A Shape Theorem for a One-Dimensional Growing Particle System with a Bounded Number of Occupants per Site
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A Shape Theorem for a One-Dimensional Growing Particle System with a Bounded Number of Occupants per Site Viktor Bezborodov1
· Luca Di Persio2 · Tyll Krueger1
Received: 30 March 2020 / Revised: 28 July 2020 © The Author(s) 2020
Abstract We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also derive a limit estimate and an exponential estimate on the fluctuations of the position of the rightmost particle. Keywords Birth process · Shape theorem · Interacting particle system Mathematics Subject Classification 60K35 · 82C22
1 Introduction In this paper we consider a one-dimensional growing particle system with a finite range of interaction. A configuration is specified by assigning to each site x ∈ Z a number of particles η(x) ∈ {0, 1, . . . , N }, n ∈ N, occupying x. The state space of the process is thus {0, 1, . . . , N }Z . Under additional assumptions such as non-degeneracy and translation invariance, we show that the system spreads linearly in time and the speed can be expressed as an average value of a certain functional over a certain measure. A respective shape theorem and a fluctuation result are given. The first shape theorem was proven in [31] for a discrete-space growth model. A general shape theorem for discrete-space attractive growth models can be found in [18,
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Viktor Bezborodov [email protected] Luca Di Persio [email protected] Tyll Krueger [email protected]
1
Faculty of Electronics, Wrocław University of Science and Technology, Wrocław, Poland
2
Department of Computer Science, The University of Verona, Verona, Italy
123
Journal of Theoretical Probability
Chapter 11]. In the continuous-space settings shape results for growth models have been obtained in [15] for a model of growing sets and in [7] for a continuous-space particle birth process. The asymptotic behavior of the position of the rightmost particle of the branching random walk under various assumptions is given in [17,20], and [16], see also references therein. A sharp condition for a shape theorem for a random walk with restriction is given in [11]. The speed of propagation for a one-dimensional discretespace supercritical branching random walk with an exponential moment condition can be found in [8]. More refined limiting properties have been obtained recently, such as the limiting law of the minimum or the limiting process seen from its tip, see [1–3,5]. Blondel [9] proves a shape result for the East model, which is a non-attractive particle system. A law of large numbers and a central limit theorem for the position of the tip were established in [14] for a stochastic combustion process with a bounded number of particles per site. In many cases the underlying stochastic model is attractive, which enables the application of a subadditive ergodic theorem. Typically shape results have been obtained using the subadditivity pro
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