Coexistence in a random growth model with competition. (English) Zbl 1442.60105

Turnbull, Shane; Turner, Amanda

Electron. Commun. Probab. 25, Paper No. 26, 14 p. (2020).

Summary: We consider a variation of the Hastings-Levitov model HL(0) for random growth in which the growing cluster consists of two competing regions. We allow the size of successive particles to depend both on the region in which the particle is attached, and the harmonic measure carried by that region. We identify conditions under which one can ensure coexistence of both regions. In particular, we consider whether it is possible for the process giving the relative harmonic measures of the regions to converge to a non-trivial ergodic limit.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F17	Functional limit theorems; invariance principles

Keywords:

random growth models; Hastings-Levitov; scaling limits; ergodic limits

References:

[1]	Stewart N. Ethier and Thomas G. Kurtz, Markov processes: Characterization and convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. · Zbl 0592.60049
[2]	M. B. Hastings and L. S. Levitov, Laplacian growth as one-dimensional turbulence, Physica D (1998). · Zbl 0962.76542
[3]	Olav Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. · Zbl 0996.60001
[4]	Harry Kesten, Hitting probabilities of random walks on \(\mathbf{Z}^d \), Stochastic Process. Appl. 25 (1987), no. 2, 165-184. · Zbl 0626.60067
[5]	Eva Löcherbach, Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity, J. Theor. Probab. (2019).
[6]	Sean P. Meyn and R. L. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab. 25 (1993), no. 3, 518-548. · Zbl 0781.60053
[7]	James Norris and Amanda Turner, Hastings-Levitov aggregation in the small-particle limit, Comm. Math. Phys. 316 (2012), no. 3, 809-841. · Zbl 1259.82026
[8]	Vittoria Silvestri, Fluctuation results for Hastings-Levitov planar growth, Probab. Theory Related Fields 167 (2017), no. 1-2, 417-460. · Zbl 1360.60074
[9]	Alan Sola, Amanda Turner, and Fredrik Viklund, One-dimensional scaling limits in a planar Laplacian random growth model, Comm. Math. Phys. 371 (2019), no. 1, 285-329. · Zbl 1426.82036
[10]	T.