arXiv subjects

Rick Durrett

Publications and source records attributed to Rick Durrett.

At least 19 recordsLinked to original sources

arXiv2026

Discontinuous phase transitions in the one-dimensional pair and triplet creation processes

Dickman and Tom\'e (1991) introduced models on $Z$ in which $k$ consecutive occupied sites give birth at rate $\lambda$, individual particles die at rate 1, and the configuration is subject to nearest neighbor stirring. They claimed that the pair creation model ($k=2$) has a continuous phase transition and was in the universality class of directed percolation, but the triplet creation model ($k=3$) has a discontinuous (or first-order) phase transition when the stirring rate is large. Hinrichsen (2000a) disputed the second claim and argued that first-order phase transitions in 1+1-dimensional nonequilibrium systems with fluctuating ordered phases are impossible. In this paper we prove rigorously that the phase transitions are discontinuous in both the pair and triplet models when the stirring rate is large. At the end of Section 2 we state some open problems and suggest reasons that the results in the physics literature differ from those presented here,

math.PR↗

arXiv2024

Critical first passage percolation on random graphs

In 1999, Zhang proved that, for first passage percolation on the square lattice $\mathbb{Z}^2$ with i.i.d. non-negative edge weights, if the probability that the passage time distribution of an edge $P(t_e = 0) =1/2 $, the critical value for bond percolation on $\mathbb{Z}^2$, then the passage time from the origin $0$ to the boundary of $[-n,n]^2$ may converge to $\infty$ or stay bounded depending on the nature of the distribution of $t_e$ close to zero. In 2017, Damron, Lam, and Wang gave an easily checkable necessary and sufficient condition for the passage time to remain bounded. Concurrently, there has been tremendous growth in the study of weak and strong disorder on random graph models. Standard first passage percolation with strictly positive edge weights provides insight in the weak disorder regime. Critical percolation on such graphs provides information on the strong disorder (namely the minimal spanning tree) regime. Here we consider the analogous problem of Zhang but now for a sequence of random graphs $\{G_n:n\geq 1\}$ generated by a supercritical configuration model with a fixed degree distribution. Let $p_c$ denote the associated critical percolation parameter, and suppose each edge $e\in E(G_n)$ has weight $t_e \sim p_c \delta_0 +(1-p_c)\delta_{F_\zeta}$ where $F_\zeta$ is the cdf of a random variable $\zeta$ supported on $(0,\infty)$. The main question of interest is: when does the passage time between two randomly chosen vertices have a limit in distribution in the large network $n\to \infty$ limit? There are interesting similarities between the answers on $\mathbb{Z}^2$ and on random graphs, but it is easier for the passage times on random graphs to stay bounded.

math.PR↗

arXiv2024

Unusual properties of contact processes on percolated graphs

In this paper we will consider the contact process in a very simple type of random environment that physicists call the random dilution model. We start with the contact process on a graph, here either $\mathbb{Z}^d$, a $d$-dimensional torus or an \ER graph, and then flip independent $(1-p)$ coins to delete edges, or delete vertices. Let $p^*$ be the threshold for percolation in the diluted graph. We will primarily be concerned with two phenomena. (i) The critical value for the contact process on the dliuted graph $\lambda_c(p)$ does not converge to $\infty$ as $p \downarrow p^*$. (ii) In contrast to the contact process on a homogeneous graph, the density of 1's starting from all sites occupied converges to 0 at a polynomial rate when $p<p^*$ (the ``Griffiths phase'') and like $c/(\log t)^a$ when $p=p^*$.

math.PR↗

arXiv2023

The phase transition in Chung-Lu graphs

In 2002, Chung and Lu introduced a version of the Erdos-Renyi model which an edge between $i$ and $j$ is present with probability $p(i,j)$. They applied this model to compute the diameter of power-law random graphs, with yielded easier proofs than those for the configuration model. In 2007 their model was brought and integrated under the umbrella of Bolobas, Janson, and Riordan's inhomogeneous random graphs. However, the properties of the Chung-Lu model were never fully explored. In this paper, we fill the gap by giving a result for the cluster sizes in the subcritical regime and the fraction of vertices in the giant component in the supercritical phase.

math.PR↗

arXiv2023

Critical behavior of two-choice rules: a class of Achlioptas processes

Achlioptas processes are a class of dynamically grown random graphs where on each step several edges are chosen at random but only one is added. The sum rule, product rule, and bounded size rules have been extensively studied. Here we introduce a new collection of rules called two-choice rules. In these systems one first pick $m$ vertices at random from the graph and chooses a vertex $v$ according to some rule based on their cluster sizes. The procedure is then repeated with a second independent sample to pick a vertex $v'$ and we add an edge from $v$ and $v'$. These systems are tractable because the cluster size distribution satisfies an analog of the Smoluchowski equation. We study the critical exponents associated with the phase transitions in five of these models. In contrast to the situation for $d$-dimensional percolation we show that all of the critical exponents can be computed if we know $\beta$, the exponent associated with the size of the giant component. When $\beta=1$ all the critical exponents are the same as for the \ER graph.

math.PR↗

arXiv2023

Triple birthday matches in the Senate: Lies, damned lies and chatGPT

Our question is ``What is the probability that at least three members of the senate share the same birthday?'' Before the pandemic, I asked this question in several popular math talks I gave at universities across the country. Inspired by ChatGPT's abysmal failure to answer the question, I have recently come back to this problem and now have a more satisfactory answer, thanks in no small part to what I learned form a page of Wolfram's Math World, which I located by a Google search.

