Interacting
Interacting diffusions
diffusions asasmarked
marked Gibbs
Gibbs point
point processes
processes
Alexander
Alexander Zass
Zass
WIAS
WIAS Berlin
BerlinRandomRandom point
point processes
processes inin statisticalstatistical physics
physics
Berlin,Berlin, June
June 30,
30, 2022
2022

1PART ONE

EXISTENCEDM.AM THE FREE SYSTEMCONFIGURATIONSS 11.11 BE A GENERALNORMED SPACE

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THE STATE SPACE FOR THESYSTEM IS

mark19,11

SPACE

WE CONSIDER SIMPLE POINT CONFIGURATIONS ON E I E
LOC FINITE RADON MEASURES OF THE FORM

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LET U BE THE SET OFALL SUCH CONFIGURATIONSWE IDENTIFY A MEASURE
J WITH ITS SUPPORTÉ

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UNDER TZ ISIra In Pois z all
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GIBBSIAN INTERACTION
EXAMPLEAWE CONSIDER AN ENERGY FUNCTIONALH
ASSUMPTION 2
SUCH THAT

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3 INFINITE VOLUMEWE WANT TO DESCRIBE A PROBABILITY MEASURE ON M
CORRESPONDING TO THECASE N IRIAu In n d usa
ISSUE NUMBER ONE NO MEANING FOR THEENERGY OF AN

INFINITECONFIGURATIONILENUMBERTO THE FAMILY OF MARGINALS Paula nNOT CONSISTENT INTHE KOLMOGOROV SENSEISAN

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GENERAL SPACES
POCH STANZESSIN 2021

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INFINITECONFIGURATIONANDANYACCRd THE
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BUT IN GENERALHA IS NOTA LOCAL FUNCTIONAL BECAUSE OF THE INTERACTION BTW

Jd AND Jac

EjPROBLEM

HOW TO GUARANTEE EXISTENCE OF

THE LIMIT4.2 TEMPERED CONFIGURATIONS AND INTERACTION RANGEWE INTRODUCE A SUBSET M TEMP OF THE CONFIGURATIONON WHICH WE CAN CONTROL THE
SPACE M
SIZE OF THE MARKSTEMPERED CONFIGURATIONSMEN Ut nut WHERE JEUT
ytimid sted

Eweectsuch THAT

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FOR ANYJE Ut For ANY belt
BIX llull j n 136,11 0

OF BALLS OF THEIR GERM GRAIN SETNJASSUMPTION4jjjjjj

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CAN INTERSECT A GIVEN BOUNDED SUBSET OFMdMt to 1

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6 EXISTENCEVIA THE ENTROPY METHODDEFINITIONLET Q Q BE TWO PROB MEASURESRE1120BOUNDED THE RELATIVE ENTROPYON MOF QW R TQ

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FOR ANYZIP 0UNDER THE ABOVE ASSUMPTIONSTHERE EXISTS AT LEASTONE INFINITEVOLUME GIBBS POINT PROCESSPSKETCHMEMSTEPONE DEFINEASEQOF STANONARISED
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2PART TWOUNIQUENESSPAIR POTENTIALANINCOMPLETE HISTORICAL NOE

CLUSTER EXPANSION

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DISAGREEMENT PERCOLATIONVANDENBERG MAES 1994
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YIH ADMITS THE FOLLOWINGPegeNPICd IRIRu

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2 THE KIRKWOOD SALSBURG OPERATORY 2113,0 AND CONSIDERTHE BANACH SPACE X OF SEQUENCESLET Crrn InFlorioIncaSTAN 1Ebrdxnilebr.I.cl il

THE ABOVE EQUATIONS CAN BE REWRITTENAS A FIXED POINT PROBLEM ON Xcr

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BB UNIQUENESS STRATEGY

1HAVE AN ESTIMATE OF THE FORM

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PROVE THAT THE OPERATOR Kz

IS A CONTRACTIONinSHOW THAT LENIN IS A Solution

OF THE FIXED POINT PROBLEM

4

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ASSUME THATInaif22Btn Bp o

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REFERENCESID RUELLE STATISTICAL MECHANICS 1969Iz D DEREUDRE INTRODUCTION TO THE
THEORY OF GIBBS POINT PROCESSES 2019INSTOCHASTIC GEOMETRY533 S RIELLY A ZMARKED GIBBS POINTPROCESSESWITH UNBOUNDED INTERACTION AN EXISTENCERESULT2020JSP43 A Z GIBBS POINT PROCESSES ON PATHSPACEEXISTENCE CLUSTER EXPANSION AND UNIQUENESS2022MPRF