Discontinuous phase transitions in the one-dimensional pair and triplet creation processes
Abstract
Dickman and Tomé (1991) introduced models on Z in which k consecutive occupied sites give birth at rate λ, individual particles die at rate 1, and the configuration is subject to nearest neighbor stirring. They claimed that the pair creation model (k=2) has a continuous phase transition and was in the universality class of directed percolation, but the triplet creation model (k=3) has a discontinuous (or first-order) phase transition when the stirring rate is large. Hinrichsen (2000a) disputed the second claim and argued that first-order phase transitions in 1+1-dimensional nonequilibrium systems with fluctuating ordered phases are impossible. In this paper we prove rigorously that the phase transitions are discontinuous in both the pair and triplet models when the stirring rate is large. At the end of Section 2 we state some open problems and suggest reasons that the results in the physics literature differ from those presented here,
Related
- Continuous phase transitions in the k-creation process without stirring in d=1
- From smooth to discontinuous kernels: a variance transfer principle for hyperuniform processes
- Generic small-scale creation in the two-dimensional Euler equation
- The discontinuous Galerkin method for solving the two-dimensional transport equation with distributed-order memory kernel on triangular meshes
- A Robust ADI-Based Direct Discontinuous Galerkin Method for Two-Dimensional Singularly Perturbed Parabolic PDEs
BibTeX
@misc{durrett2026discontinuous,
title = {{Discontinuous phase transitions in the one-dimensional pair and triplet creation processes}},
author = {Rick Durrett},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29878},
eprint = {2608.29878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29878},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discontinuous phase transitions in the one-dimensional pair and triplet creation processes
AU - Rick Durrett
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2608.29878
UR - https://arxiv.org/abs/2608.29878
AB - Dickman and Tomé (1991) introduced models on Z in which k consecutive occupied sites give birth at rate λ, individual particles die at rate 1, and the configuration is subject to nearest neighbor stirring. They claimed that the pair creation model (k=2) has a continuous phase transition and was in the universality class of directed percolation, but the triplet creation model (k=3) has a discontinuous (or first-order) phase transition when the stirring rate is large. Hinrichsen (2000a) disputed the second claim and argued that first-order phase transitions in 1+1-dimensional nonequilibrium systems with fluctuating ordered phases are impossible. In this paper we prove rigorously that the phase transitions are discontinuous in both the pair and triplet models when the stirring rate is large. At the end of Section 2 we state some open problems and suggest reasons that the results in the physics literature differ from those presented here,
ER -