math.HO↗

arXiv2022

Competitive exclusion in a model with seasonality: three species cannot coexist in an ecosystem with two seasons

Chan, Durrett, and Lanchier introduced a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into two seasons. They proved that there is an open set of the parameters for which both species can coexist when their dispersal range is sufficiently large. Numerical simulations suggested that three species can coexist in the presence of two seasons. The main point of this paper is to prove that this conjecture is incorrect. To do this we prove results for a more general ODE model and contrast its behavior with other related systems that have been studied in order to understand the competitive exclusion principle.

q-bio.PE↗

arXiv2020

A stochastic spatial model for the sterile insect control strategy

In the system we study, 1's and 0's represent occupied and vacant sites in the contact process with births at rate $\lambda$ and deaths at rate 1. $-1$'s are sterile individuals that do not reproduce but appear spontaneously on vacant sites at rate $\alpha$ and die at rate $\theta\alpha$. We show that the system (which is attractive but has no dual) dies out at the critical value and has a nontrivial stationary distribution when it is supercritical. Our most interesting results concern the asymptotics when $\alpha\to 0$. In this regime the process resembles the contact process in a random environment.

math.PR↗

arXiv2020

Controlling the spread of COVID-19 on college campuses

This research was done during the DOMath program at Duke University from May 18 to July 10, 2020. At the time, Duke and other universities across the country were wrestling with the question of how to safely welcome students back to campus in the Fall. Because of this, our project focused on using mathematical models to evaluate strategies to suppress the spread of the virus on campus, specifically in dorms and in classrooms. For dorms, we show that giving students single rooms rather than double rooms can substantially reduce virus spread. For classrooms, we show that moving classes with size above some cutoff online can make the basic reproduction number $R_0<1$, preventing a wide spread epidemic. The cutoff will depend on the contagiousness of the disease in classrooms.

math.PR↗

arXiv2020

The Life and Mathematical Legacy of Thomas M. Liggett

Thomas Milton Liggett was a world renowned UCLA probabilist, famous for his monograph Interacting Particle Systems. He passed away peacefully on May 12, 2020. This is a perspective article in memory of both Tom Liggett the person and Tom Liggett the mathematician.

math.PR↗

arXiv2020

The q-voter model on the torus

In the $q$-voter model, the voter at $x$ changes its opinion at rate $f_x^q$, where $f_x$ is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if $q<1$ and clustering if $q>1$. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for $q$ close to 1. More precisely, we show that if $q<1$, then for any $m<\infty$ the process on the three-dimensional torus with $n$ points survives for time $n^m$, and after an initial transient phase has a density that it is always close to 1/2. If $q>1$, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time $\log n$ but the stochastic process on the same time scale dies out at time $(1/3)\log n$.

math.PR↗

arXiv2020

Motion by mean curvature in interacting particle systems

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al there were two nontrivial stationary distributions.

math.PR↗

arXiv2020

Susceptible-Infected Epidemics on Evolving Graphs

The evoSIR model is a modification of the usual SIR process on a graph $G$ in which $S-I$ connections are broken at rate $\rho$ and the $S$ connects to a randomly chosen vertex. The evoSI model is the same as evoSIR but recovery is impossible. In \cite{DOMath} the critical value for evoSIR was computed and simulations showed that when $G$ is an Erd\H os-R\'enyi graph with mean degree 5, the system has a discontinuous phase transition, i.e., as the infection rate $\lambda$ decreases to $\lambda_c$, the fraction of individuals infected during the epidemic does not converge to 0. In this paper we study evoSI dynamics on graphs generated by the configuration model. We show that there is a quantity $\Delta$ determined by the first three moments of the degree distribution, so that the phase transition is discontinuous if $\Delta>0$ and continuous if $\Delta<0$.

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 on homogeneous trees the critical values $\lambda_1$ and $\lambda_2$ for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generation is $(n,a_1,\ldots, a_k)$ with $\max_i a_i \le Cn^{1-\delta}$ and $\log(a_1 \cdots a_k)/\log n \to b$ as $n\to\infty$. We show that the critical value for local survival is asymptotically $\sqrt{c (\log n)/n}$ where $c=(k-b)/2$. This supports Pemantle's claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.

math.PR↗

arXiv2019

The Symbiotic Contact Process

We consider a contact process on $Z^d$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $\mu \le 1$ for each particle at that site. We show the critical birth rate $\lambda_c(\mu)$ for weak survival is of order $\sqrt{\mu}$ as $\mu \to 0$. Mean-field calculations predict that when $\mu < 1/2$ there is a discontinuous transition as $\lambda$ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to physicists the paper that introduced the model, the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.

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 $\rho$ and rewire to a randomly chosen individual. We compute the critical infection rate $\lambda_c$ 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 $\lambda_c$.

math.PR↗

arXiv2018

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 $\lambda_2=0$ and (ii) when it is geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $\lambda_c(n)$ 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 $\lambda_c(n) \to 0$ we give estimates on the rate of convergence. Physicists tell us that $\lambda_c(n) \sim 1/\Lambda(n)$ where $\Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.

math.PR↗

arXiv2018

Spatial heterogeneity can explain the stable coexistence of savanna and forest in South America

In work with a variety of co-authors, Staver and Levin have argued that savanna and forest coexist as alternative stable states with discontinuous changes in density of trees at the boundary. Here we formulate a nonhomogeneous spatial model of the competition between forest and savanna. We prove that coexistence occurs for a time that is exponential in the size of the system, and that after an initial transient, boundaries between the alternative equilibria remain stable. This result is valid in general for systems that exhibit bistability in a homogeneous environment.

math.PR↗