Conditioning of Integrable Determinantal Point Processes

Inhoud

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