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.
Highlights
In this paper we consider a one-dimensional growing particle system with a finite range of interaction
The state space of the process is {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 general shape theorem for discrete-space attractive growth models can be found in [18, B Viktor Bezborodov viktor.bezborodov@pwr.edu.pl Luca Di Persio luca.dipersio@univr.it Tyll Krueger tyll.krueger@pwr.wroc.pl 1 Faculty of Electronics, Wrocław University of Science and Technology, Wrocław, Poland 2 Department of Computer Science, The University of Verona, Verona, Italy Journal of Theoretical Probability (2021) 34:2265–2284
Summary
In this paper we consider a one-dimensional growing particle system with a finite range of interaction. A general shape theorem for discrete-space attractive growth models can be found in [18,. Blondel [9] proves a shape result for the East model, which is a non-attractive particle system. Shape results have been obtained using the subadditivity property in one form or another This is the case for the systems of motionless particles listed above (see, among others, [7,15,18]) and for those with moving particles, see, e.g., shape theorem for the frog model [6]. A certain kind of subadditivity was used in [27], where a shape theorem for a non-attractive model involving two types of moving particles is given.
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