the same for the delSIR model in which $S-I$ connections are deleted instead of rewired. The final size of a large delSIR epidemic has a continuous transition. Simulations suggest that the final size of a large evoSIR epidemic is discontinuous at $λ_c$.

math.PR↗

arXiv2019

The Contact Process on Periodic Trees

A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that the critical values $λ_1$ and $λ_2$ for global and local survival were different. Here, we will consider the case of trees in which the degrees of vertices are periodic. We will compute bounds on $λ_1$ and $λ_2$ and for the corresponding critical values $λ_g$ and $λ_\ell$ for branching random walk. Much of what we find for period two $(a,b)$ trees was known to Pemantle. However, two significant new results give sharp asymptotics for the critical value $λ_2$ of $(1,n)$ trees and generalize that result to the $(a_1,\ldots, a_k, n)$ tree when $\max_i a_i \le n^{1-ε}$ and $a_1 \cdots a_k = n^b$. We also give results for $λ_g$ and $λ_\ell$ on $(a,b,c)$ trees. Since the values come from solving cubic equations, the explicit formulas are not pretty, but it is surprising that they depend only on $a+b+c$ and $abc$.

math.PR↗

arXiv2018

Block size in Geometric(p)-biased permutations

Fix a probability distribution $\mathbf p = (p_1, p_2, \cdots)$ on the positive integers. The first block in a $\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\infty)$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with positive probability.

math.PR↗

arXiv2018

Poisson percolation on the oriented square lattice

In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component containing the origin in the oriented case. We show that the density of occupied sites at height $y$ in the cluster is close to the percolation probability in the corresponding homogeneous percolation process, and we study the fluctuations of the boundary.

math.PR↗

arXiv2018

Diffusion limit for the partner model at the critical value

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds $I_0$) the extinction time in the critical model is of order $\sqrt{N}$. Here we improve that result by proving the convergence of $i_N(t)=I(\sqrt{N}t)/\sqrt{N}$ to a limiting diffusion. We do this by showing that within a short time, this four dimensional process collapses to two dimensions: the number of $SI$ and $II$ partnerships are constant multiples of the the number of infected singles. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude $\sqrt{N}$ on the original time scale and averages out of the limit theorem for $i_N(t)$. As a by-product of our proof we show that if $τ_N$ is the extinction time of $i_N(t)$ (on the $\sqrt{N}$ time scale) then $τ_N$ has a limit.

math.PR↗

arXiv2017

Poisson percolation on the square lattice

On the square lattice raindrops fall on an edge with midpoint $x$ at rate $\|x\|_\infty^{-α}$. The edge becomes open when the first drop falls on it. Let $ρ(x,t)$ be the probability that the edge with midpoint $x=(x_1,x_2)$ is open at time $t$ and let $n(p,t)$ be the distance at which edges are open with probability $p$ at time $t$. We show that with probability tending to 1 as $t \to \infty$: (i) the cluster containing the origin $\mathbb C_0(t)$ is contained in the square of radius $n(p_c-ε,t)$, and (ii) the cluster fills the square of radius $n(p_c+ε,t)$ with the density of points near $x$ being close to $θ(ρ(x,t))$ where $θ(p)$ is the percolation probability when bonds are open with probability $p$ on $\mathbb Z^2$. Results of Nolin suggest that if $N=n(p_c,t)$ then the boundary fluctuations of $\mathbb C_0(t)$ are of size $N^{4/7}$.

math.PR↗

arXiv2017

Temporal profiles of avalanches on networks

An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles that characterize the growth and decay of avalanches of fixed duration. At the critical point of the dynamics the average avalanche shapes for different durations can be rescaled so that they collapse onto a single universal curve. We apply Markov branching process theory to derive a simple equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology; specifically, on networks with heavy-tailed degree distributions. We give examples using numerical simulations of models for information spreading, neural dynamics, and behaviour adoption and we propose simple experimental tests to quantify whether cascading systems are in the critical state.

physics.soc-ph↗

arXiv2016

The Evolving Voter Model on Thick Graphs

In the evolving voter model, when an individual interacts with a neighbor having an opinion different from theirs, they will with probability $1-α$ imitate the neighbor but with probability $ α$ will sever the connection and choose a new neighbor at random (i) from the graph or (ii) from those with the same opinion. Durrett et al. used simulation and heuristics to study these dynamics on sparse graphs. Recently Basu and Sly have studied this system with $1-α= ν/N$ on a dense Erdős-Rényi graph $G(N,1/2)$ and rigorously proved that there is a phase transition from rapid disconnection into components with a single opinion to prolonged persistence of discordant edges as $ν$ increases. In this paper, we consider the intermediate situation of Erdős-Rényi random graphs with average degree $L=N^a$ where $0 < a < 1$. Most of the paper is devoted to a rigorous analysis of an approximation of the dynamics called the approximate master equation. Using ideas of \cite{LMR} and \cite{Silk} we are able to analyze these dynamics in great detail.

