arXiv subjects

Erin Beckman

Publications and source records attributed to Erin Beckman.

9 recordsLinked to original sources

arXiv2025

Chase-escape with conversion as a multiple sclerosis lesion model

We introduce conversion to the stochastic process known as chase-escape in an effort to model aspects of inflammatory damage from multiple sclerosis. We prove monotonicity results for aggregate damage for the model on the positive integers, trees, stars, and the complete graph. Additionally, we establish the existence and asymptotic order of a phase transition on bounded degree graphs with a non-trivial site percolation threshold.

math.PR↗

arXiv2024

Inferring birth versus death dynamics for ecological interactions in stochastic heterogeneous populations

In this paper, we study the significance of ecological interactions and separation of birth and death dynamics in stochastic heterogeneous populations via general birth-death processes. Interactions can manifest through the birth dynamics, the death dynamics, or some combination of the two. The underlying microscopic mechanisms are important but often implicit in population-level data. We propose an inference method for disambiguating the types of interaction and the birth and death processes from population size time series data of a stochastic $n$-type heterogeneous population. The interspecies interactions considered can be competitive, antagonistic, or mutualistic. We show that different pairs of birth and death rates with the same net growth rate result in different time series statistics. Then, the inference method is validated in the example of a birth-death process inspired by the two-type Lotka-Volterra interaction dynamics. Utilizing stochastic fluctuations enables us to estimate additional parameters in this stochastic Lotka-Volterra model, which are not identifiable in a deterministic model.

q-bio.PE↗

arXiv2022

The central limit theorem via doubling of variables

We give a new, self-contained proof of the multidimensional central limit theorem using the technique of ``doubling variables," which is traditionally used to prove uniqueness of solutions of partial differential equations (PDEs). Our technique also yields quantitative bounds for random variables with finite $2+γ$ moment for some $γ\in (0,1]$; when $γ=1$, this proves a version of the Berry--Esseen theorem in $\mathbb{R}^d$.

math.PR↗

arXiv2022

Symmetric cooperative motion in one dimension

We explore the relationship between recursive distributional equations and convergence results for finite difference schemes of parabolic partial differential equations (PDEs). We focus on a family of random processes called symmetric cooperative motions, which generalize the symmetric simple random walk and the symmetric hipster random walk introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, arXiv:1909.07367]. We obtain a distributional convergence result for symmetric cooperative motions and, along the way, obtain a novel proof of the Bernoulli central limit theorem. In addition, we prove a PDE result relating distributional solutions and viscosity solutions of the porous medium equation and the parabolic $p$-Laplace equation, respectively, in one dimension.

math.PR↗

arXiv2021

Asymmetric cooperative motion in one dimension

We prove distributional convergence for a family of random processes on $\mathbb{Z}$, which we call asymmetric cooperative motions. The model generalizes the "totally asymmetric hipster random walk" introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, 2020]. We present a novel approach based on connecting a temporal recurrence relation satisfied by the cumulative distribution functions of the process to the theory of finite difference schemes for Hamilton-Jacobi equations [Crandall and Lyons, 1984]. We also point out some surprising lattice effects that can persist in the distributional limit, and propose several generalizations and directions for future research.

math.PR↗

arXiv2020

Chase-escape with death on trees

Chase-escape is a competitive growth process in which red particles spread to adjacent uncolored sites, while blue particles overtake adjacent red particles. We introduce the variant in which red particles die and describe the phase diagram for the resulting process on infinite d-ary trees. A novel connection to weighted Catalan numbers makes it possible to characterize the critical behavior.

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↗

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

The frog model on trees with drift

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a $d$-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as $d$ tends to infinity along certain subsequences.

math.PR↗