Poisson Representable Processes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913868963381248 |
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| author | Forsström, Malin P. Gantert, Nina Steif, Jeffrey E. |
| author_facet | Forsström, Malin P. Gantert, Nina Steif, Jeffrey E. |
| contents | Motivated by Alain-Sol Sznitman's interlacement process, we consider the set of $\{0,1\}$-valued processes which can be constructed in an analogous way, namely as a union of sets coming from a Poisson process on a collection of sets. Our main focus is to determine which processes are representable in this way. Some of our results are as follows. (1) All positively associated Markov chains and a large class of renewal processes are so representable. (2) Whether an average of two product measures, with close densities, on $n$ variables, is representable is related to the zeroes of the polylogarithm functions. (3) Using (2), we show that a number of tree-indexed Markov chains as well as the Ising model on $ \mathbb{Z}^d ,$ $d\geq 2,$ for certain parameters are not so representable. (4) The collection of permutation invariant processes that are representable corresponds exactly to the set of infinitely divisible random variables on $[0,\infty]$ via a certain transformation. (5) The supercritical (low temperature) Curie-Weiss model is not representable for large~$n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Poisson Representable Processes Forsström, Malin P. Gantert, Nina Steif, Jeffrey E. Probability Motivated by Alain-Sol Sznitman's interlacement process, we consider the set of $\{0,1\}$-valued processes which can be constructed in an analogous way, namely as a union of sets coming from a Poisson process on a collection of sets. Our main focus is to determine which processes are representable in this way. Some of our results are as follows. (1) All positively associated Markov chains and a large class of renewal processes are so representable. (2) Whether an average of two product measures, with close densities, on $n$ variables, is representable is related to the zeroes of the polylogarithm functions. (3) Using (2), we show that a number of tree-indexed Markov chains as well as the Ising model on $ \mathbb{Z}^d ,$ $d\geq 2,$ for certain parameters are not so representable. (4) The collection of permutation invariant processes that are representable corresponds exactly to the set of infinitely divisible random variables on $[0,\infty]$ via a certain transformation. (5) The supercritical (low temperature) Curie-Weiss model is not representable for large~$n$. |
| title | Poisson Representable Processes |
| topic | Probability |
| url | https://arxiv.org/abs/2401.13412 |