arXiv subjects

Iddo Ben-Ari

Publications and source records attributed to Iddo Ben-Ari.

At least 19 recordsLinked to original sources

arXiv2025

On quasi-stationary distributions for stochastic rumor models

This paper examines the quasi-stationary behavior of stochastic rumor processes. Using the results by van Doorn and Pollett (2008), we first prove that the continuous-time Maki--Thompson model has a unique quasi-stationary distribution (QSD) given by the point mass at the state $(0, 1)$. To obtain a non-trivial QSD, we modify the absorption set by conditioning the process on not returning to the level $y=1$ after leaving the initial state $(N, 1)$. For this modified model, we establish the existence and uniqueness of a non-trivial QSD that assigns positive probability to all transient states, and then derive an explicit formula for this QSD in terms of paths and transition rates. We also discuss the ratio of expectations distribution as an alternative approach to describe the long-term behavior before absorption. The analysis is further extended to the Daley--Kendall rumor model and the stochastic SIR epidemic model.

math.PR↗

arXiv2024

Representation and Characterization of Quasistationary Distributions for Markov Chains

This work provides complete description of Quasistationary Distributions (QSDs) for Markov chains with a unique absorbing state and an irreducible set of non-absorbing states. As is well-known, every QSD has an associated absorption parameter describing the exponential tail of the absorption time under the law of the process with the QSD as the initial distribution. The analysis associated with the existence and representation of QSDs corresponding to a given parameter is according to whether the moment generating function of the absorption time starting from any non-absorbing state evaluated at the parameter is finite or infinite, the finite or infinite moment generating function regimes, respectively. For parameters in the finite regime, it is shown that when exist, all QSDs are in the convex cone of a Martin entry boundary associated with the parameter. The infinite regime corresponds to at most one parameter value and at most one QSD. In this regime, when a QSD exists, it is unique and can be represented by a renewal-type formula. Multiple applications to the findings are presented, including revisiting some of the main classical results in the area.

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arXiv2023

Can a single migrant per generation rescue a dying population?

We introduce a population model to test the hypothesis that even a single migrant per generation may rescue a dying population. Let $(c_k)$ be a sequence of real numbers in $(0,1)$. Let $X_n$ be a size of the population at time $n\geq 0$. Then, $X_{n+1}=X_n - Y_{n+1}+1$, where the conditional distribution of $Y_{n+1}$ given $X_n=k$ is a binomial random variable with parameters $(k ,c(k))$. We assume that $\lim_{k\to\infty}kc(k)=ρ$ exists. If $ρ<1$ the process is transient with speed $1-ρ$ (so yes a single migrant per generation may rescue a dying population!) and if $ρ>1$ the process is positive recurrent. In the critical case $ρ=1$ the process is recurrent or transient according to how $k c(k)$ converges to $1$. When $ρ=0$ and under some regularity conditions, the support of the increments is eventually finite.

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arXiv2022

Quasistationary Distribution for the Invasion Model on a Complete Bipartite Graph

The Invasion Model on the complete bibartitle graph was introduced and studied by physicists as a rudimentary model for opinion dynamics on complex networks. We identify the limit of the Quasistationary distribution for the model as one partition size tends to infinity. The limit is a highly dispersed measure. A distinctive feature of the model is that of two time scales with non-trivial interaction. The work and the results complement and are in sharp contrast to the analogous results on the closely related Voter Model.

math.PR↗

arXiv2021

Quasi-Stationary Distributions for the Voter Model on Complete Bipartite Graphs

We consider the discrete-time voter model on complete bipartite graphs and study the quasi-stationary distribution (QSD) for the model as the size of one of the partitions tends to infinity while the other partition remains fixed. We show that the QSDs converge weakly to a nontrivial limit which features a consensus with the exception of a random number of dissenting vertices in the "large" partition. Moreover, we explicitly calculate the law of the number of dissenters and show that it follows the heavy-tailed Sibuya distribution with parameter depending on the size of the "small" partition. Our results rely on a discrete-time analogue of the well-known duality between the continuous-time voter model and coalescing random walks which we develop in the paper.