math.PR↗

arXiv2016

Latent Voter Model on Locally Tree-Like Random Graphs

In the latent voter model, which models the spread of a technology through a social network, individuals who have just changed their choice have a latent period, which is exponential with rate $λ$, during which they will not buy a new device. We study site and edge versions of this model on random graphs generated by a configuration model in which the degrees $d(x)$ have $3 \le d(x) \le M$. We show that if the number of vertices $n \to\infty$ and $\log n \ll λ_n \ll n$ then the latent voter model has a quasi-stationary state in which each opinion has probability $\approx 1/2$ and persists in this state for a time that is $\ge n^m$ for any $m<\infty$. Thus, even a very small latent period drastically changes the behavior of the voter model.

math.PR↗

arXiv2016

Genealogies in Expanding Populations

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [25]. The first step, which is easily accomplished using results of Mueller and Tribe [38], is to show that when space and time are rescaled correctly, our biased voter model converges to a Wright-Fisher SPDE. A simple extension of a result of Durrett and Restrepo [18] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. Brunet et al. [8] have conjectured that genealogies in models of this type are described by the Bolthausen-Sznitman coalescent, see [39]. However, in the model we study there are no simultaneous coalescences. Our third and most significant result concerns "tracer dynamics" in which some of the initial particles in the biased voter model are labeled. We show that the joint distribution of the labeled and unlabeled particles converges to the solution of a system of stochastic partial differential equations. A new duality equation that generalizes the one Shiga [44] developed for the Wright-Fisher SPDE is the key to the proof of that result.

math.PR↗

arXiv2015

Evolutionary Games on the Torus with Weak Selection

We study evolutionary games on the torus with $N$ points in dimensions $d\ge 3$. The matrices have the form $\bar G = {\bf 1} + w G$, where ${\bf 1}$ is a matrix that consists of all 1's, and $w$ is small. As in Cox Durrett and Perkins \cite{CDP} we rescale time and space and take a limit as $N\to\infty$ and $w\to 0$. If (i) $w \gg N^{-2/d}$ then the limit is a PDE on ${\bf R}^d$. If (ii) $N^{-2/d} \gg w \gg N^{-1}$, then the limit is an ODE. If (iii) $w \ll N^{-1}$ then the effect of selection vanishes in the limit. In regime (ii) if we introduce a mutation $μ$ so that $μ/w \to \infty$ slowly enough then we arrive at Tarnita's formula that describes how the equilibrium frequencies are shifted due to selection.

math.PR↗

arXiv2015

Coexistence of grass, saplings and trees in the Staver-Levin forest model

In this paper, we consider two attractive stochastic spatial models in which each site can be in state 0, 1 or 2: Krone's model in which 0${}={}$vacant, 1${}={}$juvenile and 2${}={}$a mature individual capable of giving birth, and the Staver-Levin forest model in which 0${}={}$grass, 1${}={}$sapling and 2${}={}$tree. Our first result shows that if $(0,0)$ is an unstable fixed point of the mean-field ODE for densities of 1's and 2's then when the range of interaction is large, there is positive probability of survival starting from a finite set and a stationary distribution in which all three types are present. The result we obtain in this way is asymptotically sharp for Krone's model. However, in the Staver-Levin forest model, if $(0,0)$ is attracting then there may also be another stable fixed point for the ODE, and in some of these cases there is a nontrivial stationary distribution.

math.PR↗

arXiv2014

Spatial Evolutionary Games with small selection coefficients

Here we will use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $Z^d$, $d\ge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $σ^2I$. The games we consider have payoff matrices of the form ${\bf 1} + wG$ where ${\bf 1}$ is matrix of all 1's and $w$ is small and positive. Since our population size $N=\infty$, we call our selection small rather than weak which usually means $w =O(1/N)$. The key to studying these games is the fact that when the dynamics are suitably rescaled in space and time they convergence to solutions of a reaction diffusion equation (RDE). Inspired by work of Ohtsuki and Nowak and Tarnita et al we show that the reaction term is the replicator equation for a modified game matrix and the modifications of the game matrix depend on the interaction kernel only through the values of two or three simple probabilities for an associated coalescing random walk. Two strategy games lead to an RDE with a cubic nonlinearity, so we can describe the phase diagram completely. Three strategy games lead to a pair of coupled RDE, but using an idea from our earlier work, we are able to show that if there is a repelling function for the replicator equation for the modified game, then there is coexistence in the spatial game when selection is small. This enables us to prove coexistence in the spatial model in a wide variety of examples where the replicator equation of the modified game has an attracting equilibrium with all components positive. Using this result we are able to analyze the behavior of four evolutionary games that have recently been used in cancer modeling.

math.PR↗

arXiv2014

Spatial networks evolving to reduce length

Motivated by results of Henry, Pralat and Zhang (PNAS 108.21 (2011): 8605-8610), we propose a general scheme for evolving spatial networks in order to reduce their total edge lengths. We study the properties of the equilbria of two networks from this class, which interpolate between three well studied objects: the Erdős-Rényi random graph, the random geometric graph, and the minimum spanning tree. The first of our two evolutions can be used as a model for a social network where individuals have fixed opinions about a number of issues and adjust their ties to be connected to people with similar views. The second evolution which preserves the connectivity of the network has potential applications in the design of transportation networks and other distribution systems.

physics.soc-ph↗