A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains. (English) Zbl 1154.60075
The authors depart from their previous [Ann. Probab. 33, No. 6, 2402–2462 (2005; Zbl 1111.60074)], apparently unrelated paper on the time rate of the spread of infection in a moving population. Once interpreted as a specific growth model, based on the concept of continuous-time symmetric random walks, one may justifiably transcribe some of its results to the one-dimensional diffusion-limited aggregation. If compared to the standard DLA considerations, where one adds one particle at a time, in the present analysis one has at disposal an inifinity of the like from the start. All particles perform independent continuous-time random walks with a fixed jump rate, until being absorbed by the aggregate. An explicit construction of the growth process in terms of color-chainging particles, allows to give estimates on the asymptotic behavior of the growth process (Theorems 2 and 3). Open problems are mentioned. In view of difficulties with a rigorous proof of the existence of a nontrivial invariant measure in some cases, suitable caricature (toy) models were proposed to support the positive recurrence property validity.
Reviewer: Piotr Garbaczewski (Opole)
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 60G50 | Sums of independent random variables; random walks |
Keywords:
diffusion-limited aggregation; growth model; time rate of aggregation; Markov chains; continuous-time random walk; positive recurrence; Poisson random variablesCitations:
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