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arXiv2021

Self-similarity in an exchangeable site-dynamics model

We consider a model for which every site of $\mathbb{N}$ is assigned a fitness in $[0,1]$. At every discrete time all the sites are updated and each site samples a uniform on $[0,1]$, independently of everything else. At every discrete time and independently of the past the environment is good with probability $p$ or bad with probability $1-p$. The fitness of each site is then updated to the maximum or the minimum between its present fitness and the sampled uniform, according to whether the environment is good or bad. Assuming the initial fitness distribution is exchangeable over the site indexing, the empirical fitness distribution is a probability-valued Markov process. We show that this Markov process converges to an explicitly-identified stationary distribution exhibiting a self-similar structure.

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arXiv2020

Quasi-Limiting Behavior of Drifted Brownian Motion

A Quasi-Stationary Distribution (QSD)for a Markov process with an almost surely hit absorbing state is a time-invariant initial distribution for the process conditioned on not being absorbed by any given time. An initial distribution for the process is in the domain of attraction of some QSD $ν$ if the distribution of the process a time $t$, conditioned not to be absorbed by time $t$ converges to $ν$. In this work study mostly Brownian motion with constant drift on the half line $[0,\infty)$ absorbed at $0$. Previous work by Martinez et al. identifies all QSDs and provides a nearly complete characterization for their domain of attraction. Specifically, it was shown that if the distribution a well-defined exponential tail (including the case of lighter than any exponential tail), then it is in the domain of attraction of a QSD determined by the exponent. In this work we 1. Obtain a new approach to existing results, explaining the direct relation between a QSD and an initial distribution in its domain of attraction. 2. Study the behavior under a wide class of initial distributions whose tail is heavier than exponential, and obtain no-trivial limits under appropriate scaling.

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arXiv2020

Power-Law Tails in a Fitness-Driven Model for Biological Evolution

We study a discrete-time stochastic process that can also be interpreted as a model for a viral evolution. A distinguishing feature of our process is power-law tails due to dynamics that resembles preferential attachment models. In the model we study, a population is partitioned into sites, with each site labeled by a uniquely-assigned real number in the interval $[0,1]$ known as fitness. The population size is a discrete-time transient birth-and-death process with probability $p$ of birth and $1-p$ of death. The fitness is assigned at birth according to the following rule: the new member of the population either "mutates" with probability $r$, creating a new site uniformly distributed on $[0,1]$ or "inherits" with probability $1-r$, joining an existing site with probability proportional to the site's size. At each death event, a member from the site with the lowest fitness is killed. The number of sites eventually tends to infinity if and only if $pr>1-p$. Under this assumption, we show that as time tends to infinity, the joint empirical measure of site size and fitness (proportion of population in sites of size and fitness in given ranges) converges a.s. to the product of a modified Yule distribution and the uniform distribution on $[(1-p)/(pr),1]$. Our approach is based on the method developed in \cite{similar-but-different}. The model and the results were independently obtained by Roy and Tanemura in [RT].

math.PR↗

arXiv2020

Finite-Memory Elephant Random Walk and the Central Limit Theorem for Additive Functionals

The Central Limit Theorem (CLT) for additive functionals of Markov chains is a well known result with a long history. In this paper we present applications to two finite-memory versions of the Elephant Random Walk, solving a problem from arXiv:1812.01915. We also present a derivation of the CLT for additive functionals of finite state Markov chains, which is based on positive recurrence, the CLT for IID sequences and some elementary linear algebra, and which focuses on characterization of the variance.

math.PR↗

arXiv2019

On Transformations of Markov Chains and Poisson Boundary

A discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded harmonic function with respect to the transition function of the original chain is harmonic with respect to the transition function of the transformed chain. The reverse inclusion is in general not true. Our main result provides a sufficient condition on the stopping time which guarantees that the space of bounded harmonic functions for the transformed chain embeds in the space of bounded harmonic sequences for the original chain. We also obtain a similar result on positive unbounded harmonic functions, under some additional conditions. Our work was motivated by and is analogous to Forghani-Kaimanovich, the well-studied case when the Markov chain is a random walk on a discrete group.

