Conditioning of Integrable Determinantal Point Processes
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Thèse présentée en vue de l'obtention du grade de docteur en sciences. Lire la suite
In this thesis, we emphasise the role of a particular "integrable" structure in the study of determinantal point processes, namely kernels of integrable form.
We introduce an observation of a point process involving the notions of marketing and thinning, and then condition that point process on the resulting observation. This is shown to preserve important subclasses of point processes :
- determinantal point processes, i.e. point processes whose correlation functions can be written as determinants of a so-called correlation kernel: the transformation makes use of Fredholm determinants and minors ;
- determinantal point processes induced by kernels of projections ;
- determinantal point processes with kernels of integrable form : the transformation can be characterised by the solution to a Riemann-Hilbert problem (RHP).
We then study Jãnossy densities associated to observations of the Airy point process, which are, informally speaking, likelihoods of observations. We prove that we can apply the conditioning transformation to this process and that the kernel of the conditioned point process can be characterized by an RHP. Moreover :
- that RHP's solution satisfies linear differential equations and undergoes isomonodromic deformations ;
- from the compatibility solutions of these differential equations we obtain an isospectral deformation of the Stark operator whose potential is expressible in terms of the Jãnossy density and satisfies the cylindrical Kortreweg-de Vries equation ;
- we prove a trace formula implying that the Stark operator's eigenfunctions are solutions to an infinitely coupled version of the Painlevé II differential equation.
Gabriel Glesner obtained his Master in Mathematics in 2018 at UCLouvain. He then started a Ph.D in Mathematics under Pr. Tom Claeys's supervision within the GPP centre at UCLouvain, researching Riemann-Hilbert problems methods applied to integrable probabilistic models.
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Spécifications
- Éditeur
- Presses universitaires de Louvain
- Auteur
- Gabriel Glesner,
- Collection
- Thèses de la Faculté des sciences
- Langue
- français
- Code publique Onix
- 06 Professionnel et académique
- Date de première publication du titre
- 21 août 2023
- Type d'ouvrage
- Thèse
- Avec
- Bibliographie, Appendices
Livre broché
- Date de publication
- 23 août 2023
- ISBN-13
- 9782390613770
- Ampleur
- Nombre absolu de pages : 178
- Code interne
- 105809
- Format
- 16 x 24 cm
- Poids
- 80 grammes
- Prix
- 16,00 €
- ONIX XML
- Version 2.1, Version 3
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Sommaire
Determinantal Point Processes and Operators of Integrable Form iii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Historical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Traces and determinants of operators . . . . . . . . . . . . . . . xiv
Operators of integrable form and Riemann-Hilbert problems . . xv
Precise definitions and fundamental results . . . . . . . . . . . . . . xix
1 Conditioning of DPPs 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Background and motivation . . . . . . . . . . . . . . . . 1
1.1.2 DPPs: generalities and main examples . . . . . . . . . . 3
1.1.3 Marking and conditioning: informal construction and
statement of results . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.5 Orthogonal polynomial ensembles . . . . . . . . . . . . . 11
1.1.6 DPPs with integrable kernels and Riemann-Hilbert prob-
lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Construction of marked and conditional processes . . . . . . . . 11
1.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Bernoulli marking . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Conditioning on an empty observation . . . . . . . . . . 15
1.2.4 Conditioning on a finite mark 1 configuration ξ1 . . . . 18
1.3 Number rigidity and DPPs corresponding to projection operators 23
1.3.1 DPPs induced by orthogonal projections . . . . . . . . . 23
1.3.2 Disintegration . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.3 Marking rigidity . . . . . . . . . . . . . . . . . . . . . . 26
1.4 OPEs on the real line or on the unit circle . . . . . . . . . . . . 28
1.4.1 OPEs on the real line . . . . . . . . . . . . . . . . . . . 28
1.4.2 OPEs on the unit circle . . . . . . . . . . . . . . . . . . 28
1.4.3 Conditional ensembles associated to OPEs . . . . . . . . 29
1.4.4 Unitary invariant ensembles and scaling limits . . . . . 30
1.4.5 Marginal distribution of mark 0 points with known num-
ber of mark 1 points. . . . . . . . . . . . . . . . . . . . . 30
1.5 Integrable DPPs . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5.1 General integrable kernels . . . . . . . . . . . . . . . . . 32
1.5.2 Integrable kernels characterized by a RH problem . . . . 37
2 Jánossy Densities of the Airy Kernel DPP 43
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Preliminaries on Jánossy densities . . . . . . . . . . . . . . . . 55
2.2.1 Operator preliminaries . . . . . . . . . . . . . . . . . . . 55
2.2.2 Conditional ensembles . . . . . . . . . . . . . . . . . . . 56
2.2.3 Factorizations of Jánossy densities . . . . . . . . . . . . 58
2.3 RH characterization of Jánossy densities . . . . . . . . . . . . . 60
2.3.1 RH problems . . . . . . . . . . . . . . . . . . . . . . . . 60
2.3.2 Stark equation . . . . . . . . . . . . . . . . . . . . . . . 65
2.3.3 Asymptotics as s → +∞ . . . . . . . . . . . . . . . . . . 68
2.3.4 Proofs of Theorems I and II . . . . . . . . . . . . . . . . 71
2.3.5 Comparison with inverse scattering for the Stark operator 73
2.3.6 Connection with the theory of Schlesinger transformations 74
2.3.7 Isospectral deformation and cKdV:
proof of Theorem III . . . . . . . . . . . . . . . . . . . . 76
2.3.8 Generalisation to discontinuous σ's . . . . . . . . . . . . 79
2.4 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.4.2 Right tail: XT − 1
3 → ∞ . . . . . . . . . . . . . . . . . . 81
2.4.3 Left tail: X/T → −∞ . . . . . . . . . . . . . . . . . . . 83
2.4.4 Intermediate regimes: −KT ≤ X ≤ MT 1
3 . . . . . . . . 90
3 Asymptotics in Classical Orthogonal Ensembles 95
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.2 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2.1 Symbols with Fisher-Hartwig singularities . . . . . . . . 99
3.2.2 Symbols with a gap or an emerging gap . . . . . . . . . 102
3.2.3 Gap probabilities and global rigidity . . . . . . . . . . . 105
3.2.4 Possible generalisations . . . . . . . . . . . . . . . . . . 108
3.3 Proof of Proposition 3.1.1 . . . . . . . . . . . . . . . . . . . . . 109
3.4 Symbols with Fisher-Hartwig singularities . . . . . . . . . . . . 111
3.4.1 Asymptotics for ΦN (±1) . . . . . . . . . . . . . . . . . . 111
3.4.2 Proofs of Theorem 3.2.1 and Theorem 3.2.2 . . . . . . . 120
3.5 Symbols with a gap or an emerging gap . . . . . . . . . . . . . 120
3.5.1 Asymptotics for ΦN (±1) . . . . . . . . . . . . . . . . . . 121
3.5.2 Proof of Theorem 3.2.5 . . . . . . . . . . . . . . . . . . 128
3.6 Gap probabilities and global rigidity . . . . . . . . . . . . . . . 128
3.6.1 Proof of Corollary 3.2.6 . . . . . . . . . . . . . . . . . . 128
3.6.2 Proof of Corollaries 3.2.8 and 3.2.10 . . . . . . . . . . . 129
3.6.3 Proof of Theorem 3.2.12 . . . . . . . . . . . . . . . . . . 129
Outlook of Further Research 133