Poisson representable processes: In celebration of geoffrey grimmett’s 70th birthday
Artikel i vetenskaplig tidskrift, 2025

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 Zd, 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,∞] via a certain transformation. (5) The supercritical (low temperature) Curie-Weiss model is not representable for large n.

Markov chain

Poisson process

Poisson representation

Författare

Malin Palö Forsström

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Göteborgs universitet

N. Gantert

SoCIT

Jeffrey Steif

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Probability Theory and Related Fields

0178-8051 (ISSN) 1432-2064 (eISSN)

Vol. In Press

Interagerande partikelsystem, cellulära automater, kvasilokalitet och färgrepresentationer.

Vetenskapsrådet (VR) (2020-03763), 2021-01-01 -- 2024-12-31.

Ämneskategorier (SSIF 2025)

Sannolikhetsteori och statistik

DOI

10.1007/s00440-025-01391-8

Mer information

Senast uppdaterat

2025-06-27