A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains
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.
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