Two-type annihilating systems on the complete and star graph. (English) Zbl 1475.60194
Summary: Red and blue particles are placed in equal proportion throughout either the complete or star graph and iteratively sampled to take simple random walk steps. Mutual annihilation occurs when particles with different colors meet. We compare the time it takes to extinguish every particle to the analogous time in the (simple to analyze) one-type setting. Additionally, we study the effect of asymmetric particle speeds.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 05C81 | Random walks on graphs |
References:
| [1] | Arratia, Richard, Limiting point processes for rescalings of coalescing and annihilating random walks on \(\mathbb{Z}^d\), Ann. Probab., 909-936 (1981) · Zbl 0496.60098 |
| [2] | Arratia, Richard, Site recurrence for annihilating random walks on \(\mathbb{Z}^d\), Ann. Probab., 11, 3, 706-713 (1983) · Zbl 0517.60075 |
| [3] | Bahl, Riti; Barnet, Philip; Johnson, Tobias; Junge, Matthew, Diffusion-limited annihilating systems and the increasing convex order (2021), ArXiv ID: 2104.12797 |
| [4] | Bramson, Maury; Griffeath, David, Asymptotics for interacting particle systems on \(\mathbb{Z}^d\), Z. Wahrscheinlichkeitstheor. Verwandte Geb., 53, 2, 183-196 (1980) · Zbl 0417.60097 |
| [5] | Bramson, Maury; Lebowitz, Joel L., Asymptotic behavior of densities in diffusion-dominated annihilation reactions, Phys. Rev. Lett., 61, 21, 2397 (1988) |
| [6] | Bramson, Maury; Lebowitz, Joel L., Asymptotic behavior of densities in diffusion dominated two-particle reactions, Physica A, 168, 1, 88-94 (1990) |
| [7] | Bramson, Maury; Lebowitz, Joel L., Asymptotic behavior of densities for two-particle annihilating random walks, J. Stat. Phys., 62, 1-2, 297-372 (1991) · Zbl 0739.60091 |
| [8] | Bramson, Maury; Lebowitz, Joel L., Spatial structure in diffusion-limited two-particle reactions, Ann. Appl. Probab., 11, 1, 121-181 (2001) · Zbl 1016.60088 |
| [9] | Cabezas, M.; Rolla, L. T.; Sidoravicius, V., Non-equilibrium phase transitions: Activated random walks at criticality, J. Stat. Phys., 155, 6, 1112-1125 (2014) · Zbl 1297.82025 |
| [10] | Cabezas, M.; Rolla, L. T.; Sidoravicius, V., Recurrence and density decay for diffusion-limited annihilating systems, Probab. Theory Related Fields, 170, 3, 587-615 (2018) · Zbl 1429.60071 |
| [11] | Cooper, Colin; Elsasser, Robert; Ono, Hirotaka; Radzik, Tomasz, Coalescing random walks and voting on connected graphs, SIAM J. Discrete Math., 27, 4, 1748-1758 (2013) · Zbl 1361.05120 |
| [12] | Cooper, Colin; Frieze, Alan; Radzik, Tomasz, Multiple random walks and interacting particle systems, (International Colloquium on Automata, Languages, and Programming (2009), Springer), 399-410 · Zbl 1247.05222 |
| [13] | Cooper, Colin; Frieze, Alan; Radzik, Tomasz, Multiple random walks in random regular graphs, SIAM J. Discrete Math., 23, 4, 1738-1761 (2009) · Zbl 1207.05179 |
| [14] | Cox, J. Theodore, Coalescing random walks and voter model consensus times on the torus in \(\mathbb{Z}^d\), Ann. Probab., 17, 4, 1333-1366 (1989) · Zbl 0685.60100 |
| [15] | Damron, Michael; Gravner, Janko; Junge, Matthew; Lyu, Hanbaek; Sivakoff, David, Parking on transitive unimodular graphs, Ann. Appl. Probab., 29, 4, 2089-2113 (2019) · Zbl 1466.60201 |
| [16] | Durrett, Rick, Probability: Theory and Examples, Vol. 49 (2019), Cambridge University Press · Zbl 1440.60001 |
| [17] | Erdos, P.; Ney, P., Some problems on random intervals and annihilating particles, Ann. Probab., 2, 5, 828-839 (1974) · Zbl 0297.60052 |
| [18] | Goldschmidt, Christina; Przykucki, Michał, Parking on a random tree, Combin. Probab. Comput., 28, 1, 23-45 (2019) · Zbl 1434.60043 |
| [19] | Johnson, Tobias; Junge, Matthew; Lyu, Hanbaek; Sivakoff, David, Particle density in diffusion-limited annihilating systems (2020), arXiv e-prints |
| [20] | Kanade, Varun; Mallmann-Trenn, Frederik; Sauerwald, Thomas, On coalescence time in graphs: When is coalescing as fast as meeting?, (Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (2019), SIAM), 956-965 · Zbl 1434.05137 |
| [21] | Kang, K.; Redner, S., Scaling approach for the kinetics of recombination processes, Phys. Rev. Lett., 52, 955-958 (1984) |
| [22] | Konheim, Alan G.; Weiss, Benjamin, An occupancy discipline and applications, SIAM J. Appl. Math., 14, 6, 1266-1274 (1966) · Zbl 0201.50204 |
| [23] | Lackner, Marie-Louise; Panholzer, Alois, Parking functions for mappings, J. Combin. Theory Ser. A, 142, 1-28 (2016) · Zbl 1337.05048 |
| [24] | Lee, Benjamin P.; Cardy, John, Renormalization group study of the \(A + B \to 0\) diffusion-limited reaction, J. Stat. Phys., 80, 5, 971-1007 (1995) · Zbl 1081.82599 |
| [25] | Lootgieter, J.-C., Problèmes de récurrence concernant des mouvements aléatoires de particules sur z avec destruction, (Annales de l’IHP Probabilités et statistiques, Vol. 13, No. 2 (1977)), 127-139 · Zbl 0383.60080 |
| [26] | Ovchinnikov, A. A.; Zeldovich, Ya B., Role of density fluctuations in bimolecular reaction kinetics, Chem. Phys., 28, 1-2, 215-218 (1978) |
| [27] | Przykucki, Michał; Roberts, Alexander; Scott, Alex, Parking on the integers (2019), ArXiv preprint arXiv:1907.09437 |
| [28] | Schwartz, Diane, On hitting probabilities for an annihilating particle model, Ann. Probab., 398-403 (1978) · Zbl 0404.60098 |
| [29] | Toussaint, Doug; Wilczek, Frank, Particle-antiparticle annihilation in diffusive motion, J. Chem. Phys., 78, 5, 2642-2647 (1983) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
