Year: 2025
Type: article
Abstract: Motivated by Alain-Sol Sznitman’s interlacement process, we consider the set of $$\{0,1\}$$ { 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 $$ \mathbb {Z}^d ,$$ Z d , $$ d\ge 2,$$ d ≥ 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 $$[0,\infty ]$$ [ 0 , ∞ ] via a certain transformation. (5) The supercritical (low temperature) Curie-Weiss model is not representable for large n .
Institutions: Chalmers University of Technology, University of Gothenburg
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4411412663