Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of 2 + 1 -dimensional discrete-height random surfaces in statistical mechanics. We prove that, un...

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Bibliographic Details
Published inProbability theory and related fields Vol. 170; no. 1-2; pp. 11 - 47
Main Authors Ioffe, Dmitry, Velenik, Yvan, Wachtel, Vitali
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2018
Springer Nature B.V
Subjects
Constraining
Economics
Finance
Insurance
Management
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Probability
Probability Theory and Stochastic Processes
Quantitative Finance
Random walk
Rescaling
Statistical mechanics
Statistics for Business
Theoretical
Doob transforms
Sturm–Liouville operators
Dyson diffusions
Slater determinants
Ordered random walks
Ferrari–Spohn diffusions
Invariance principles
82B41
82B20
Self-potentials
60K35
60F17
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Summary:We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of 2 + 1 -dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari–Spohn diffusions associated with limiting Sturm–Liouville operators. In particular, the limiting invariant measures are given by the squares of the corresponding Slater determinants.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-016-0751-z