Abelian oil and water dynamics does not have an absorbing-state phase transition
Authors:
Elisabetta Candellero, Alexandre Stauffer and Lorenzo Taggi
Journal:
Trans. Amer. Math. Soc. 374 (2021), 2733-2752
MSC (2020):
Primary 60K35, 82C22, 82C26
DOI:
https://doi.org/10.1090/tran/8276
Published electronically:
January 20, 2021
MathSciNet review:
4223031
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Abstract | References | Similar Articles | Additional Information
Abstract: The oil and water model is an interacting particle system with two types of particles and a dynamics that conserves the number of particles, which belongs to the so-called class of Abelian networks. Widely studied processes in this class are sandpiles models and activated random walks, which are known (at least for some choice of the underlying graph) to undergo an absorbing-state phase transition. This phase transition characterizes the existence of two regimes, depending on the particle density: a regime of fixation at low densities, where the dynamics converges towards an absorbing state and each particle jumps only finitely many times, and a regime of activity at large densities, where particles jump infinitely often and activity is sustained indefinitely. In this work we show that the oil and water model is substantially different than sandpiles models and activated random walks, in the sense that it does not undergo an absorbing-state phase transition and is in the regime of fixation at all densities. Our result works in great generality: for any graph that is vertex transitive and for a large class of initial configurations.
- Riddhipratim Basu, Shirshendu Ganguly, and Christopher Hoffman, Non-fixation for conservative stochastic dynamics on the line, Comm. Math. Phys. 358 (2018), no. 3, 1151–1185. MR 3778354, DOI https://doi.org/10.1007/s00220-017-3059-7
- Benjamin Bond and Lionel Levine, Abelian networks I. Foundations and examples, SIAM J. Discrete Math. 30 (2016), no. 2, 856–874. MR 3493110, DOI https://doi.org/10.1137/15M1030984
- Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59 (1987), 381–384.
- Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality, Phys. Rev. A (3) 38 (1988), no. 1, 364–374. MR 949160, DOI https://doi.org/10.1103/PhysRevA.38.364
- Elisabetta Candellero, Shirshendu Ganguly, Christopher Hoffman, and Lionel Levine, Oil and water: a two-type internal aggregation model, Ann. Probab. 45 (2017), no. 6A, 4019–4070. MR 3729622, DOI https://doi.org/10.1214/16-AOP1157
- Deepak Dhar, The abelian sandpile and related models, Phy. A: Stat. Mech. Appl. 263 (1999), no. 1, 4–25, Proceedings of the 20th IUPAP International Conference on Statistical Physics.
- C. Hoffman, J. Richey, and L. T. Rolla, Active phase for activated random walk on $\mathbb {Z}$, arXiv:2009.09491 (2020).
- Antal A. Járai, Sandpile models, Probab. Surv. 15 (2018), 243–306. MR 3857602, DOI https://doi.org/10.1214/14-PS228
- Russell Lyons and Yuval Peres, Probability on trees and networks, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016. MR 3616205
- Miguel A. Muñoz, Ronald Dickman, Romualdo Pastor-Satorras, Alessandro Vespignani, and Stefano Zapperi, Sandpiles and absorbing-state phase transitions: Recent results and open problems, AIP Conf. Proc. 574 (2001), no. 1, 102–110.
- Romualdo Pastor-Satorras and Alessandro Vespignani, Field theory of absorbing phase transitions with a nondiffusive conserved field, Phys. Rev. E 62 (2000), R5875–R5878.
- Michela Rossi, Romualdo Pastor-Satorras, and Alessandro Vespignani, Universality class of absorbing phase transitions with a conserved field, Phys. Rev. Lett. 85 (2000), 1803–1806.
- Leonardo T. Rolla and Vladas Sidoravicius, Absorbing-state phase transition for driven-dissipative stochastic dynamics on ${\Bbb Z}$, Invent. Math. 188 (2012), no. 1, 127–150. MR 2897694, DOI https://doi.org/10.1007/s00222-011-0344-5
- Vladas Sidoravicius and Augusto Teixeira, Absorbing-state transition for stochastic sandpiles and activated random walks, Electron. J. Probab. 22 (2017), Paper No. 33, 35. MR 3646059, DOI https://doi.org/10.1214/17-EJP50
- Alexandre Stauffer and Lorenzo Taggi, Critical density of activated random walks on transitive graphs, Ann. Probab. 46 (2018), no. 4, 2190–2220. MR 3813989, DOI https://doi.org/10.1214/17-AOP1224
- Lorenzo Taggi, Absorbing-state phase transition in biased activated random walk, Electron. J. Probab. 21 (2016), Paper No. 13, 15. MR 3485355, DOI https://doi.org/10.1214/16-EJP4275
- Lorenzo Taggi, Active phase for activated random walks on $\Bbb {Z}^d$, $d\geq 3$, with density less than one and arbitrary sleeping rate, Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 3, 1751–1764 (English, with English and French summaries). MR 4010950, DOI https://doi.org/10.1214/18-aihp933
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Additional Information
Elisabetta Candellero
Affiliation:
Dip. di Matematica e Fisica, Università Roma Tre, Largo S. Murialdo 1, 00146, Rome, Italy
MR Author ID:
968125
ORCID:
0000-0003-2424-8695
Email:
ecandellero@mat.uniroma3.it
Alexandre Stauffer
Affiliation:
Dip. di Matematica e Fisica, Università Roma Tre, Largo S. Murialdo 1, 00146, Rome, Italy; and Department of Mathematical Sciences, University of Bath, BA2 7AY Bath, United Kingdom
MR Author ID:
781199
Email:
a.stauffer@bath.ac.uk
Lorenzo Taggi
Affiliation:
Weierstrass Institute for applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
Address at time of publication:
Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00186, Roma, Italy
MR Author ID:
1106315
Email:
lorenzo.taggi@uniroma1.it
Received by editor(s):
January 30, 2019
Received by editor(s) in revised form:
July 23, 2020
Published electronically:
January 20, 2021
Additional Notes:
The first author was partially supported by the project “Programma per Giovani Ricercatori Rita Levi Montalcini” awarded by the Italian Ministry of Education (MIUR). The first author also acknowledges partial support by “INdAM – GNAMPA Project 2019”.
The second and third authors acknowledge support from EPSRC Early Career Fellowship EP/N004566/1.
The third author acknowledges support from DFG German Research Foundation BE 5267/1.
Article copyright:
© Copyright 2021
American Mathematical Society