Time-Scaling, Ergodicity and Covariance Decay of Interacting Particle Systems
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929412292739072 |
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| author | Głuchowski, Maciej Menz, Georg |
| author_facet | Głuchowski, Maciej Menz, Georg |
| contents | The main focus of this article is the study of ergodicity of Interacting Particle Systems (IPS). We present a simple lemma showing that scaling time is equivalent to taking the convex combination of the transition matrix of the IPS with the identity. As a consequence, the ergodic properties of IPS are invariant under this transformation. Surprisingly, this simple observation has non-trivial implications: It allows to extend any result that does not respect this invariance, which we demonstrate with examples. Additionally, we develop a recursive method to deduce decay of correlations for IPS with alphabets of arbitrary (finite) size, and apply the Time-Scaling Lemma to that as well. As an application of this new criterion we show that certain one-dimensional IPS are ergodic answering an open question of Toom et al.. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06935 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Time-Scaling, Ergodicity and Covariance Decay of Interacting Particle Systems Głuchowski, Maciej Menz, Georg Probability 60K35, 82C22 The main focus of this article is the study of ergodicity of Interacting Particle Systems (IPS). We present a simple lemma showing that scaling time is equivalent to taking the convex combination of the transition matrix of the IPS with the identity. As a consequence, the ergodic properties of IPS are invariant under this transformation. Surprisingly, this simple observation has non-trivial implications: It allows to extend any result that does not respect this invariance, which we demonstrate with examples. Additionally, we develop a recursive method to deduce decay of correlations for IPS with alphabets of arbitrary (finite) size, and apply the Time-Scaling Lemma to that as well. As an application of this new criterion we show that certain one-dimensional IPS are ergodic answering an open question of Toom et al.. |
| title | Time-Scaling, Ergodicity and Covariance Decay of Interacting Particle Systems |
| topic | Probability 60K35, 82C22 |
| url | https://arxiv.org/abs/2312.06935 |