math.PR↗

arXiv2018

The subcritical phase for a homopolymer model

We study a model of continuous-time nearest-neighbor random walk on $\mathbb{Z}^d$ penalized by its occupation time at the origin, also known as a homopolymer. For a fixed real parameter $β$ and time $t>0$, we consider the probability measure on paths of the random walk starting from the origin whose Radon-Nikodym derivative is proportional to the exponent of the product $β$ times the occupation time at the origin up to time $t$. The case $β>0$ was studied previously by Cranston and Molchanov arXiv:1508.06915. We consider the case $β<0$, which is intrinsically different only when the underlying walk is recurrent, that is $d=1,2$. Our main result is a scaling limit for the distribution of the homopolymer on the time interval $[0,t]$, as $t\to\infty$, a result that coincides with the scaling limit for penalized Brownian motion due to Roynette and Yor. In two dimensions, the penalizing effect is asymptotically diminished, and the homopolymer scales to standard Brownian motion. Our approach is based on potential analytic and martingale approximation for the model. We also apply our main result to recover a scaling limit for a wetting model. We study the model through analysis of resolvents.

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arXiv2017

A random walk with catastrophes

Random population dynamics with catastrophes (events pertaining to possible elimination of a large portion of the population) has a long history in the mathematical literature. In this paper we study an ergodic model for random population dynamics with linear growth and binomial catastrophes: in a catastrophe, each individual survives with some fixed probability, independently of the rest. Through a coupling construction, we obtain sharp two-sided bounds for the rate of convergence to stationarity which are applied to show that the model exhibits a cutoff phenomenon.

math.PR↗

arXiv2017

On a Local Version of the Bak-Sneppen Model

A major difficulty in studying the Bak-Sneppen model is in effectively comparing it with well-understood models. This stems from the use of two geometries: complete graph geometry to locate the global fitness minimizer, and graph geometry to replace the species in the neighborhood of the minimizer. We present a variant in which only the graph geometry is used. This allows to obtain the stationary distribution through random walk dynamics. We use this to show that for constant-degree graphs, the stationary fitness distribution converges to an IID law as the number of vertices tends to infinity. We also discuss exponential ergodicity through coupling, and avalanches for the model.

math.PR↗

arXiv2016

A Probabilistic Approach to Generalized Zeckendorf Decompositions

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.

math.PR↗

arXiv2014

Efficient Coupling for Random Walk with Redistribution

What can one say on convergence to stationarity of a finite state Markov chain that behaves "locally" like a nearest neighbor random walk on ${\mathbb Z}$ ? The model we consider is a version of nearest neighbor lazy random walk on the state space $ \{0,\dots,N\}$: the probability for staying put at each site is $\frac 12$, the transition to the nearest neighbors, one on the right and one on the left, occurs with probability $\frac14$ each, where we identify two sites, $J_0$ and $J_N$ as, respectively, the neighbor of $0$ from the left and the neighbor of $N$ from the right (but $0$ is not a neighbor of $J_0$ and $N$ is not neighbor of $J_N$). This model is a discrete version of diffusion with redistribution on an interval studied by several authors in recent past, and for which the the exponential rates of convergence to stationarity were computed analytically, but had no intuitive or probabilistic interpretation, except for the case where the jumps from the endpoints are identical (or more generally have the same distribution). We study convergence to stationarity probabilistically, by finding an efficient coupling. The coupling identifies the "bottlenecks" responsible for the rates of convergence and also gives tight computable bounds on the total variation norm of the process between two starting points. The adaptation to the diffusion case is straightforward.

math.PR↗

arXiv2012

Principal Eigenvalue for Brownian Motion on a Bounded Interval with Degenerate Instantaneous Jumps

We consider a model of Brownian motion on a bounded open interval with instantaneous jumps. The jumps occur at a spatially dependent rate given by a positive parameter times a continuous function positive on the interval and vanishing on its boundary. At each jump event the process is redistributed uniformly in the interval. We obtain sharp asymptotic bounds on the principal eigenvalue for the generator of the process as the parameter tends to infinity. Our work answers a question posed by Arcusin and Pinsky.

math.PR↗

arXiv2010

On a species survival model

In this paper we provide some sharp asymptotic results for a stochastic model of species survival recently proposed by Guiol, Marchado, and Schinazi.

math.PR↗

arXiv2007

A random walk on Z with drift driven by its occupation time at zero

We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes according to the rate of decay of the drift. In particular, when the rate is sufficiently slow, the position of the random walk, properly normalized, converges to a symmetric exponential law. In this regime, in contrast to the classical case, the range of the walk scales differently from its position.

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