Motivated by Alain-Sol Sznitman’s interlacement process, we consider the set of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{0,1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \mathbb {Z}^d ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ d\ge 2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> for certain parameters are not so representable. (4) The collection of permutation invariant processes that are representable corresponds exactly to the set of infinitely divisible random variables on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$[0,\infty ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> via a certain transformation. (5) The supercritical (low temperature) Curie-Weiss model is not representable for large n.

Poisson representable processes

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2025年6月18日

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Malin P. Forsström,
Nina Gantert,
Jeffrey E. Steif

摘要

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,$ $d \geq 2,$ 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 $[0,\infty ]$ $[0, \infty]$ via a certain transformation. (5) The supercritical (low temperature) Curie-Weiss model is not representable for large n.