Random walk on random walks: higher dimensions

Blondel, Oriane; Hilário, Marcelo R.; Santos, Renato Soares dos; Sidoravičius, Vladas; Teixeira, Augusto

doi:10.1214/19-ejp337

Cited by 8 publications

(17 citation statements)

References 32 publications

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“…Let us finally precise that this model or similar ones, have been studied in dimension 2 and above, see for instance [BHSST19,SS18]. A fair amount of our proof only applies in dimension 1, in particular the proof of the item (2) above.…”

Section: 2mentioning

confidence: 89%

Random Walk on the Simple Symmetric Exclusion Process

Hilário

Kious

Teixeira

2020

Commun. Math. Phys.

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We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $$\rho \in [0, 1]$$ ρ ∈ [ 0 , 1 ] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $$\rho $$ ρ except for at most two values $$\rho _-, \rho _+ \in [0, 1]$$ ρ - , ρ + ∈ [ 0 , 1 ] . The asymptotic speed we obtain in our LLN is a monotone function of $$\rho $$ ρ . Also, $$\rho _-$$ ρ - and $$\rho _+$$ ρ + are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

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Section: 2mentioning

confidence: 89%

Random Walk on the Simple Symmetric Exclusion Process

Hilário

Kious

Teixeira

2020

Commun. Math. Phys.

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“…The present article is a continuation of the works [10,12] concerning the behaviour of a random walker in a dynamic random environment (RWDRE) given by a system of independent simple symmetric random walks. These works are focused on the high density regime in one and higher dimensions, respectively.…”

Section: S:intromentioning

confidence: 93%

“…1. Theorems 1.2 and 1.3 are proved with the help of a renormalization scheme taken from [10]. In fact, given the setup developed therein, our problem is reduced to proving two triggering theorems, which are key a priori estimates on the probability of certain undesired events (cf.…”

Section: Lazy Environmentmentioning

confidence: 99%

“…Here we will consider the low density regime in one dimension, and also the case of a strong local drift on particles. As indicated in [10,12], the main challenge in this model stems from the relatively poor mixing properties of the random environment. In fact, these properties become even worse as the density decreases, which poses additional difficulties in our setting.…”

Section: S:intromentioning

confidence: 99%

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Random walk on random walks: low densities

Blondel¹,

Hilário²,

Santos³

et al. 2017

Preprint

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We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or nonlazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.

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“…While physical objects are constrained to three or fewer dimensions, the dimensions of social, technological, biological and other networks span a wide spectrum [5] , [6] , [7] . A proper understanding of dynamical network phenomena, such as random walks [8] , [9] , synchronization and consensus [10] , [11] , [12] , [13] , heat conduction [14] , [15] , information dissemination [16] , [17] , [18] , [19] , [20] , [21] , competition and cooperation [22] , [23] , [24] , [25] and the spread of rumours [26] , [27] , opinions [28] and diseases [29] , [30] , [31] , [32] , [33] , [34] , [35] including the ongoing COVID-19 pandemic [36] , [37] , requires assessment of the process across a broad range of dimensions and other network properties. This requires the development and application of flexible higher-dimensional network models.…”

Section: Introductionmentioning

confidence: 99%

Epidemic dynamics on higher-dimensional small world networks

Wang

Moore

Small

et al. 2022

Applied Mathematics and Computation

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Dimension governs dynamical processes on networks. The social and technological networks which we encounter in everyday life span a wide range of dimensions, but studies of spreading on finite-dimensional networks are usually restricted to one or two dimensions. To facilitate investigation of the impact of dimension on spreading processes, we define a flexible higher-dimensional small world network model and characterize the dependence of its structural properties on dimension. Subsequently, we derive mean field, pair approximation, intertwined continuous Markov chain and probabilistic discrete Markov chain models of a COVID-19-inspired susceptible-exposed-infected-removed (SEIR) epidemic process with quarantine and isolation strategies, and for each model identify the basic reproduction number , which determines whether an introduced infinitesimal level of infection in an initially susceptible population will shrink or grow. We apply these four continuous state models, together with discrete state Monte Carlo simulations, to analyse how spreading varies with model parameters. Both network properties and the outcome of Monte Carlo simulations vary substantially with dimension or rewiring rate, but predictions of continuous state models change only slightly. A different trend appears for epidemic model parameters: as these vary, the outcomes of Monte Carlo change less than those of continuous state methods. Furthermore, under a wide range of conditions, the four continuous state approximations present similar deviations from the outcome of Monte Carlo simulations. This bias is usually least when using the pair approximation model, varies only slightly with network size, and decreases with dimension or rewiring rate. Finally, we characterize the discrepancies between Monte Carlo and continuous state models by simultaneously considering network efficiency and network size.

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Random walk on random walks: higher dimensions

Cited by 8 publications

References 32 publications

Random Walk on the Simple Symmetric Exclusion Process

Random Walk on the Simple Symmetric Exclusion Process

Random walk on random walks: low densities

Epidemic dynamics on higher-dimensional small world networks

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